Stieltjes moment sequences for pattern-avoiding permutations
Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, Jean-Marie, Maillard

TL;DR
This paper investigates whether certain pattern-avoiding permutation sequences are Stieltjes moment sequences, explicitly finds their density functions, links them to modular forms and random walks, and provides bounds on their growth constants.
Contribution
It explicitly characterizes the density functions for classical pattern-avoiding sequences and explores their connections to modular forms, random walks, and growth constant bounds.
Findings
Sequences for increasing patterns are Stieltjes moment sequences.
Density functions are explicitly or numerically determined.
Bounds on the growth constant of the $Av(1324)$ sequence are improved.
Abstract
A small set of combinatorial sequences have coefficients that can be represented as moments of a nonnegative measure on . Such sequences are known as Stieltjes moment sequences. This article focuses on some classical sequences in enumerative combinatorics, denoted , and counting permutations of that avoid some given pattern . For increasing patterns , we recall that the corresponding sequences, , 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 show that the generating functions of the sequences and correspond, up to simple rational functions, to an order-one linear differential operator acting on a classical modular…
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