Stieltjes moment sequences for pattern-avoiding permutations
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A small set 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. This article focuses on 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), we recall that the corresponding sequences, Av(123… 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 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… 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.
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