enumerating eulerian orientations. - monash...
TRANSCRIPT
![Page 1: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/1.jpg)
Enumerating Eulerian Orientations.
Andrew Elvey PriceJoint work with Tony Guttmann and Mireille Bousquet-Melou
The University of Melbourne
20/11/2017
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 2: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/2.jpg)
ROOTED PLANAR EULERIAN ORIENTATIONS
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 3: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/3.jpg)
ROOTED PLANAR EULERIAN ORIENTATIONS
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 4: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/4.jpg)
ROOTED PLANAR EULERIAN ORIENTATIONS
A planar orientation is a directed planar map (a directed graphembedded in the plane).
It is Eulerian if each vertex has equal in degree and out degree.
Rooted means that one vertex and one incident half-edge arechosen as the root vertex and root edge. In my diagrams, the rootvertex is drawn at the bottom, and the root half-edge is theleftmost half-edge incident to the vertex.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 5: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/5.jpg)
ROOTED PLANAR EULERIAN ORIENTATIONS
A planar orientation is a directed planar map (a directed graphembedded in the plane).
It is Eulerian if each vertex has equal in degree and out degree.
Rooted means that one vertex and one incident half-edge arechosen as the root vertex and root edge. In my diagrams, the rootvertex is drawn at the bottom, and the root half-edge is theleftmost half-edge incident to the vertex.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 6: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/6.jpg)
ROOTED PLANAR EULERIAN ORIENTATIONS
A planar orientation is a directed planar map (a directed graphembedded in the plane).
It is Eulerian if each vertex has equal in degree and out degree.
Rooted means that one vertex and one incident half-edge arechosen as the root vertex and root edge. In my diagrams, the rootvertex is drawn at the bottom, and the root half-edge is theleftmost half-edge incident to the vertex.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 7: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/7.jpg)
ROOTED PLANAR EULERIAN ORIENTATIONS
A planar orientation is a directed planar map (a directed graphembedded in the plane).
It is Eulerian if each vertex has equal in degree and out degree.
Rooted means that one vertex and one incident half-edge arechosen as the root vertex and root edge.
In my diagrams, the rootvertex is drawn at the bottom, and the root half-edge is theleftmost half-edge incident to the vertex.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 8: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/8.jpg)
ROOTED PLANAR EULERIAN ORIENTATIONS
A planar orientation is a directed planar map (a directed graphembedded in the plane).
It is Eulerian if each vertex has equal in degree and out degree.
Rooted means that one vertex and one incident half-edge arechosen as the root vertex and root edge. In my diagrams, the rootvertex is drawn at the bottom, and the root half-edge is theleftmost half-edge incident to the vertex.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 9: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/9.jpg)
ONE EDGE ROOTED PLANAR EULERIAN ORIENTATIONS
There are two planar rooted Eulerian orientations with one edge.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 10: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/10.jpg)
ONE EDGE ROOTED PLANAR EULERIAN ORIENTATIONS
There are two planar rooted Eulerian orientations with one edge.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 11: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/11.jpg)
TWO EDGE ROOTED PLANAR EULERIAN ORIENTATIONS
There are 10 planar rooted Eulerian orientations with two edges.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 12: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/12.jpg)
TWO EDGE ROOTED PLANAR EULERIAN ORIENTATIONS
There are 10 planar rooted Eulerian orientations with two edges.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 13: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/13.jpg)
COUNTING ROOTED PLANAR EULERIAN ORIENTATIONS
Let an be the number of rooted planar Eulerian orientations withn edges.
a1 = 2.
a2 = 10.
Aim: Find a formula for an.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 14: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/14.jpg)
COUNTING ROOTED PLANAR EULERIAN ORIENTATIONS
Let an be the number of rooted planar Eulerian orientations withn edges.
a1 = 2.
a2 = 10.
Aim: Find a formula for an.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 15: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/15.jpg)
COUNTING ROOTED PLANAR EULERIAN ORIENTATIONS
Let an be the number of rooted planar Eulerian orientations withn edges.
a1 = 2.
a2 = 10.
Aim: Find a formula for an.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 16: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/16.jpg)
COUNTING ROOTED PLANAR EULERIAN ORIENTATIONS
Let an be the number of rooted planar Eulerian orientations withn edges.
a1 = 2.
a2 = 10.
Aim: Find a formula for an.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 17: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/17.jpg)
COUNTING ROOTED PLANAR EULERIAN ORIENTATIONS
Let an be the number of rooted planar Eulerian orientations withn edges.
a1 = 2.
a2 = 10.
Aim: Find a formula for an.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 18: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/18.jpg)
BACKGROUND ON THE PROBLEM
In 2016, Bonichon, Bousquet-Melou, Dorbec and Pennarunposed the problem of enumerating planar rooted Eulerianorientations with a given number of edges.
They computed the number an of these orientations for n ≤ 15.
They also proved that the growth rate
µ = limn→∞
n√
an
exists and lies in the interval (11.56, 13.005)
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 19: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/19.jpg)
BACKGROUND ON THE PROBLEM
In 2016, Bonichon, Bousquet-Melou, Dorbec and Pennarunposed the problem of enumerating planar rooted Eulerianorientations with a given number of edges.
They computed the number an of these orientations for n ≤ 15.
They also proved that the growth rate
µ = limn→∞
n√
an
exists and lies in the interval (11.56, 13.005)
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 20: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/20.jpg)
BACKGROUND ON THE PROBLEM
In 2016, Bonichon, Bousquet-Melou, Dorbec and Pennarunposed the problem of enumerating planar rooted Eulerianorientations with a given number of edges.
They computed the number an of these orientations for n ≤ 15.
They also proved that the growth rate
µ = limn→∞
n√
an
exists and lies in the interval (11.56, 13.005)
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 21: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/21.jpg)
BACKGROUND ON THE PROBLEM
In 2016, Bonichon, Bousquet-Melou, Dorbec and Pennarunposed the problem of enumerating planar rooted Eulerianorientations with a given number of edges.
They computed the number an of these orientations for n ≤ 15.
They also proved that the growth rate
µ = limn→∞
n√
an
exists and lies in the interval (11.56, 13.005)
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 22: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/22.jpg)
4-VALENT PLANAR ROOTED EULERIAN ORIENTATIONS
Let bn be the number of 4-valent rooted planar Eulerianorientations with n vertices.
Bonichon et al also posed the problem of enumerating these.
This is equivalent to the ice type model on a random lattice, aproblem in mathematical physics.
It is also the sum of the Tutte polynomials TΓ(0,−2) over all4-valent rooted planar maps Γ with n vertices.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 23: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/23.jpg)
4-VALENT PLANAR ROOTED EULERIAN ORIENTATIONS
Let bn be the number of 4-valent rooted planar Eulerianorientations with n vertices.
Bonichon et al also posed the problem of enumerating these.
This is equivalent to the ice type model on a random lattice, aproblem in mathematical physics.
It is also the sum of the Tutte polynomials TΓ(0,−2) over all4-valent rooted planar maps Γ with n vertices.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 24: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/24.jpg)
4-VALENT PLANAR ROOTED EULERIAN ORIENTATIONS
Let bn be the number of 4-valent rooted planar Eulerianorientations with n vertices.
Bonichon et al also posed the problem of enumerating these.
This is equivalent to the ice type model on a random lattice, aproblem in mathematical physics.
It is also the sum of the Tutte polynomials TΓ(0,−2) over all4-valent rooted planar maps Γ with n vertices.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 25: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/25.jpg)
4-VALENT PLANAR ROOTED EULERIAN ORIENTATIONS
Let bn be the number of 4-valent rooted planar Eulerianorientations with n vertices.
Bonichon et al also posed the problem of enumerating these.
This is equivalent to the ice type model on a random lattice, aproblem in mathematical physics.
It is also the sum of the Tutte polynomials TΓ(0,−2) over all4-valent rooted planar maps Γ with n vertices.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 26: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/26.jpg)
4-VALENT PLANAR ROOTED EULERIAN ORIENTATIONS
Let bn be the number of 4-valent rooted planar Eulerianorientations with n vertices.
Bonichon et al also posed the problem of enumerating these.
This is equivalent to the ice type model on a random lattice, aproblem in mathematical physics.
It is also the sum of the Tutte polynomials TΓ(0,−2) over all4-valent rooted planar maps Γ with n vertices.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 27: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/27.jpg)
BIJECTION TO NUMBERED MAPS (N-MAPS)
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 28: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/28.jpg)
BIJECTION TO NUMBERED MAPS (N-MAPS)
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 29: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/29.jpg)
BIJECTION TO NUMBERED MAPS (N-MAPS)
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 30: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/30.jpg)
BIJECTION TO NUMBERED MAPS (N-MAPS)
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 31: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/31.jpg)
BIJECTION TO NUMBERED MAPS (N-MAPS)
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 32: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/32.jpg)
BIJECTION TO NUMBERED MAPS (N-MAPS)
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 33: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/33.jpg)
N-MAPS
N-maps are rooted planar maps with numbered vertices such that:
The root vertex is numbered 0.
Numbers on adjacent vertices differ by 1.
By the bijection, an is the number of N-maps with n edges.
This bijection sends vertices with degree k to faces with degree k.
Hence, 4-valent orientations are sent to quadrangulations.
So, bn is the number of numbered quadrangulations(N-quadrangulations) with n faces.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 34: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/34.jpg)
N-MAPS
N-maps are rooted planar maps with numbered vertices such that:
The root vertex is numbered 0.
Numbers on adjacent vertices differ by 1.
By the bijection, an is the number of N-maps with n edges.
This bijection sends vertices with degree k to faces with degree k.
Hence, 4-valent orientations are sent to quadrangulations.
So, bn is the number of numbered quadrangulations(N-quadrangulations) with n faces.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 35: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/35.jpg)
N-MAPS
N-maps are rooted planar maps with numbered vertices such that:
The root vertex is numbered 0.
Numbers on adjacent vertices differ by 1.
By the bijection, an is the number of N-maps with n edges.
This bijection sends vertices with degree k to faces with degree k.
Hence, 4-valent orientations are sent to quadrangulations.
So, bn is the number of numbered quadrangulations(N-quadrangulations) with n faces.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 36: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/36.jpg)
N-MAPS
N-maps are rooted planar maps with numbered vertices such that:
The root vertex is numbered 0.
Numbers on adjacent vertices differ by 1.
By the bijection, an is the number of N-maps with n edges.
This bijection sends vertices with degree k to faces with degree k.
Hence, 4-valent orientations are sent to quadrangulations.
So, bn is the number of numbered quadrangulations(N-quadrangulations) with n faces.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 37: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/37.jpg)
N-MAPS
N-maps are rooted planar maps with numbered vertices such that:
The root vertex is numbered 0.
Numbers on adjacent vertices differ by 1.
By the bijection, an is the number of N-maps with n edges.
This bijection sends vertices with degree k to faces with degree k.
Hence, 4-valent orientations are sent to quadrangulations.
So, bn is the number of numbered quadrangulations(N-quadrangulations) with n faces.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 38: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/38.jpg)
N-MAPS
N-maps are rooted planar maps with numbered vertices such that:
The root vertex is numbered 0.
Numbers on adjacent vertices differ by 1.
By the bijection, an is the number of N-maps with n edges.
This bijection sends vertices with degree k to faces with degree k.
Hence, 4-valent orientations are sent to quadrangulations.
So, bn is the number of numbered quadrangulations(N-quadrangulations) with n faces.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 39: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/39.jpg)
N-MAPS
N-maps are rooted planar maps with numbered vertices such that:
The root vertex is numbered 0.
Numbers on adjacent vertices differ by 1.
By the bijection, an is the number of N-maps with n edges.
This bijection sends vertices with degree k to faces with degree k.
Hence, 4-valent orientations are sent to quadrangulations.
So, bn is the number of numbered quadrangulations(N-quadrangulations) with n faces.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 40: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/40.jpg)
N-MAPS
N-maps are rooted planar maps with numbered vertices such that:
The root vertex is numbered 0.
Numbers on adjacent vertices differ by 1.
By the bijection, an is the number of N-maps with n edges.
This bijection sends vertices with degree k to faces with degree k.
Hence, 4-valent orientations are sent to quadrangulations.
So, bn is the number of numbered quadrangulations(N-quadrangulations) with n faces.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 41: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/41.jpg)
COUNTING N-QUADRANGULATIONS
For the rest of the talk I will focus on the problem of countingN-quadrangulations with a fixed number of faces.
As I mentioned, this is equivalent to enumerating 4-valent rootedplanar Eulerian orientations.
We want to find a way to decompose all largeN-quadrangulations into smaller N-quadrangulations.
That will hopefully lead to a recursive formula for calculatingthe numbers bn.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 42: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/42.jpg)
COUNTING N-QUADRANGULATIONS
For the rest of the talk I will focus on the problem of countingN-quadrangulations with a fixed number of faces.
As I mentioned, this is equivalent to enumerating 4-valent rootedplanar Eulerian orientations.
We want to find a way to decompose all largeN-quadrangulations into smaller N-quadrangulations.
That will hopefully lead to a recursive formula for calculatingthe numbers bn.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 43: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/43.jpg)
COUNTING N-QUADRANGULATIONS
For the rest of the talk I will focus on the problem of countingN-quadrangulations with a fixed number of faces.
As I mentioned, this is equivalent to enumerating 4-valent rootedplanar Eulerian orientations.
We want to find a way to decompose all largeN-quadrangulations into smaller N-quadrangulations.
That will hopefully lead to a recursive formula for calculatingthe numbers bn.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 44: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/44.jpg)
COUNTING N-QUADRANGULATIONS
For the rest of the talk I will focus on the problem of countingN-quadrangulations with a fixed number of faces.
As I mentioned, this is equivalent to enumerating 4-valent rootedplanar Eulerian orientations.
We want to find a way to decompose all largeN-quadrangulations into smaller N-quadrangulations.
That will hopefully lead to a recursive formula for calculatingthe numbers bn.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 45: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/45.jpg)
COUNTING N-QUADRANGULATIONS
For the rest of the talk I will focus on the problem of countingN-quadrangulations with a fixed number of faces.
As I mentioned, this is equivalent to enumerating 4-valent rootedplanar Eulerian orientations.
We want to find a way to decompose all largeN-quadrangulations into smaller N-quadrangulations.
That will hopefully lead to a recursive formula for calculatingthe numbers bn.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 46: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/46.jpg)
CONTRACTION IDEA
The most important decomposition we use works as follows:
Choose an N-quadrangulation Γ.
Choose a connected subgraph τ of Γ with positive integervertices.
Contract τ to a single vertex.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 47: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/47.jpg)
CONTRACTION IDEA
The most important decomposition we use works as follows:
Choose an N-quadrangulation Γ.
Choose a connected subgraph τ of Γ with positive integervertices.
Contract τ to a single vertex.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 48: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/48.jpg)
CONTRACTION IDEA
The most important decomposition we use works as follows:
Choose an N-quadrangulation Γ.
Choose a connected subgraph τ of Γ with positive integervertices.
Contract τ to a single vertex.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 49: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/49.jpg)
CONTRACTION IDEA
The most important decomposition we use works as follows:
Choose an N-quadrangulation Γ.
Choose a connected subgraph τ of Γ with positive integervertices.
Contract τ to a single vertex.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 50: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/50.jpg)
CONTRACTION IDEA
The most important decomposition we use works as follows:
Choose an N-quadrangulation Γ.
Choose a connected subgraph τ of Γ with positive integervertices.
Contract τ to a single vertex.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 51: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/51.jpg)
CONTRACTION EXAMPLE
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 52: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/52.jpg)
CONTRACTION EXAMPLE
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 53: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/53.jpg)
CONTRACTION
Choose an N-quadrangulation Γ.
Choose a connected subgraph τ of Γ with positive integervertices.
Contract τ to a single vertex, to form a new N-quadrangulationΓ′.
We call τ the patch.
We call Γ′ the contracted map.
To use this, we need to enumerate patches. In patches the outer facemay have any (even) degree.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 54: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/54.jpg)
CONTRACTION
Choose an N-quadrangulation Γ.
Choose a connected subgraph τ of Γ with positive integervertices.
Contract τ to a single vertex, to form a new N-quadrangulationΓ′.
We call τ the patch.
We call Γ′ the contracted map.
To use this, we need to enumerate patches. In patches the outer facemay have any (even) degree.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 55: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/55.jpg)
CONTRACTION
Choose an N-quadrangulation Γ.
Choose a connected subgraph τ of Γ with positive integervertices.
Contract τ to a single vertex, to form a new N-quadrangulationΓ′.
We call τ the patch.
We call Γ′ the contracted map.
To use this, we need to enumerate patches. In patches the outer facemay have any (even) degree.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 56: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/56.jpg)
CONTRACTION
Choose an N-quadrangulation Γ.
Choose a connected subgraph τ of Γ with positive integervertices.
Contract τ to a single vertex, to form a new N-quadrangulationΓ′.
We call τ the patch.
We call Γ′ the contracted map.
To use this, we need to enumerate patches.
In patches the outer facemay have any (even) degree.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 57: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/57.jpg)
CONTRACTION
Choose an N-quadrangulation Γ.
Choose a connected subgraph τ of Γ with positive integervertices.
Contract τ to a single vertex, to form a new N-quadrangulationΓ′.
We call τ the patch.
We call Γ′ the contracted map.
To use this, we need to enumerate patches. In patches the outer facemay have any (even) degree.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 58: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/58.jpg)
T-MAPS
In order to count N-quadrangulations we introduce a specialisationcalled T-maps, which are N-maps in which:
Every inner face has degree 4.
All vertices adjacent to the root vertex v0 are numbered 1.
The vertices around the outer are alternately numbered 0 and 1.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 59: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/59.jpg)
T-MAPS
In order to count N-quadrangulations we introduce a specialisationcalled T-maps, which are N-maps in which:
Every inner face has degree 4.
All vertices adjacent to the root vertex v0 are numbered 1.
The vertices around the outer are alternately numbered 0 and 1.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 60: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/60.jpg)
T-MAPS
In order to count N-quadrangulations we introduce a specialisationcalled T-maps, which are N-maps in which:
Every inner face has degree 4.
All vertices adjacent to the root vertex v0 are numbered 1.
The vertices around the outer are alternately numbered 0 and 1.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 61: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/61.jpg)
T-MAPS
In order to count N-quadrangulations we introduce a specialisationcalled T-maps, which are N-maps in which:
Every inner face has degree 4.
All vertices adjacent to the root vertex v0 are numbered 1.
The vertices around the outer are alternately numbered 0 and 1.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 62: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/62.jpg)
T-MAPS
In order to count N-quadrangulations we introduce a specialisationcalled T-maps, which are N-maps in which:
Every inner face has degree 4.
All vertices adjacent to the root vertex v0 are numbered 1.
The vertices around the outer are alternately numbered 0 and 1.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 63: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/63.jpg)
COUNTING T-MAPS
We count the T-maps using the generating function
T(t, a, b) =∑
Γ
t|V(Γ)|ad(v0)bf (Γ),
where the sum is over all T-maps Γ.In the above equation:
d(v0) denotes the degree of v0.
f (Γ) denotes the degree of the outer face of Γ.
Then bn = 2[tn+2][a1][b4]T(t, a, b)Now we need a way to decompose T-maps into smaller maps.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 64: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/64.jpg)
COUNTING T-MAPS
We count the T-maps using the generating function
T(t, a, b) =∑
Γ
t|V(Γ)|ad(v0)bf (Γ),
where the sum is over all T-maps Γ.
In the above equation:
d(v0) denotes the degree of v0.
f (Γ) denotes the degree of the outer face of Γ.
Then bn = 2[tn+2][a1][b4]T(t, a, b)Now we need a way to decompose T-maps into smaller maps.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 65: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/65.jpg)
COUNTING T-MAPS
We count the T-maps using the generating function
T(t, a, b) =∑
Γ
t|V(Γ)|ad(v0)bf (Γ),
where the sum is over all T-maps Γ.In the above equation:
d(v0) denotes the degree of v0.
f (Γ) denotes the degree of the outer face of Γ.
Then bn = 2[tn+2][a1][b4]T(t, a, b)Now we need a way to decompose T-maps into smaller maps.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 66: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/66.jpg)
COUNTING T-MAPS
We count the T-maps using the generating function
T(t, a, b) =∑
Γ
t|V(Γ)|ad(v0)bf (Γ),
where the sum is over all T-maps Γ.In the above equation:
d(v0) denotes the degree of v0.
f (Γ) denotes the degree of the outer face of Γ.
Then bn = 2[tn+2][a1][b4]T(t, a, b)Now we need a way to decompose T-maps into smaller maps.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 67: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/67.jpg)
COUNTING T-MAPS
We count the T-maps using the generating function
T(t, a, b) =∑
Γ
t|V(Γ)|ad(v0)bf (Γ),
where the sum is over all T-maps Γ.In the above equation:
d(v0) denotes the degree of v0.
f (Γ) denotes the degree of the outer face of Γ.
Then bn = 2[tn+2][a1][b4]T(t, a, b)
Now we need a way to decompose T-maps into smaller maps.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 68: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/68.jpg)
COUNTING T-MAPS
We count the T-maps using the generating function
T(t, a, b) =∑
Γ
t|V(Γ)|ad(v0)bf (Γ),
where the sum is over all T-maps Γ.In the above equation:
d(v0) denotes the degree of v0.
f (Γ) denotes the degree of the outer face of Γ.
Then bn = 2[tn+2][a1][b4]T(t, a, b)Now we need a way to decompose T-maps into smaller maps.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 69: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/69.jpg)
T-MAP DECOMPOSITION EXAMPLE
Enumerating Eulerian Orientations. Andrew Elvey Price
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T-MAP DECOMPOSITION EXAMPLE
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 71: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/71.jpg)
T-MAP DECOMPOSITION EXAMPLE
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 72: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/72.jpg)
T-MAP DECOMPOSITION EXAMPLE
Enumerating Eulerian Orientations. Andrew Elvey Price
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T-MAP DECOMPOSITION EXAMPLE
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 74: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/74.jpg)
T-MAP DECOMPOSITION EXAMPLE
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 75: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/75.jpg)
T-MAP DECOMPOSITION EXAMPLE
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 76: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/76.jpg)
T-MAP DECOMPOSITION EXAMPLE
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 77: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/77.jpg)
FORMULA FOR T-MAPS
Using the decomposition shown, we get a formula relating thegenerating function for T-maps to itself:
T(t, a, b) =1
1− [x−1]aT(t, 1/x, b)T(t, a, 1/(1− x)).
Along with some initial conditions, this is enough to uniquelydetermine the power series T .
Moreover, This allows us to calculate the coefficients of T inpolynomial time.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 78: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/78.jpg)
FORMULA FOR T-MAPS
Using the decomposition shown, we get a formula relating thegenerating function for T-maps to itself:
T(t, a, b) =1
1− [x−1]aT(t, 1/x, b)T(t, a, 1/(1− x)).
Along with some initial conditions, this is enough to uniquelydetermine the power series T .
Moreover, This allows us to calculate the coefficients of T inpolynomial time.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 79: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/79.jpg)
FORMULA FOR T-MAPS
Using the decomposition shown, we get a formula relating thegenerating function for T-maps to itself:
T(t, a, b) =1
1− [x−1]aT(t, 1/x, b)T(t, a, 1/(1− x)).
Along with some initial conditions, this is enough to uniquelydetermine the power series T .
Moreover, This allows us to calculate the coefficients of T inpolynomial time.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 80: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/80.jpg)
FORMULA FOR T-MAPS
Using the decomposition shown, we get a formula relating thegenerating function for T-maps to itself:
T(t, a, b) =1
1− [x−1]aT(t, 1/x, b)T(t, a, 1/(1− x)).
Along with some initial conditions, this is enough to uniquelydetermine the power series T .
Moreover, This allows us to calculate the coefficients of T inpolynomial time.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 81: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/81.jpg)
THE ALGORITHM
Yay! We have a polynomial time algorithm for calculating thenumber bn = 2[tn+2][a1][b4]T(t, a, b) of N-quadrangulationswith n faces.
bn is also the number of 4-valent rooted planar Eulerianorientations with n vertices.
Using this algorithm we computed bn for n < 100.
Using a similar algorithm, we computed an for n < 90.
Enumerating Eulerian Orientations. Andrew Elvey Price
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THE ALGORITHM
Yay! We have a polynomial time algorithm for calculating thenumber bn = 2[tn+2][a1][b4]T(t, a, b) of N-quadrangulationswith n faces.
bn is also the number of 4-valent rooted planar Eulerianorientations with n vertices.
Using this algorithm we computed bn for n < 100.
Using a similar algorithm, we computed an for n < 90.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 83: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/83.jpg)
THE ALGORITHM
Yay! We have a polynomial time algorithm for calculating thenumber bn = 2[tn+2][a1][b4]T(t, a, b) of N-quadrangulationswith n faces.
bn is also the number of 4-valent rooted planar Eulerianorientations with n vertices.
Using this algorithm we computed bn for n < 100.
Using a similar algorithm, we computed an for n < 90.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 84: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/84.jpg)
THE ALGORITHM
Yay! We have a polynomial time algorithm for calculating thenumber bn = 2[tn+2][a1][b4]T(t, a, b) of N-quadrangulationswith n faces.
bn is also the number of 4-valent rooted planar Eulerianorientations with n vertices.
Using this algorithm we computed bn for n < 100.
Using a similar algorithm, we computed an for n < 90.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 85: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/85.jpg)
THE ALGORITHM
Yay! We have a polynomial time algorithm for calculating thenumber bn = 2[tn+2][a1][b4]T(t, a, b) of N-quadrangulationswith n faces.
bn is also the number of 4-valent rooted planar Eulerianorientations with n vertices.
Using this algorithm we computed bn for n < 100.
Using a similar algorithm, we computed an for n < 90.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 86: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/86.jpg)
SERIES ANALYSIS
We want to guess the growth rate of the sequence b0, b1, . . .using only the known 100 terms.
The simplest way to try to do this is to plot the ratiosrn = bn/bn−1 against 1/n.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 87: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/87.jpg)
SERIES ANALYSIS
We want to guess the growth rate of the sequence b0, b1, . . .using only the known 100 terms.
The simplest way to try to do this is to plot the ratiosrn = bn/bn−1 against 1/n.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 88: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/88.jpg)
SERIES ANALYSIS
We want to guess the growth rate of the sequence b0, b1, . . .using only the known 100 terms.
The simplest way to try to do this is to plot the ratiosrn = bn/bn−1 against 1/n.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 89: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/89.jpg)
PLOT OF RATIOS bn/bn−1
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 90: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/90.jpg)
SERIES ANALYSIS
We want to guess the growth rate of the sequence b0, b1, . . .using only the known 100 terms.
The simplest way to try to do this is to plot the ratiosrn = bn/bn−1 against 1/n.
The growth rate is where this line intersects with 1/n = 0.
This way we estimate the growth rate µ ≈ 21.6.
But we can do better!
First, we approximately extend the series.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 91: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/91.jpg)
SERIES ANALYSIS
We want to guess the growth rate of the sequence b0, b1, . . .using only the known 100 terms.
The simplest way to try to do this is to plot the ratiosrn = bn/bn−1 against 1/n.
The growth rate is where this line intersects with 1/n = 0.
This way we estimate the growth rate µ ≈ 21.6.
But we can do better!
First, we approximately extend the series.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 92: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/92.jpg)
SERIES ANALYSIS
We want to guess the growth rate of the sequence b0, b1, . . .using only the known 100 terms.
The simplest way to try to do this is to plot the ratiosrn = bn/bn−1 against 1/n.
The growth rate is where this line intersects with 1/n = 0.
This way we estimate the growth rate µ ≈ 21.6.
But we can do better!
First, we approximately extend the series.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 93: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/93.jpg)
SERIES ANALYSIS
We want to guess the growth rate of the sequence b0, b1, . . .using only the known 100 terms.
The simplest way to try to do this is to plot the ratiosrn = bn/bn−1 against 1/n.
The growth rate is where this line intersects with 1/n = 0.
This way we estimate the growth rate µ ≈ 21.6.
But we can do better!
First, we approximately extend the series.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 94: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/94.jpg)
SERIES ANALYSIS
We want to guess the growth rate of the sequence b0, b1, . . .using only the known 100 terms.
The simplest way to try to do this is to plot the ratiosrn = bn/bn−1 against 1/n.
The growth rate is where this line intersects with 1/n = 0.
This way we estimate the growth rate µ ≈ 21.6.
But we can do better!
First, we approximately extend the series.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 95: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/95.jpg)
DIFFERENTIAL APPROXIMANTS
This is a summary of Tony’s method for approximately extending theseries:
Let B(t) = b0 + b1t + b2t2 + . . .
Choose a random sequence of positive integers L,M, d0, . . . , dM
which sum to 100 (where M = 2 or 3 and no two values of di
differ by more than 2).
Calculate the unique polynomials P,Q0,Q1, . . . ,QM (up toscaling) of degrees L,M, d0, . . . , dM such that the first 100coefficients of
P(t)−M∑
k=0
Qk(t)(
tddt
)k
B(t)
are all 0.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 96: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/96.jpg)
DIFFERENTIAL APPROXIMANTS
This is a summary of Tony’s method for approximately extending theseries:
Let B(t) = b0 + b1t + b2t2 + . . .
Choose a random sequence of positive integers L,M, d0, . . . , dM
which sum to 100 (where M = 2 or 3 and no two values of di
differ by more than 2).
Calculate the unique polynomials P,Q0,Q1, . . . ,QM (up toscaling) of degrees L,M, d0, . . . , dM such that the first 100coefficients of
P(t)−M∑
k=0
Qk(t)(
tddt
)k
B(t)
are all 0.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 97: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/97.jpg)
DIFFERENTIAL APPROXIMANTS
This is a summary of Tony’s method for approximately extending theseries:
Let B(t) = b0 + b1t + b2t2 + . . .
Choose a random sequence of positive integers L,M, d0, . . . , dM
which sum to 100 (where M = 2 or 3 and no two values of di
differ by more than 2).
Calculate the unique polynomials P,Q0,Q1, . . . ,QM (up toscaling) of degrees L,M, d0, . . . , dM such that the first 100coefficients of
P(t)−M∑
k=0
Qk(t)(
tddt
)k
B(t)
are all 0.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 98: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/98.jpg)
DIFFERENTIAL APPROXIMANTS
This is a summary of Tony’s method for approximately extending theseries:
Let B(t) = b0 + b1t + b2t2 + . . .
Choose a random sequence of positive integers L,M, d0, . . . , dM
which sum to 100 (where M = 2 or 3 and no two values of di
differ by more than 2).
Calculate the unique polynomials P,Q0,Q1, . . . ,QM (up toscaling) of degrees L,M, d0, . . . , dM such that the first 100coefficients of
P(t)−M∑
k=0
Qk(t)(
tddt
)k
B(t)
are all 0.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 99: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/99.jpg)
DIFFERENTIAL APPROXIMANTS
Approximate B by the solution B̃ of
M∑k=0
Qk(t)(
tddt
)k
B̃(t) = P(t).
Repeat these steps for every possible sequenceP,Q0,Q1, . . . ,QM to obtain many approximations B̃.
For each ratio rn = bn+1/bn we get a range of approximations,which give us an expected value (given by the mean of most ofthe approximation) and error estimate (given by the standarddeviation of the approximations).
Surprisingly, these estimates generally seem to be very accurate.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 100: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/100.jpg)
DIFFERENTIAL APPROXIMANTS
Approximate B by the solution B̃ of
M∑k=0
Qk(t)(
tddt
)k
B̃(t) = P(t).
Repeat these steps for every possible sequenceP,Q0,Q1, . . . ,QM to obtain many approximations B̃.
For each ratio rn = bn+1/bn we get a range of approximations,which give us an expected value (given by the mean of most ofthe approximation) and error estimate (given by the standarddeviation of the approximations).
Surprisingly, these estimates generally seem to be very accurate.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 101: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/101.jpg)
DIFFERENTIAL APPROXIMANTS
Approximate B by the solution B̃ of
M∑k=0
Qk(t)(
tddt
)k
B̃(t) = P(t).
Repeat these steps for every possible sequenceP,Q0,Q1, . . . ,QM to obtain many approximations B̃.
For each ratio rn = bn+1/bn we get a range of approximations,which give us an expected value (given by the mean of most ofthe approximation) and error estimate (given by the standarddeviation of the approximations).
Surprisingly, these estimates generally seem to be very accurate.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 102: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/102.jpg)
DIFFERENTIAL APPROXIMANTS
Approximate B by the solution B̃ of
M∑k=0
Qk(t)(
tddt
)k
B̃(t) = P(t).
Repeat these steps for every possible sequenceP,Q0,Q1, . . . ,QM to obtain many approximations B̃.
For each ratio rn = bn+1/bn we get a range of approximations,which give us an expected value (given by the mean of most ofthe approximation) and error estimate (given by the standarddeviation of the approximations).
Surprisingly, these estimates generally seem to be very accurate.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 103: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/103.jpg)
DIFFERENTIAL APPROXIMANTS
Approximate B by the solution B̃ of
M∑k=0
Qk(t)(
tddt
)k
B̃(t) = P(t).
Repeat these steps for every possible sequenceP,Q0,Q1, . . . ,QM to obtain many approximations B̃.
For each ratio rn = bn+1/bn we get a range of approximations,which give us an expected value (given by the mean of most ofthe approximation) and error estimate (given by the standarddeviation of the approximations).
Surprisingly, these estimates generally seem to be very accurate.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 104: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/104.jpg)
SERIES ANALYSIS
Using differential approximants, we approximate 1000 furtherratios, which we estimate to be accurate to at least 10 significantdigits.
so now we have a sequence of ratios and approximate ratiosr1, r2, . . . , r1100.
when we plot these against 1/n they don’t seem completelylinear, but plotted against 1/(n log(n)2) they do seem prettylinear.
Using the line between adjacent points in this plot and takingtheir intercept with the y-axis gives better approximations for thegrowth rate µ.
Now we plot these approximations.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 105: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/105.jpg)
SERIES ANALYSIS
Using differential approximants, we approximate 1000 furtherratios, which we estimate to be accurate to at least 10 significantdigits.
so now we have a sequence of ratios and approximate ratiosr1, r2, . . . , r1100.
when we plot these against 1/n they don’t seem completelylinear, but plotted against 1/(n log(n)2) they do seem prettylinear.
Using the line between adjacent points in this plot and takingtheir intercept with the y-axis gives better approximations for thegrowth rate µ.
Now we plot these approximations.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 106: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/106.jpg)
SERIES ANALYSIS
Using differential approximants, we approximate 1000 furtherratios, which we estimate to be accurate to at least 10 significantdigits.
so now we have a sequence of ratios and approximate ratiosr1, r2, . . . , r1100.
when we plot these against 1/n they don’t seem completelylinear, but plotted against 1/(n log(n)2) they do seem prettylinear.
Using the line between adjacent points in this plot and takingtheir intercept with the y-axis gives better approximations for thegrowth rate µ.
Now we plot these approximations.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 107: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/107.jpg)
SERIES ANALYSIS
Using differential approximants, we approximate 1000 furtherratios, which we estimate to be accurate to at least 10 significantdigits.
so now we have a sequence of ratios and approximate ratiosr1, r2, . . . , r1100.
when we plot these against 1/n they don’t seem completelylinear, but plotted against 1/(n log(n)2) they do seem prettylinear.
Using the line between adjacent points in this plot and takingtheir intercept with the y-axis gives better approximations for thegrowth rate µ.
Now we plot these approximations.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 108: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/108.jpg)
SERIES ANALYSIS
Using differential approximants, we approximate 1000 furtherratios, which we estimate to be accurate to at least 10 significantdigits.
so now we have a sequence of ratios and approximate ratiosr1, r2, . . . , r1100.
when we plot these against 1/n they don’t seem completelylinear, but plotted against 1/(n log(n)2) they do seem prettylinear.
Using the line between adjacent points in this plot and takingtheir intercept with the y-axis gives better approximations for thegrowth rate µ.
Now we plot these approximations.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 109: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/109.jpg)
SERIES ANALYSIS
Using differential approximants, we approximate 1000 furtherratios, which we estimate to be accurate to at least 10 significantdigits.
so now we have a sequence of ratios and approximate ratiosr1, r2, . . . , r1100.
when we plot these against 1/n they don’t seem completelylinear, but plotted against 1/(n log(n)2) they do seem prettylinear.
Using the line between adjacent points in this plot and takingtheir intercept with the y-axis gives better approximations for thegrowth rate µ.
Now we plot these approximations.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 110: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/110.jpg)
PLOT OF RATIOS APPROXIMATIONS FOR µ
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 111: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/111.jpg)
PLOT OF RATIOS APPROXIMATIONS FOR µ
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 112: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/112.jpg)
SERIES ANALYSIS
Based on this graph, we estimate that the growth rateµ ≈ 21.7656.
This is suspiciously close to 4√
3π = 21.76559 . . .
We do the same analysis for the sequence a0, a1, a2, . . . thenumbers of rooted planar Eulerian orientations
In this case we find that the growth rate is approximately 4π.
We conjecture that 4π and 4√
3π are the exact growth rates.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 113: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/113.jpg)
SERIES ANALYSIS
Based on this graph, we estimate that the growth rateµ ≈ 21.7656.
This is suspiciously close to 4√
3π = 21.76559 . . .
We do the same analysis for the sequence a0, a1, a2, . . . thenumbers of rooted planar Eulerian orientations
In this case we find that the growth rate is approximately 4π.
We conjecture that 4π and 4√
3π are the exact growth rates.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 114: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/114.jpg)
SERIES ANALYSIS
Based on this graph, we estimate that the growth rateµ ≈ 21.7656.
This is suspiciously close to 4√
3π = 21.76559 . . .
We do the same analysis for the sequence a0, a1, a2, . . . thenumbers of rooted planar Eulerian orientations
In this case we find that the growth rate is approximately 4π.
We conjecture that 4π and 4√
3π are the exact growth rates.
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 115: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/115.jpg)
SERIES ANALYSIS
Based on this graph, we estimate that the growth rateµ ≈ 21.7656.
This is suspiciously close to 4√
3π = 21.76559 . . .
We do the same analysis for the sequence a0, a1, a2, . . . thenumbers of rooted planar Eulerian orientations
In this case we find that the growth rate is approximately 4π.
We conjecture that 4π and 4√
3π are the exact growth rates.
Enumerating Eulerian Orientations. Andrew Elvey Price
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SERIES ANALYSIS
Based on this graph, we estimate that the growth rateµ ≈ 21.7656.
This is suspiciously close to 4√
3π = 21.76559 . . .
We do the same analysis for the sequence a0, a1, a2, . . . thenumbers of rooted planar Eulerian orientations
In this case we find that the growth rate is approximately 4π.
We conjecture that 4π and 4√
3π are the exact growth rates.
Enumerating Eulerian Orientations. Andrew Elvey Price
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MORE CONJECTURES
The growth rate 4√
3π pointed us in the direction of looking atother combinatorial sequences with this growth rate.
Using this we have conjectured an the exact solution for thegenerating function
B(t) = b0 + b1t + b2t2,
which agrees with the 100 terms that we have computed exactly.
Assuming that this conjectures are correct, this solution isD-algebraic but not D-finite.
Using this we can produce thousands of conjectured terms bn. Itturn out that our approximate ratios were all correct to 30significant digits!
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 118: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/118.jpg)
MORE CONJECTURES
The growth rate 4√
3π pointed us in the direction of looking atother combinatorial sequences with this growth rate.
Using this we have conjectured an the exact solution for thegenerating function
B(t) = b0 + b1t + b2t2,
which agrees with the 100 terms that we have computed exactly.
Assuming that this conjectures are correct, this solution isD-algebraic but not D-finite.
Using this we can produce thousands of conjectured terms bn. Itturn out that our approximate ratios were all correct to 30significant digits!
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 119: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/119.jpg)
MORE CONJECTURES
The growth rate 4√
3π pointed us in the direction of looking atother combinatorial sequences with this growth rate.
Using this we have conjectured an the exact solution for thegenerating function
B(t) = b0 + b1t + b2t2,
which agrees with the 100 terms that we have computed exactly.
Assuming that this conjectures are correct, this solution isD-algebraic but not D-finite.
Using this we can produce thousands of conjectured terms bn. Itturn out that our approximate ratios were all correct to 30significant digits!
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 120: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/120.jpg)
MORE CONJECTURES
The growth rate 4√
3π pointed us in the direction of looking atother combinatorial sequences with this growth rate.
Using this we have conjectured an the exact solution for thegenerating function
B(t) = b0 + b1t + b2t2,
which agrees with the 100 terms that we have computed exactly.
Assuming that this conjectures are correct, this solution isD-algebraic but not D-finite.
Using this we can produce thousands of conjectured terms bn. Itturn out that our approximate ratios were all correct to 30significant digits!
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 121: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/121.jpg)
MORE CONJECTURES
The growth rate 4√
3π pointed us in the direction of looking atother combinatorial sequences with this growth rate.
Using this we have conjectured an the exact solution for thegenerating function
B(t) = b0 + b1t + b2t2,
which agrees with the 100 terms that we have computed exactly.
Assuming that this conjectures are correct, this solution isD-algebraic but not D-finite.
Using this we can produce thousands of conjectured terms bn.
Itturn out that our approximate ratios were all correct to 30significant digits!
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 122: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/122.jpg)
MORE CONJECTURES
The growth rate 4√
3π pointed us in the direction of looking atother combinatorial sequences with this growth rate.
Using this we have conjectured an the exact solution for thegenerating function
B(t) = b0 + b1t + b2t2,
which agrees with the 100 terms that we have computed exactly.
Assuming that this conjectures are correct, this solution isD-algebraic but not D-finite.
Using this we can produce thousands of conjectured terms bn. Itturn out that our approximate ratios were all correct to 30significant digits!
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 123: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/123.jpg)
CONJECTURES
In the same way found a conjectured D-algebraic form for thegenerating function A(t) for a0, a1, a2, . . .
Collaborating with Mireille Bousquet-Melou, we have proventhis.
We are still working on the conjecture for b0, b1, . . .
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 124: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/124.jpg)
CONJECTURES
In the same way found a conjectured D-algebraic form for thegenerating function A(t) for a0, a1, a2, . . .
Collaborating with Mireille Bousquet-Melou, we have proventhis.
We are still working on the conjecture for b0, b1, . . .
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 125: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/125.jpg)
CONJECTURES
In the same way found a conjectured D-algebraic form for thegenerating function A(t) for a0, a1, a2, . . .
Collaborating with Mireille Bousquet-Melou, we have proventhis.
We are still working on the conjecture for b0, b1, . . .
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 126: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/126.jpg)
FURTHER QUESTIONS
Can we count rooted planar Eulerian orientations by edges andvertices?
Can we determine ∑Γ:|V(Γ)|=n
TΓ(x, y),
for other specific values of x, y?
For all x, y??
Enumerating Eulerian Orientations. Andrew Elvey Price
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FURTHER QUESTIONS
Can we count rooted planar Eulerian orientations by edges andvertices?
Can we determine ∑Γ:|V(Γ)|=n
TΓ(x, y),
for other specific values of x, y?
For all x, y??
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 128: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/128.jpg)
FURTHER QUESTIONS
Can we count rooted planar Eulerian orientations by edges andvertices?
Can we determine ∑Γ:|V(Γ)|=n
TΓ(x, y),
for other specific values of x, y?
For all x, y??
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 129: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/129.jpg)
FURTHER QUESTIONS
Can we count rooted planar Eulerian orientations by edges andvertices?
Can we determine ∑Γ:|V(Γ)|=n
TΓ(x, y),
for other specific values of x, y?
For all x, y??
Enumerating Eulerian Orientations. Andrew Elvey Price
![Page 130: Enumerating Eulerian Orientations. - Monash …users.monash.edu/~gfarr/research/slides/ElveyPrice...Numbers on adjacent vertices differ by 1. By the bijection, a n is the number of](https://reader034.vdocuments.net/reader034/viewer/2022042407/5f216a6d14a1af7eec1cc7d2/html5/thumbnails/130.jpg)
THANK YOU
Thank You!
Enumerating Eulerian Orientations. Andrew Elvey Price