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 $[0,\infty )$. 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 $Av\left(𝒫\right)$, 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 $Av\left(1234\right)$ and $Av\left(12345\right)$, correspond to an order-one linear differential operator acting on a classical modular form given as a pullback of a Gaussian ${}_2{F}_1$ 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 ${}_2{F}_1$ 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 $Av\left(1324\right)$ 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.