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 . 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 , and counting permutations of that avoid some given pattern . For increasing patterns , they showed that the corresponding sequences, , 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 and , correspond to an order-one linear differential operator acting on a classical modular form given as a pullback of a Gaussian 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 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 unit steps in random directions.
As a bonus, they studied the challenging case of the 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.