About the Moments of the Generalized Ulam Problem
Samen Hossein, Shannon Starr
TL;DR
This work addresses the generalized Ulam problem, focusing on the moments and large-deviation behavior of $Z_{n,k}$, the number of increasing subsequences of length $k$ in a random permutation. It derives an exact second-moment decomposition $\mathbb{E}[Z_{n,k}Z_{n,\ell}] = \sum_{j=0}^{k\wedge\ell} \mathbb{E}[Z_{n,k+\ell-j}] \mathcal{A}(k-j,\ell-j,j)$ with a combinatorial factor $\mathcal{A}$ whose multivariate generating function is $\sum \mathcal{A}(k,\ell,j) x^k y^\ell z^j = 1/\sqrt{1-2(x+y)+(x-y)^2}-z$, enabling a multivariate saddle-point analysis in the critical regime $k,\ell \sim \mathcal{O}(\sqrt{n})$. The paper also introduces an exactly solvable model for sums of i.i.d. exponentials, develops a replica-symmetric ansatz and a replica-to-zero trick to access leading-order behavior of higher moments, and reveals limitations of replica-symmetric predictions via a Poisson-point-process description near the minimum. Extending to the third moment, it employs diagonal generating-function methods to obtain a generating function for $r=3$ that reduces to an elliptic integral, and discusses interpretations in terms of random-walk intersections in $d=2$ and $d=3$. Overall, the results connect combinatorial moment calculations with analytic and probabilistic physics-inspired techniques, and lay groundwork for future rigorous validation and extensions to higher moments.
Abstract
Given $π\in S_n$, let $Z_{n,k}(π)=\sum_{1\leq i_1<\dots<i_k\leq n} \mathbf{1}(\{ π_{i_1}<\dots<π_{i_k}\}$ denote the number of increasing subsequences of length $k$. Consider the "generalized Ulam problem," studying the distribution of $Z_{n,k}$ for general $k$ and $n$. For the 2nd moment, Ross Pinsky initiated a combinatorial study by considering a pair of subsequences $i^{(r)}_1<\dots<i^{(r)}_k$ for $r \in \{1,2\}$, and conditioning on the size of the intersection $j = |\{i_1^{(1)},\dots,i^{(1)}_k\} \cap \{i^{(2)}_1,\dots,i^{(2)}_k\}|$. We obtain the exact large deviation rate function for $\mathbf{E}[Z_{n,k} Z_{n,\ell}]$ in the asymptotic regime $k\sim κn^{1/2}$, $\ell \sim λn^{1/2}$ as $n \to \infty$, for $κ,λ\in (0,\infty)$. This uses multivariate generating function techniques, as found in the textbook of Pemantle and Wilson. The requisite generating function enumerates pairs of up-right paths in $d=2$, which both end at $(k,\ell)$ with a given number of intersections. We also evaluate the analogous generating function for pairs of $(+\boldsymbol{i},+\boldsymbol{j},+\boldsymbol{k})$ paths in $d=3$, which both end at $(k,\ell,m)$, which has some utility in calculating the 3rd moment. Finally, we consider a simpler problem involving partitions instead of permutations, where all moments are calculable and the replica symmetric ansatz can be stated if not proved.