enumerating (2+2)-free posets by the number of minimal elements and other statistics sergey kitaev...
TRANSCRIPT
![Page 1: Enumerating (2+2)-free posets by the number of minimal elements and other statistics Sergey Kitaev Reykjavik University Joint work with Jeff Remmel University](https://reader036.vdocuments.net/reader036/viewer/2022062515/56649f4d5503460f94c6e925/html5/thumbnails/1.jpg)
Enumerating (2+2)-free posets by the number of minimal elements and other statistics
Sergey KitaevReykjavik University
Joint work with
Jeff RemmelUniversity of California, San Diego
![Page 2: Enumerating (2+2)-free posets by the number of minimal elements and other statistics Sergey Kitaev Reykjavik University Joint work with Jeff Remmel University](https://reader036.vdocuments.net/reader036/viewer/2022062515/56649f4d5503460f94c6e925/html5/thumbnails/2.jpg)
Unlabeled (2+2)-free posets
A partially ordered set is called (2+2)-free if it contains no induced sub-posets isomorphic to (2+2) =
Such posets arise as interval orders (Fishburn):
P. C. Fishburn, Intransitive indifference with unequal indifference intervals, J. Math.Psych. 7 (1970) 144–149.
bad guy good guy
![Page 3: Enumerating (2+2)-free posets by the number of minimal elements and other statistics Sergey Kitaev Reykjavik University Joint work with Jeff Remmel University](https://reader036.vdocuments.net/reader036/viewer/2022062515/56649f4d5503460f94c6e925/html5/thumbnails/3.jpg)
Ascent sequencesNumber of ascents in a word: asc(0, 0, 2, 1, 1, 0, 3, 1, 2, 3) = 4
(0,0,2,1,1,0,3,1,2,3) is not an ascent sequence, whereas (0,0,1,0,1,3,0) is.
![Page 4: Enumerating (2+2)-free posets by the number of minimal elements and other statistics Sergey Kitaev Reykjavik University Joint work with Jeff Remmel University](https://reader036.vdocuments.net/reader036/viewer/2022062515/56649f4d5503460f94c6e925/html5/thumbnails/4.jpg)
Mireille Bousquet-Mélou
Anders Claesson
Mark Dukes
SK
Unlabeled (2+2)-free posets, ascent sequences, and pattern avoiding permutations
![Page 5: Enumerating (2+2)-free posets by the number of minimal elements and other statistics Sergey Kitaev Reykjavik University Joint work with Jeff Remmel University](https://reader036.vdocuments.net/reader036/viewer/2022062515/56649f4d5503460f94c6e925/html5/thumbnails/5.jpg)
Mireille Bousquet-Mélou
Anders Claesson
Mark Dukes
SK
Unlabeled (2+2)-free posets, ascent sequences, and pattern avoiding permutations
Robert Parviainen
![Page 6: Enumerating (2+2)-free posets by the number of minimal elements and other statistics Sergey Kitaev Reykjavik University Joint work with Jeff Remmel University](https://reader036.vdocuments.net/reader036/viewer/2022062515/56649f4d5503460f94c6e925/html5/thumbnails/6.jpg)
Mireille Bousquet-MélouAnders Claesson
Mark Dukes
SK
Unlabeled (2+2)-free posets, ascent sequences, and pattern avoiding permutations
Svante Linusson
Invited talk at the AMS-MAA joint mathematics meeting
![Page 7: Enumerating (2+2)-free posets by the number of minimal elements and other statistics Sergey Kitaev Reykjavik University Joint work with Jeff Remmel University](https://reader036.vdocuments.net/reader036/viewer/2022062515/56649f4d5503460f94c6e925/html5/thumbnails/7.jpg)
Mireille Bousquet-MélouAnders Claesson
Mark Dukes
SK
Unlabeled (2+2)-free posets, ascent sequences, and pattern avoiding permutations
Jeff Remmel
The present talk
![Page 8: Enumerating (2+2)-free posets by the number of minimal elements and other statistics Sergey Kitaev Reykjavik University Joint work with Jeff Remmel University](https://reader036.vdocuments.net/reader036/viewer/2022062515/56649f4d5503460f94c6e925/html5/thumbnails/8.jpg)
Mireille Bousquet-MélouAnders Claesson
Mark Dukes
SK
Unlabeled (2+2)-free posets, ascent sequences, and pattern avoiding permutations
A direct encoding of Stoimenow’s matchings as ascent sequences
![Page 9: Enumerating (2+2)-free posets by the number of minimal elements and other statistics Sergey Kitaev Reykjavik University Joint work with Jeff Remmel University](https://reader036.vdocuments.net/reader036/viewer/2022062515/56649f4d5503460f94c6e925/html5/thumbnails/9.jpg)
Overview of results by Bousquet-Mélou et al. (2008)
Bijections (respecting several statistics) between the following objects
unlabeled (2+2)-free posets on n elements
pattern-avoiding permutations of length n
ascent sequences of length n
linearized chord diagrams with n chords = certain involutions
Closed form for the generating function for these classes of objects
Pudwell’s conjecture (on permutations avoiding 31524) is settled using modified ascent sequences
_ _
![Page 10: Enumerating (2+2)-free posets by the number of minimal elements and other statistics Sergey Kitaev Reykjavik University Joint work with Jeff Remmel University](https://reader036.vdocuments.net/reader036/viewer/2022062515/56649f4d5503460f94c6e925/html5/thumbnails/10.jpg)
Unlabeled (2+2)-free posets
Theorem. (easy to prove) A poset is (2+2)-free iff the collection of strict down-sets may be linearly ordered by inclusion.
![Page 11: Enumerating (2+2)-free posets by the number of minimal elements and other statistics Sergey Kitaev Reykjavik University Joint work with Jeff Remmel University](https://reader036.vdocuments.net/reader036/viewer/2022062515/56649f4d5503460f94c6e925/html5/thumbnails/11.jpg)
Unlabeled (2+2)-free posets
How can one decompose a (2+2)-free poset?
![Page 12: Enumerating (2+2)-free posets by the number of minimal elements and other statistics Sergey Kitaev Reykjavik University Joint work with Jeff Remmel University](https://reader036.vdocuments.net/reader036/viewer/2022062515/56649f4d5503460f94c6e925/html5/thumbnails/12.jpg)
Unlabeled (2+2)-free posets
2
![Page 13: Enumerating (2+2)-free posets by the number of minimal elements and other statistics Sergey Kitaev Reykjavik University Joint work with Jeff Remmel University](https://reader036.vdocuments.net/reader036/viewer/2022062515/56649f4d5503460f94c6e925/html5/thumbnails/13.jpg)
Unlabeled (2+2)-free posets
1 1 3
1 0 1
Read labels backwards: (0, 1, 0, 1, 3, 1, 1, 2) – an ascent sequence!
Removing last point gives one extra 0.
![Page 14: Enumerating (2+2)-free posets by the number of minimal elements and other statistics Sergey Kitaev Reykjavik University Joint work with Jeff Remmel University](https://reader036.vdocuments.net/reader036/viewer/2022062515/56649f4d5503460f94c6e925/html5/thumbnails/14.jpg)
Theorem. There is a 1-1 correspondence between unlabeled (2+2)-free posets on n elements and ascent sequences of length n.
(0, 1, 0, 1, 3, 1, 1, 2)
![Page 15: Enumerating (2+2)-free posets by the number of minimal elements and other statistics Sergey Kitaev Reykjavik University Joint work with Jeff Remmel University](https://reader036.vdocuments.net/reader036/viewer/2022062515/56649f4d5503460f94c6e925/html5/thumbnails/15.jpg)
Some statistics preserved under the bijection
(0, 1, 0, 1, 3, 1, 1, 2)
(0, 1, 0, 1, 3, 1, 1, 2 )
(0, 1, 0, 1, 3, 1, 1, 2)
min zeros
min maxlevel
last element
(0, 3, 0, 1, 4, 1, 1, 2)
Level distri-bution
letter distributionin modif. sequence
![Page 16: Enumerating (2+2)-free posets by the number of minimal elements and other statistics Sergey Kitaev Reykjavik University Joint work with Jeff Remmel University](https://reader036.vdocuments.net/reader036/viewer/2022062515/56649f4d5503460f94c6e925/html5/thumbnails/16.jpg)
Some statistics preserved under the bijection
(0, 1, 0, 1, 3, 1, 1, 2)
(0, 1, 0, 1, 3, 1, 1, 2)
(0, 1, 0, 1, 3, 1, 1, 2)
highestlevel
number of ascents
(0, 3, 0, 1, 4, 1, 1, 2)
right-to-left maxin mod. sequencemax
compo-nents
Components inmodif. sequence
(0, 3, 0, 1, 4, 1, 1, 2)
![Page 17: Enumerating (2+2)-free posets by the number of minimal elements and other statistics Sergey Kitaev Reykjavik University Joint work with Jeff Remmel University](https://reader036.vdocuments.net/reader036/viewer/2022062515/56649f4d5503460f94c6e925/html5/thumbnails/17.jpg)
![Page 18: Enumerating (2+2)-free posets by the number of minimal elements and other statistics Sergey Kitaev Reykjavik University Joint work with Jeff Remmel University](https://reader036.vdocuments.net/reader036/viewer/2022062515/56649f4d5503460f94c6e925/html5/thumbnails/18.jpg)
A generalization of the generating function
lds=size of last non-trivial downset
...
minmaxmin
![Page 19: Enumerating (2+2)-free posets by the number of minimal elements and other statistics Sergey Kitaev Reykjavik University Joint work with Jeff Remmel University](https://reader036.vdocuments.net/reader036/viewer/2022062515/56649f4d5503460f94c6e925/html5/thumbnails/19.jpg)
The main result in this talk (SK & J. Remmel, 2009):
The corresponding posets:
![Page 20: Enumerating (2+2)-free posets by the number of minimal elements and other statistics Sergey Kitaev Reykjavik University Joint work with Jeff Remmel University](https://reader036.vdocuments.net/reader036/viewer/2022062515/56649f4d5503460f94c6e925/html5/thumbnails/20.jpg)
A conjecture (SK & J. Remmel, 2009):
Compare to
![Page 21: Enumerating (2+2)-free posets by the number of minimal elements and other statistics Sergey Kitaev Reykjavik University Joint work with Jeff Remmel University](https://reader036.vdocuments.net/reader036/viewer/2022062515/56649f4d5503460f94c6e925/html5/thumbnails/21.jpg)
Posets avoiding and
Ascent sequences are restricted as follows:
m-1, where m is the max element here
Catalan many
Catalan many
Hilmar HaukurGuðmundsson
![Page 22: Enumerating (2+2)-free posets by the number of minimal elements and other statistics Sergey Kitaev Reykjavik University Joint work with Jeff Remmel University](https://reader036.vdocuments.net/reader036/viewer/2022062515/56649f4d5503460f94c6e925/html5/thumbnails/22.jpg)
Posets avoiding and
Self modified ascent sequences
Bayoumi, El-Zahar, Khamis (1989)
![Page 23: Enumerating (2+2)-free posets by the number of minimal elements and other statistics Sergey Kitaev Reykjavik University Joint work with Jeff Remmel University](https://reader036.vdocuments.net/reader036/viewer/2022062515/56649f4d5503460f94c6e925/html5/thumbnails/23.jpg)
Thank you for your attention!