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Section: New Results

Densities of Stieltjes moment sequences for pattern-avoiding permutations

A small subset of combinatorial sequences have coefficients that can be represented as moments of a nonnegative measure on [0,). Such sequences are known as Stieltjes moment sequences. They have a number of useful properties, such as log-convexity, which in turn enables one to rigorously bound their growth constant from below.

In [12], Alin Bostan together with Andrew Elvey Price, Anthony Guttmann and Jean-Marie Maillard, studied some classical sequences in enumerative combinatorics, denoted Av(𝒫), and counting permutations of {1,2,...,n} that avoid some given pattern 𝒫. For increasing patterns 𝒫=(12...k), they showed that the corresponding sequences, Av(123...k), are Stieltjes moment sequences, and explicitly determined the underlying density function, either exactly or numerically, by using the Stieltjes inversion formula as a fundamental tool.

They showed that the densities for Av(1234) and Av(12345), correspond to an order-one linear differential operator acting on a classical modular form given as a pullback of a Gaussian 2F1 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 2F1 hypergeometric function. Moreover, these density functions are 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.

As a bonus, they studied the challenging case of the Av(1324) sequence and gave compelling numerical evidence that this too is a Stieltjes moment sequence. Accepting this, they proved new lower bounds on the growth constant of this sequence, which are stronger than existing bounds. A further unproven assumption leads to even better bounds, which can be extrapolated to give a good estimate of the (unknown) growth constant.