Problem 60-2, On a Binomial Identity Arising from a Sorting ProblemAuthor(s): Paul BrockSource: SIAM Review, Vol. 2, No. 1 (Jan., 1960), p. 40Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2028060 .

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40 PROBLEMS AND SOLUTIONS

The fluids have approximately equal densities and the heavier was flowed under AB so that the equilibrium would be stable.

The steady-state distribution of the fluids is assumed to be given by

(1) as = a +2s + as2s

where S(x, y) denotes the fractional amount of the lower fluid present at the point (x, y). The equation was obtained under the following assumptions (for an analogous problem, see H. Bateman, Partial Differential Equations, p. 343):

(1) Streamline motion and molecular diffusion cause the dispersion. (2) The coefficient a reflects the dispersion of flow caused by velocity varia-

tions along each streamline and molecular diffusion in the direction of flow.

(3) The coefficient A reflects the dispersion perpendicular to the direction of flow caused by diffusion between streamlines.

The boundary conditions to be satisfied are

(a) as = 0 on the impermeable boundaries, ay (b) S-Ofory > 0,x--oo ,and

(c) S -1 fory < 0,x --oo.

Solve equation (1) for an infinite medium. Here boundary condition (a) is replaced by

(a') as = Ofory = O,x < 0. Oy

Problem 60-2, On a Binomial Identity Arising from a Sorting Problem, by PAUL

BROCK (Hughes Aircraft Company).

If

H(A,B) = f (i+j) (A-i + ) (B + -)(A-+B -i)

show that

H(A, B) - H(A - 1, B) - H(A, B -1 ) = (A + B).

This problem has arisen by considering the N! permutations of the integers 1 through N; each permutation will have a maximum length monotonic increas- ing and decreasing subsequence (see Problem 59-3). If one tries to count the number of those permutations whose maximum monotonic increasing subse- quence is of length r and whose maximum monotonic decreasing subsequence is

of length N - r + 1, the total count is ( 1) sequences. The proof of this

can be shown to be equivalent to demonstrating the above identity.

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