Complexity Problems in Enumerative Combinatorics

“…This is known under the name of the Noonan-Zeilberger conjecture (1996). The conjecture was disproved by Garrabrant and Pak [54,93] using tools from complexity/computability theory; their striking result is that there exists a family of patterns which each have length 80, such that the corresponding generating function is not D-finite 6 . Note that this existential result is not yet complemented by any concrete example, ideally of a single avoided permutation, such as (1324).…”

Section: Related Workmentioning

confidence: 99%

Stieltjes Moment Sequences for Pattern-Avoiding Permutations

Bostan

Price²,

Guttmann³

et al. 2020

Electron. J. Combin.

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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. They have a number of nice properties, such as log-convexity, which are useful to rigorously bound their growth constant from below. 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 first illustrate our approach on two basic examples, $Av(123)$ and $Av(1342)$, whose generating functions are algebraic. We next investigate the general (transcendental) case of $Av(123\ldots k)$, which counts permutations whose longest increasing subsequences have length at most $k-1$. 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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“…Combinatorics in recent years, see Pak's ICM paper [18] for a recent survey. Yet, this perspective is still largely missing in f -vector theory.…”

Section: Computational Complexity Computational Complexity Gains Impmentioning

confidence: 99%