Generating Function for Pinsky's Combinatorial Second Moment Formula for the Generalized Ulam Problem

Abstract

Given a uniform random permutation $\pi \in S_n$, let $Z_{n,k}$ be equal to the number of increasing subsequences of length $k$: so $Z_{n,k}=|\{(i_1,\dots,i_k) \in \mathbb{Z}^k\, :\, 1\leq i_1<\dots<i_k\leq n\, ,\ \pi_{i_1}<\dots<\pi_{i_k}\}|$. In an important paper, Ross Pinsky proved $\mathbf{E}\big[Z_{n,k}^2\big]$ is equal to $\sum_{i} A(k-i,i)B(n,2k-i)$, where for any nonnegative integers $N$ and $j$, we have $B(N,j) = \binom{N}{j}/j!$ and $A(N,j)$ is a particular nonnegative integer, which Pinsky characterized in two different ways. One characterization of $A(N,j)$ involves the occupation time of the $x$-axis prior to a first return to the origin. Using this, he proved a law of large numbers for the sequence $Z_{n,k_n}$ when $k_n=o(n^{2/5})$ as $n \to \infty$. In a follow-up paper, he also proved the sequence $Z_{n,k_n}$ fails to obey a law of large numbers when $1/k_n = o(1/n^{4/9})$ as $n \to \infty$. Here, we return to his combinatorial formula for the the second moment of $Z_{n,k}$, and we obtain a generating function for the $A(N,j)$ triangular array. We are motivated by the hope of applying spin glass techniques to the well-known Ulam's problem to see if this gives a new perspective.