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Increasing and decreasing subsequences and their variants
 PROCEEDINGS OF INTERNATIONAL CONGRESS OF MATHEMATICAL SOCIETY
, 2006
"... 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 ob ..."
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Cited by 33 (2 self)
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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 VershikKerov and (almost) by LoganShepp. 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.
A Survey of Alternating Permutations
, 2009
"... Abstract. A permutation a1a2 · · · an of 1, 2,..., n is alternating if a1> a2 < a3> a4 < · · ·. We survey some aspects of the theory of alternating permutations, beginning with the famous result of André that if En is the number of alternating permutations of 1, 2,..., n, then P xn n ..."
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Cited by 29 (1 self)
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Abstract. A permutation a1a2 · · · an of 1, 2,..., n is alternating if a1> a2 < a3> a4 < · · ·. We survey some aspects of the theory of alternating permutations, beginning with the famous result of André that if En is the number of alternating permutations of 1, 2,..., n, then P xn n≥0 En = sec x + tan x. n! Topics include refinements and qanalogues of En, various occurrences of En in mathematics, longest alternating subsequences of permutations, umbral enumeration of special classes of alternating permutations, and the connection between alternating permutations and the cdindex of the symmetric group. Dedicated to Reza Khosrovshahi on the occasion of his 70th birthday 1. Basic enumerative properties. Let Sn denote the symmetric group of all permutations of [n]: = {1, 2,..., n}. A permutation w = a1a2 · · · an ∈ Sn is called alternating if a1> a2 < a3> a4 < · · ·. In other words, ai < ai+1 for i even, and ai> ai+1 for i odd. Similarly w is reverse alternating if a1 < a2> a3 < a4> · · ·. (Some authors reverse these definitions.) Let En denote the number of alternating permutations in Sn. (Set
On Growth Rates of Closed Permutation Classes
, 2003
"... A class of permutations is called closed if 2 implies 2 , where the relation is the natural containment of permutations. Let n be the set of all permutations of 1; 2; : : : ; n belonging to . We investigate the counting functions n 7! j n j of closed classes. Our main result says that if ..."
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Cited by 25 (0 self)
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A class of permutations is called closed if 2 implies 2 , where the relation is the natural containment of permutations. Let n be the set of all permutations of 1; 2; : : : ; n belonging to . We investigate the counting functions n 7! j n j of closed classes. Our main result says that if j n j < 2 for at least one n 1, then there is a unique k 1 such that F n;k j n j F n;k n holds for all n 1 with a constant c > 0. Here F n;k are the generalized Fibonacci numbers which grow like powers of the largest positive root of x 1. We characterize also the constant and the polynomial growth of closed permutation classes and give two more results on these.
The limit of a StanleyWilf sequence is not always an integer, and layered patterns beat monotone patterns
, 2004
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Longest Alternating Subsequences of Permutations
"... The length is(w) of the longest increasing subsequence of a permutation w in the symmetric group Sn has been the object of much investigation. We develop comparable results for the length as(w) of the longest alternating subsequence of w, where a sequence a,b,c,d,... is alternating if a> b < ..."
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Cited by 22 (4 self)
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The length is(w) of the longest increasing subsequence of a permutation w in the symmetric group Sn has been the object of much investigation. We develop comparable results for the length as(w) of the longest alternating subsequence of w, where a sequence a,b,c,d,... is alternating if a> b < c> d < · · ·. For instance, the expected value (mean) of as(w) for w ∈ Sn is exactly (4n + 1)/6 if n ≥ 2.
On some properties of permutation tableaux
, 2006
"... Abstract. We consider the relation between various permutation statistics and properties of permutation tableaux. We answer some of the questions of Steingrímsson and Williams [9], in particular, on the distribution of the bistatistic of numbers of rows and essential ones in permutation tableaux. We ..."
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Abstract. We consider the relation between various permutation statistics and properties of permutation tableaux. We answer some of the questions of Steingrímsson and Williams [9], in particular, on the distribution of the bistatistic of numbers of rows and essential ones in permutation tableaux. We also consider and enumerate sets of permutation tableaux related to some pattern restrictions on permutations. 1.
Profile Classes and Partial WellOrder for Permutations
"... It is known that the set of permutations, under the pattern containment ordering, is not a partial wellorder. Characterizing the partially wellordered closed sets (equivalently: down sets or ideals) in this poset remains a wideopen problem. Given a 0/±1 matrix M, we define a closed set of permuta ..."
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Cited by 21 (6 self)
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It is known that the set of permutations, under the pattern containment ordering, is not a partial wellorder. Characterizing the partially wellordered closed sets (equivalently: down sets or ideals) in this poset remains a wideopen problem. Given a 0/±1 matrix M, we define a closed set of permutations called the profile class of M. These sets are generalizations of sets considered by Atkinson, Murphy, and Ruˇskuc. We show that the profile class of M is partially wellordered if and only if a related graph is a forest. Related to the antichains we construct to prove one of the directions of this result, we construct exotic fundamental antichains, which lack the periodicity exhibited by all previously known fundamental antichains of permutations.
ENUMERATION SCHEMES FOR PERMUTATIONS AVOIDING BARRED PATTERNS
"... Abstract. We give the first comprehensive collection of enumeration results for permutations that avoid barred patterns of length ≤ 4. We then use the method of prefix enumeration schemes to find recurrences counting permutations that avoid a barred pattern of length> 4 or a set of barred pattern ..."
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Cited by 16 (4 self)
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Abstract. We give the first comprehensive collection of enumeration results for permutations that avoid barred patterns of length ≤ 4. We then use the method of prefix enumeration schemes to find recurrences counting permutations that avoid a barred pattern of length> 4 or a set of barred patterns. 1.
Real zeros and normal distribution for statistics on Stirling permutations defined by Gessel and Stanley
 ACCEPTED FOR PUBLICATION IN SIAM JOURNAL OF DISCRETE MATHEMATICS
, 2007
"... We study Stirling permutations defined by Gessel and Stanley in [4]. We prove that their generating function according to the number of descents has real roots only. We use that fact to prove that the distribution of these descents, and other, equidistributed statistics on these objects converge to ..."
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Cited by 15 (0 self)
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We study Stirling permutations defined by Gessel and Stanley in [4]. We prove that their generating function according to the number of descents has real roots only. We use that fact to prove that the distribution of these descents, and other, equidistributed statistics on these objects converge to a normal distribution.
The multiplicity conjecture for barycentric subdivisions
"... For a simplicial complex ∆ we study the effect of barycentric subdivision on ring theoretic invariants of its StanleyReisner ring. In particular, for StanleyReisner rings of barycentric subdivisions we verify a conjecture by Huneke and Herzog & Srinivasan, that relates the multiplicity of a st ..."
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Cited by 14 (2 self)
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For a simplicial complex ∆ we study the effect of barycentric subdivision on ring theoretic invariants of its StanleyReisner ring. In particular, for StanleyReisner rings of barycentric subdivisions we verify a conjecture by Huneke and Herzog & Srinivasan, that relates the multiplicity of a standard graded kalgebra to the product of the maximal and minimal shifts in its minimal free resolution up to the height. On the way to proving the conjecture we develop new and list well known results on behavior of dimension, Hilbert series, multiplicity, local cohomology, depth and regularity when passing from the StanleyReisner ring of ∆ to the one of its barycentric subdivision.