{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,10]],"date-time":"2025-11-10T13:48:12Z","timestamp":1762782492342,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>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.\r\nThis 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.\r\nWe 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.\r\nWe 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.\r\nFinally, 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.<\/jats:p>","DOI":"10.37236\/9402","type":"journal-article","created":{"date-parts":[[2020,10,30]],"date-time":"2020-10-30T08:55:25Z","timestamp":1604048125000},"source":"Crossref","is-referenced-by-count":4,"title":["Stieltjes Moment Sequences for Pattern-Avoiding Permutations"],"prefix":"10.37236","volume":"27","author":[{"given":"Alin","family":"Bostan","sequence":"first","affiliation":[]},{"given":"Andrew","family":"Elvey Price","sequence":"additional","affiliation":[]},{"given":"Anthony John","family":"Guttmann","sequence":"additional","affiliation":[]},{"given":"Jean-Marie","family":"Maillard","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2020,10,30]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i4p20\/8202","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i4p20\/8202","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,10,30]],"date-time":"2020-10-30T08:55:25Z","timestamp":1604048125000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v27i4p20"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,10,30]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2020,10,2]]}},"URL":"https:\/\/doi.org\/10.37236\/9402","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2020,10,30]]},"article-number":"P4.20"}}