35 research outputs found

    Exact enumeration of 1342-avoiding permutations: A close link with labeled trees and planar maps

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    Solving the first nonmonotonic, longer-than-three instance of a classic enumeration problem, we obtain the generating function H(x)H(x) of all 1342-avoiding permutations of length nn as well as an {\em exact} formula for their number Sn(1342)S_n(1342). While achieving this, we bijectively prove that the number of indecomposable 1342-avoiding permutations of length nn equals that of labeled plane trees of a certain type on nn vertices recently enumerated by Cori, Jacquard and Schaeffer, which is in turn known to be equal to the number of rooted bicubic maps enumerated by Tutte in 1963. Moreover, H(x)H(x) turns out to be algebraic, proving the first nonmonotonic, longer-than-three instance of a conjecture of Zeilberger and Noonan. We also prove that Sn(1342)n\sqrt[n]{S_n(1342)} converges to 8, so in particular, limnā†’āˆž(Sn(1342)/Sn(1234))=0lim_{n\rightarrow \infty}(S_n(1342)/S_n(1234))=0

    Generalized permutation patterns - a short survey

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    An occurrence of a classical pattern p in a permutation Ļ€ is a subsequence of Ļ€ whose letters are in the same relative order (of size) as those in p. In an occurrence of a generalized pattern, some letters of that subsequence may be required to be adjacent in the permutation. Subsets of permutations characterized by the avoidanceā€”or the prescribed number of occurrencesā€” of generalized patterns exhibit connections to an enormous variety of other combinatorial structures, some of them apparently deep. We give a short overview of the state of the art for generalized patterns

    Restricted non-separable planar maps and some pattern avoiding permutations

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    Tutte founded the theory of enumeration of planar maps in a series of papers in the 1960s. Rooted non-separable planar maps are in bijection with West-2-stack-sortable permutations, beta(1,0)-trees introduced by Cori, Jacquard and Schaeffer in 1997, as well as a family of permutations defined by the avoidance of two four letter patterns. In this paper we give upper and lower bounds on the number of multiple-edge-free rooted non-separable planar maps. We also use the bijection between rooted non-separable planar maps and a certain class of permutations, found by Claesson, Kitaev and Steingrimsson in 2009, to show that the number of 2-faces (excluding the root-face) in a map equals the number of occurrences of a certain mesh pattern in the permutations. We further show that this number is also the number of nodes in the corresponding beta(1,0)-tree that are single children with maximum label. Finally, we give asymptotics for some of our enumerative results.Comment: 18 pages, 14 figure

    Generalized permutation patterns -- a short survey

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    An occurrence of a classical pattern p in a permutation \pi is a subsequence of \pi whose letters are in the same relative order (of size) as those in p. In an occurrence of a generalized pattern, some letters of that subsequence may be required to be adjacent in the permutation. Subsets of permutations characterized by the avoidance--or the prescribed number of occurrences--of generalized patterns exhibit connections to an enormous variety of other combinatorial structures, some of them apparently deep. We give a short overview of the state of the art for generalized patterns.Comment: 11 pages. Added a section on asymptotics (Section 8), added more examples of barred patterns equal to generalized patterns (Section 7) and made a few other minor additions. To appear in ``Permutation Patterns, St Andrews 2007'', S.A. Linton, N. Ruskuc, V. Vatter (eds.), LMS Lecture Note Series, Cambridge University Pres

    A self-dual poset on objects counted by the Catalan numbers and a type-B analogue

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    We introduce two partially ordered sets, PnAP^A_n and PnBP^B_n, of the same cardinalities as the type-A and type-B noncrossing partition lattices. The ground sets of PnAP^A_n and PnBP^B_n are subsets of the symmetric and the hyperoctahedral groups, consisting of permutations which avoid certain patterns. The order relation is given by (strict) containment of the descent sets. In each case, by means of an explicit order-preserving bijection, we show that the poset of restricted permutations is an extension of the refinement order on noncrossing partitions. Several structural properties of these permutation posets follow, including self-duality and the strong Sperner property. We also discuss posets QnAQ^A_n and QnBQ^B_n similarly associated with noncrossing partitions, defined by means of the excedence sets of suitable pattern-avoiding subsets of the symmetric and hyperoctahedral groups.Comment: 15 pages, 2 figure
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