### Stieltjes Moment Sequences for Pattern-Avoiding Permutations

Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, Jean-Marie Maillard
2020 Electronic Journal of Combinatorics
A small set of combinatorial sequences have coefficients that can be represented as moments of a nonnegative measure on $[0, \infty)$. Such sequences are known as Stieltjes moment sequences. They have a number of nice properties, such as log-convexity, which are useful to rigorously bound their growth constant from below. This article focuses on some classical sequences in enumerative combinatorics, denoted $Av(\mathcal{P})$, and counting permutations of $\{1, 2, \ldots, n \}$ that avoid some
more » ... $that avoid some given pattern$\mathcal{P}$. For increasing patterns$\mathcal{P}=(12\ldots k)$, we recall that the corresponding sequences,$Av(123\ldots k)$, are Stieltjes moment sequences, and we explicitly find the underlying density function, either exactly or numerically, by using the Stieltjes inversion formula as a fundamental tool. We first illustrate our approach on two basic examples,$Av(123)$and$Av(1342)$, whose generating functions are algebraic. We next investigate the general (transcendental) case of$Av(123\ldots k)$, which counts permutations whose longest increasing subsequences have length at most$k-1$. We show that the generating functions of the sequences$\, Av(1234)$and$\, Av(12345)$correspond, up to simple rational functions, to an order-one linear differential operator acting on a classical modular form given as a pullback of a Gaussian$\, _2F_1$hypergeometric function, respectively to an order-two linear differential operator acting on the square of a classical modular form given as a pullback of a$\, _2F_1$hypergeometric function. We demonstrate that the density function for the Stieltjes moment sequence$Av(123\ldots k)$is closely, but non-trivially, related to the density attached to the distance traveled by a walk in the plane with$k-1$unit steps in random directions. Finally, we study the challenging case of the$Av(1324)\$ sequence and give compelling numerical evidence that this too is a Stieltjes moment sequence. Accepting this, we show how rigorous lower bounds on the growth constant of this sequence can be constructed, which are stronger than existing bounds. A further unproven assumption leads to even better bounds, which can be extrapolated to give an estimate of the (unknown) growth constant.