On the Stanley–Wilf limit of 4231-avoiding permutations and a conjecture of Arratia

Albert, Michael H.; Elder, Murray; Rechnitzer, Andrew; Westcott, P.; Zabrocki, Mike

doi:10.1016/j.aam.2005.05.007

Cited by 40 publications

(76 citation statements)

References 22 publications

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“…It is not known whether [3], but it is not known which permutations do achieve the maximum or minimum. An interesting aspect of pattern avoidance was considered by Albert [1].…”

Section: Pattern Avoidancementioning

confidence: 99%

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Increasing and decreasing subsequences and their variants

Stanley¹

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

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Abstract. We survey the theory of increasing and decreasing subsequences of permutations. Enumeration problems in this area are closely related to the RSK algorithm. The asymptotic behavior of the expected value of the length is(w) of the longest increasing subsequence of a permutation w of 1, 2, . . . , n was obtained by Vershik-Kerov and (almost) by Logan-Shepp. The entire limiting distribution of is(w) was then determined by Baik, Deift, and Johansson. These techniques can be applied to other classes of permutations, such as involutions, and are related to the distribution of eigenvalues of elements of the classical groups. A number of generalizations and variations of increasing/decreasing subsequences are discussed, including the theory of pattern avoidance, unimodal and alternating subsequences, and crossings and nestings of matchings and set partitions. Mathematics Subject Classification (2000). Primary 05A05, 05A06; Secondary 60C05.

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“…It is not known whether [3], but it is not known which permutations do achieve the maximum or minimum. An interesting aspect of pattern avoidance was considered by Albert [1].…”

Section: Pattern Avoidancementioning

confidence: 99%

“…Hence the left-hand side of equation (8) follows. The right-hand side is then a consequence of the hook-length formula (3). 2…”

Section: Corollarymentioning

confidence: 99%

Increasing and decreasing subsequences and their variants

Stanley¹

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

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“…Figures 2 and 4 contain illustrations of large 1324-avoiders. 1 As is noted by Flajolet & Sedgewick ([8] p.169), the fact that a single example can be used to illustrate the asymptotic structure of a large random combinatorial object can be attributed to concentration of distributions, of which we make much use below in determining our lower bound. Observe the cigar-shaped boundary regions consisting of numerous small subtrees, and also the relative scarcity of points in the interior, which tend to be partitioned into a few paths connecting the two boundaries.…”

Section: Introductionmentioning

confidence: 99%

“…As far as lower bounds go, Albert, Elder, Rechnitzer, Westcott & Zabrocki [1] have established that the growth rate is at least 9.47, by using the insertion encoding of 1324-avoiders to construct a sequence of finite automata that accept subclasses of Av(1324). The growth rate of a subclass is then determined from the transition matrix of the corresponding automaton.…”

Section: Introductionmentioning

confidence: 99%

Permutations avoiding 1324 and patterns in Łukasiewicz paths

Bevan

2015

J. London Math. Soc.

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The class Av(1324), of permutations avoiding the pattern 1324, is one of the simplest sets of combinatorial objects to define that has, thus far, failed to reveal its enumerative secrets. By considering certain large subsets of the class, which consist of permutations with a particularly regular structure, we prove that the growth rate of the class exceeds 9.81. This improves on a previous lower bound of 9.47. Central to our proof is an examination of the asymptotic distributions of certain substructures in the Hasse graphs of the permutations. In this context, we consider occurrences of patterns in Łukasiewicz paths and prove that in the limit they exhibit a concentrated Gaussian distribution.

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“…In the ordered version of Turán theory, the question is: what is the maximum number edges of an ordered bipartite graph with parts of size p and q with no subgraph isomorphic to a given ordered bipartite graph? More results on this problem and its variations are given in [1,3,4,8,9]. As another variation, interval minors were recently introduced by Fox in [5] in the study of Stanley-Wilf limits.…”

Section: Introductionmentioning

confidence: 99%