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26
Generalized permutation patterns and a classification of the Mahonian statistics
 Sém. Lothar. Combin
, 2000
"... We introduce generalized permutation patterns, where we allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We show that essentially all Mahonian permutation statistics in the literature can be written as linear combinations of such patterns. Almost ..."
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Cited by 121 (1 self)
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We introduce generalized permutation patterns, where we allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We show that essentially all Mahonian permutation statistics in the literature can be written as linear combinations of such patterns. Almost all known Mahonian permutation statistics can be written as linear combinations of patterns of length at most 3. There are only fourteen possible such Mahonian statistics, which we list. Of these, eight are known and we give proofs for another three. The remaining three we conjecture to be Mahonian. We also give an explicit numerical description of the combinations of patterns a Mahonian statistic must have, depending on the maximal length of its patterns. 1 Introduction and preliminaries The simplest, and best known, Mahonian permutation statistic is the number of inversions. Its distribution, which is the defining criterion of a Mahonian statistic, was given already in 1839, by Rod...
Exact Enumeration Of 1342Avoiding Permutations A Close Link With Labeled Trees And Planar Maps
, 1997
"... Solving the first nonmonotonic, longerthanthree instance of a classic enumeration problem, we obtain the generating function H(x) of all 1342avoiding permutations of length n as well as an exact formula for their number Sn (1342). While achieving this, we bijectively prove that the number of inde ..."
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Cited by 84 (7 self)
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Solving the first nonmonotonic, longerthanthree instance of a classic enumeration problem, we obtain the generating function H(x) of all 1342avoiding permutations of length n as well as an exact formula for their number Sn (1342). While achieving this, we bijectively prove that the number of indecomposable 1342avoiding permutations of length n equals that of labeled plane trees of a certain type on n 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) turns out to be algebraic, proving the first nonmonotonic, longerthanthree instance of a conjecture of Zeilberger and Noonan. We also prove that n p Sn (1342) converges to 8, so in particular, limn!1 (Sn (1342)=Sn (1234)) = 0.
Restricted permutations, continued fractions, and Chebyshev polynomials
 J. COMBIN
, 1999
"... Let fr n (k) be the number of 132avoiding permutations on n letters that contain exactly r occurrences of 12... k, and let Fr(x; k) and F(x,y; k) be the generating functions defined by Fr(x; k) = ∑ We find an explcit expression for F(x, y; k) in the form of a continued fraction. This allows us to ..."
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Cited by 43 (24 self)
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Let fr n (k) be the number of 132avoiding permutations on n letters that contain exactly r occurrences of 12... k, and let Fr(x; k) and F(x,y; k) be the generating functions defined by Fr(x; k) = ∑ We find an explcit expression for F(x, y; k) in the form of a continued fraction. This allows us to express Fr(x; k) for 1 � r � k via Chebyshev polynomials of the second kind. n�0 fr n (k)xn and F(x,y; k) = ∑ r�0 Fr(x; k)y r.
Counting occurrences of 132 in a permutation
 Adv. Appl. Math
"... Abstract. We study the generating function for the number of permutations on n letters containing exactly r � 0 occurrences of 132. It is shown that finding this function for a given r amounts to a routine check of all permutations in S2r. ..."
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Cited by 25 (9 self)
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Abstract. We study the generating function for the number of permutations on n letters containing exactly r � 0 occurrences of 132. It is shown that finding this function for a given r amounts to a routine check of all permutations in S2r.
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 23 (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.
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 19 (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
Decomposing simple permutations, with enumerative consequences. arXiv:math.CO/0606186
, 2006
"... We prove that every sufficiently long simple permutation contains two long almost disjoint simple subsequences. This result has applications to the enumeration of restricted permutations. For example, it immediately implies a result of Bóna and (independently) Mansour and Vainshtein that for any r, ..."
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Cited by 17 (8 self)
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We prove that every sufficiently long simple permutation contains two long almost disjoint simple subsequences. This result has applications to the enumeration of restricted permutations. For example, it immediately implies a result of Bóna and (independently) Mansour and Vainshtein that for any r, the number of permutations with at most r copies of 132 has an algebraic generating function. 1. STATEMENT OF THEOREM Simplicity, under a variety of names 1, has been studied for a wide range of combinatorial objects. Our main result concerns simple permutations; possible analogues for other contexts are discussed in the conclusion. An interval in the permutation π is a set of contiguous indices I = [a,b] such that the set of values π(I) = {π(i) : i ∈ I} also forms an interval of natural numbers. Every permutation π of [n] = {1,2,...,n} has intervals of size 0, 1, and n; π is said to be simple if it has no other intervals. Figure 1 shows the plots of two simple permutations. Intervals of permutations are interesting in their own right and have applications to biomathematics; see Corteel, Louchard, and Pemantle [10], where among other results it is proved that the number of simple permutations of [n] is asymptotic to n!/e 2. More precise asymptotics are given by Albert, Atkinson, and Klazar [2].
Counting occurrences of some subword patterns
 Discr. Math. Theor. Comp. Sci
"... Abstract. We find generating functions for the number of strings (words) containing a specified number of occurrences of certain types of orderisomorphic classes of substrings called subword patterns. In particular, we find generating functions for the number of strings containing a specified numbe ..."
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Cited by 14 (9 self)
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Abstract. We find generating functions for the number of strings (words) containing a specified number of occurrences of certain types of orderisomorphic classes of substrings called subword patterns. In particular, we find generating functions for the number of strings containing a specified number of occurrences of a given 3letter subword pattern. 1.
Permutations Containing and Avoiding 123 and 132 Patterns
, 1999
"... this article we work towards the following generalization: How many permutations of length n avoid patterns p i , for i 0, and contain r j p j patterns, for j 1, r j 1? We will first consider the permutations of length n which avoid 132patterns, but contain exactly one 123pattern. We then defi ..."
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Cited by 11 (1 self)
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this article we work towards the following generalization: How many permutations of length n avoid patterns p i , for i 0, and contain r j p j patterns, for j 1, r j 1? We will first consider the permutations of length n which avoid 132patterns, but contain exactly one 123pattern. We then define a natural bijection between these permutations and the permutations of length n which avoid 123patterns, but contain exactly one 132pattern. Finally, we will calculate the number of permutations which contain one 123pattern and one 132pattern. These results address questions first raised in [NZ]
Counting Occurrences of a Pattern of Type (1,2) or (2,1
 in Permutations, Advances in Applied Mathematics
, 2002
"... Abstract. Babson and Steingrímsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Claesson presented a complete solution for the number of permutations avoiding any single pattern of type (1, 2) or (2 ..."
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Cited by 8 (0 self)
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Abstract. Babson and Steingrímsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Claesson presented a complete solution for the number of permutations avoiding any single pattern of type (1, 2) or (2, 1). For eight of these twelve patterns the answer is given by the Bell numbers. For the remaining four the answer is given by the Catalan numbers. With respect to being equidistributed there are three different classes of patterns of type (1, 2) or (2, 1). We present a recursion for the number of permutations containing exactly one occurrence of a pattern of the first or the second of the aforementioned classes, and we also find an ordinary generating function for these numbers. We prove these results both combinatorially and analytically. Finally, we give the distribution of any pattern of the third class in the form of a continued fraction, and we also give explicit formulas for the number of permutations containing exactly r occurrences of a pattern of the third class when r ∈ {1, 2, 3}. 1. Introduction and