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Article Dans Une Revue The Electronic Journal of Combinatorics Année : 2020

Stieltjes moment sequences for pattern-avoiding permutations

Résumé

A small set 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. This article focuses on some classical sequences in enumerative combinatorics, denoted $Av(\mathcal{P})$, and counting permutations of $\{1, 2, \ldots, n \}$ that avoid some given pattern $\mathcal{P}$. For increasing patterns $\mathcal{P}=(12\ldots k)$, we recall that the corresponding sequences, $Av(123\ldots k)$, 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 $Av(123\ldots k)$ 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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Dates et versions

hal-02425917 , version 1 (31-12-2019)
hal-02425917 , version 2 (03-03-2020)
hal-02425917 , version 3 (17-10-2020)

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Paternité - Pas de modifications

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Alin Bostan, Andrew Elvey-Price, Anthony John Guttmann, Jean-Marie Maillard. Stieltjes moment sequences for pattern-avoiding permutations. The Electronic Journal of Combinatorics, 2020, 27 (4), pp.1-59. ⟨10.37236/xxxx⟩. ⟨hal-02425917v3⟩
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