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87
Simple Permutations and Pattern Restricted Permutations
, 2003
"... A simple permutation is one that does not map any nontrivial interval onto an interval. It is shown that, if the number of simple permutations in a pattern restricted class of permutations is finite, the class has an algebraic generating function and is defined by a finite set of restrictions. ..."
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Cited by 49 (9 self)
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A simple permutation is one that does not map any nontrivial interval onto an interval. It is shown that, if the number of simple permutations in a pattern restricted class of permutations is finite, the class has an algebraic generating function and is defined by a finite set of restrictions. Some partial results on classes with an infinite number of simple permutations are given. Examples of results obtainable by the same techniques are given; in particular it is shown that every pattern restricted class properly contained in the 132avoiding permutations has a rational generating function.
On the number of permutations avoiding a given pattern
 J. Comb. Theory, Ser. A
, 1999
"... Let σ ∈ Sk and τ ∈ Sn be permutations. We say τ contains σ if there exist 1 ≤ x1 < x2 <... < xk ≤ n such that τ(xi) < τ(xj) if and only if σ(i) < σ(j). If τ does not contain σ we say τ avoids σ. Let F (n, σ) = {τ ∈ Sn  τ avoids σ}. Stanley and Wilf conjectured that for any σ ∈ Sk ..."
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Cited by 36 (0 self)
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Let σ ∈ Sk and τ ∈ Sn be permutations. We say τ contains σ if there exist 1 ≤ x1 < x2 <... < xk ≤ n such that τ(xi) < τ(xj) if and only if σ(i) < σ(j). If τ does not contain σ we say τ avoids σ. Let F (n, σ) = {τ ∈ Sn  τ avoids σ}. Stanley and Wilf conjectured that for any σ ∈ Sk there exists a constant c = c(σ) such that F (n, σ) ≤ c n for all n. Here we prove the following weaker statement: For every fixed σ ∈ Sk, F (n, σ) ≤ c nγ ∗ (n) , where c = c(σ) and γ ∗ (n) is an extremely slow growing function, related to the Ackermann hierarchy. 1
The permutation classes equinumerous to the Smooth class
 J. Combin
, 1998
"... We determine all permutation classes defined by pattern avoidance which are equinumerous to the class of permutations whose Schubert variety is smooth. We also provide a lattice path interpretation for the numbers of such permutations. 1 Introduction Let q =(q 1 ,q 2 ,...,q k ) # S k be a permut ..."
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Cited by 35 (0 self)
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We determine all permutation classes defined by pattern avoidance which are equinumerous to the class of permutations whose Schubert variety is smooth. We also provide a lattice path interpretation for the numbers of such permutations. 1 Introduction Let q =(q 1 ,q 2 ,...,q k ) # S k be a permutation, and let k # n. We say that the permutation p =(p 1 ,p 2 ,,p n )#S n contains a subsequence (or pattern) of type q if there is a set of indices 1 # i q1 <i q 2 < <i q k #nsuch that p(i 1 ) <p(i 2 )< <p(i k ). Otherwise we say that p is qavoiding. For example, a permutation is 132avoiding if it doesn't contain three (not necessarily consecutive) elements among which the leftmost is the smallest and the middle one is the largest. The enumeration of permutations of length n (or, in what follows, npermutations) avoiding one given pattern q is a di#cult problem and has recently generated a fairly extensive research. See [2] [3] for an overview of these results. 1 the electronic jour...
Four Classes of PatternAvoiding Permutations under one Roof: Generating Trees with Two Labels
, 2003
"... Many families of patternavoiding permutations can be described by a generating tree in which each node carries one integer label, computed recursively via a rewriting rule. A typical example is that of 123avoiding permutations. The rewriting rule automatically gives a functional equation satis ..."
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Cited by 34 (5 self)
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Many families of patternavoiding permutations can be described by a generating tree in which each node carries one integer label, computed recursively via a rewriting rule. A typical example is that of 123avoiding permutations. The rewriting rule automatically gives a functional equation satis ed by the bivariate generating function that counts the permutations by their length and the label of the corresponding node of the tree. These equations are now well understood, and their solutions are always algebraic series.
Multiavoidance of generalised patterns
 Discrete Math
, 2002
"... Abstract Recently, Babson and Steingr'imsson introduced generalised permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We investigate simultaneous avoidance of two or more 3patterns without internal dashes, that is, where ..."
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Cited by 31 (14 self)
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Abstract Recently, Babson and Steingr'imsson introduced generalised permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We investigate simultaneous avoidance of two or more 3patterns without internal dashes, that is, where the pattern corresponds to a contiguous subword in a permutation.
A Survey of StackSorting Disciplines
, 2004
"... We review the various ways that stacks, their variations and their combinations, have been used as sorting devices. In particular, we show that they have been a key motivator for the study of permutation patterns. We also show that they have connections to other areas in combinatorics such as You ..."
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Cited by 28 (0 self)
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We review the various ways that stacks, their variations and their combinations, have been used as sorting devices. In particular, we show that they have been a key motivator for the study of permutation patterns. We also show that they have connections to other areas in combinatorics such as Young tableau, planar graph theory, and simplicial complexes.
Counting patternfree set partitions I: A generalization of Stirling numbers of the second kind
, 2000
"... A partition u of [k] = f1; 2; : : : ; kg is contained in another partition v of [l] if [l] has a ksubset on which v induces u. We are interested in counting partitions v not containing a given partition u or a given set of partitions R. This concept is related to that of forbidden permutations. ..."
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Cited by 23 (11 self)
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A partition u of [k] = f1; 2; : : : ; kg is contained in another partition v of [l] if [l] has a ksubset on which v induces u. We are interested in counting partitions v not containing a given partition u or a given set of partitions R. This concept is related to that of forbidden permutations. A strengthening of StanleyWilf conjecture is proposed.
Simple permutations and algebraic generating functions
 In preparation
, 2006
"... A simple permutation is one that never maps a nontrivial contiguous set of indices contiguously. Given a set of permutations that is closed under taking subpermutations and contains only finitely many simple permutations, we provide a framework for enumerating ..."
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Cited by 22 (9 self)
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A simple permutation is one that never maps a nontrivial contiguous set of indices contiguously. Given a set of permutations that is closed under taking subpermutations and contains only finitely many simple permutations, we provide a framework for enumerating
Enumeration of permutations containing a prescribed number of occurrences of a pattern of length 3
, 2001
"... We consider the problem of enumerating the permutations containing exactly k occurrences of a pattern of length 3. This enumeration has received a lot of interest recently, and there are a lot of known results. This paper presents an alternative approach to the problem, which yields a proof for a f ..."
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Cited by 22 (0 self)
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We consider the problem of enumerating the permutations containing exactly k occurrences of a pattern of length 3. This enumeration has received a lot of interest recently, and there are a lot of known results. This paper presents an alternative approach to the problem, which yields a proof for a formula which so far only was conjectured (by Noonan and Zeilberger). This approach is based on bijections from permutations to certain lattice paths with “jumps”, which were first considered by Krattenthaler.
Asymptotic enumeration of permutations avoiding generalized patterns
 Advances in Applied Mathematics 36
, 2006
"... Abstract. Motivated by the recent proof of the StanleyWilf conjecture, we study the asymptotic behavior of the number of permutations avoiding a generalized pattern. Generalized patterns allow the requirement that some pairs of letters must be adjacent in an occurrence of the pattern in the permuta ..."
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Cited by 20 (4 self)
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Abstract. Motivated by the recent proof of the StanleyWilf conjecture, we study the asymptotic behavior of the number of permutations avoiding a generalized pattern. Generalized patterns allow the requirement that some pairs of letters must be adjacent in an occurrence of the pattern in the permutation, and consecutive patterns are a particular case of them. We determine the asymptotic behavior of the number of permutations avoiding a consecutive pattern, showing that they are an exponentially small proportion of the total number of permutations. For some other generalized patterns we give partial results, showing that the number of permutations avoiding them grows faster than for classical patterns but more slowly than for consecutive patterns. 1.