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26
Permutations with restricted patterns and Dyck paths
 Adv. Appl. Math
"... Abstract. We exhibit a bijection between 132avoiding permutations and Dyck paths. Using this bijection, it is shown that all the recently discovered results on generating functions for 132avoiding permutations with a given number of occurrences of the pattern 12... k follow directly from old resul ..."
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Cited by 126 (3 self)
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Abstract. We exhibit a bijection between 132avoiding permutations and Dyck paths. Using this bijection, it is shown that all the recently discovered results on generating functions for 132avoiding permutations with a given number of occurrences of the pattern 12... k follow directly from old results on the enumeration of Motzkin paths, among which is a continued fraction result due to Flajolet. As a bonus, we use these observations to derive further results and a precise asymptotic estimate for the number of 132avoiding permutations of {1, 2,..., n} with exactly r occurrences of the pattern 12... k. Second, we exhibit a bijection between 123avoiding permutations and Dyck paths. When combined with a result of Roblet and Viennot, this bijection allows us to express the generating function for 123avoiding permutations with a given number of occurrences of the pattern (k − 1)(k − 2)...1k in form of a continued fraction and to derive further results for these permutations.
Restricted 132avoiding permutations
 Adv. in Appl. Math
"... Abstract. We study generating functions for the number of permutations on n letters avoiding 132 and an arbitrary permutation τ on k letters, or containing τ exactly once. In several interesting cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second ..."
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Cited by 47 (23 self)
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Abstract. We study generating functions for the number of permutations on n letters avoiding 132 and an arbitrary permutation τ on k letters, or containing τ exactly once. In several interesting cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind.
Permutations avoiding an increasing number of lengthincreasing forbidden subsequences
 Discrete Math. Theor. Comput. Sci
, 2000
"... A permutation is said to ¡ be –avoiding if it does not contain any subsequence having all the same pairwise comparisons ¡ as. This paper concerns the characterization and enumeration of permutations which avoid a set ¢¤ £ of subsequences increasing both in number and in length at the same time. Le ..."
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Cited by 28 (2 self)
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A permutation is said to ¡ be –avoiding if it does not contain any subsequence having all the same pairwise comparisons ¡ as. This paper concerns the characterization and enumeration of permutations which avoid a set ¢¤ £ of subsequences increasing both in number and in length at the same time. Let ¢ £ be the set of subsequences of the “¥§¦©¨�������¦©¨����� � form ¥ ”, being any permutation ��������������¨� � on. ¨��� � For the only subsequence in ¢�� ���� � is and ���� � the –avoiding permutations are enumerated by the Catalan numbers; ¨��� � for the subsequences in ¢� � are, ������ � and the (������������������ � –avoiding permutations are enumerated by the Schröder numbers; for each other value ¨ of greater � than the subsequences in ¢ £ ¨� � are and their length ¦©¨����� � is; the permutations avoiding ¨�� these subsequences are enumerated by a number ������ � �� � � sequence such �������������� � that �� � , being � the –th Catalan number. For ¨ each we determine the generating function of permutations avoiding the subsequences in ¢� £ , according to the length, to the number of left minima and of noninversions.
Restricted permutations and Chebyshev polynomials
 SÉMINAIRE LOTHARINGIEN DE COMBINATOIRE 47 (2002) ARTICLE B47C
, 2002
"... We study generating functions for the number of permutations in Sn subject to two restrictions. One of the restrictions belongs to S3, while the other to Sk. It turns out that in a large variety of cases the answer can be expressed via Chebyshev polynomials of the second kind. ..."
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Cited by 27 (13 self)
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We study generating functions for the number of permutations in Sn subject to two restrictions. One of the restrictions belongs to S3, while the other to Sk. It turns out that in a large variety of cases the answer can be expressed via Chebyshev polynomials of the second kind.
Permutations containing and avoiding certain patterns
 Proc. 12th Conference on Formal Power Series and Algebraic Combinatorics
, 2000
"... Let T m k = {σ ∈ Sk  σ1 = m}. We prove that the number of permutations which avoid all patterns in T m k equals (k − 2)!(k − 1)n+1−k for k ≤ n. We then prove that for any τ ∈ T 1 k (or any τ ∈ T k k), the number of permutations which avoid all patterns in T 1 k (or in T k k) except for τ and contai ..."
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Cited by 22 (10 self)
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Let T m k = {σ ∈ Sk  σ1 = m}. We prove that the number of permutations which avoid all patterns in T m k equals (k − 2)!(k − 1)n+1−k for k ≤ n. We then prove that for any τ ∈ T 1 k (or any τ ∈ T k k), the number of permutations which avoid all patterns in T 1 k (or in T k k) except for τ and contain τ exactly once equals (n + 1 − k)(k − 1) n−k for k ≤ n. Finally, for any τ ∈ T m k, 2 ≤ m ≤ k − 1, this number equals (k − 1) n−k for k ≤ n. These results generalize recent results due to Robertson concerning permutations avoiding 123pattern and containing 132pattern exactly once. 1
Restricted 132alternating permutations and Chebyshev polynomials
 Annals of Combinatorics
"... A permutation is said to be alternating if it starts with rise and then descents and rises come in turn. In this paper we study the generating function for the number of alternating permutations on n letters that avoid or contain exactly once 132 and also avoid or contain exactly once an arbitrary p ..."
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Cited by 16 (2 self)
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A permutation is said to be alternating if it starts with rise and then descents and rises come in turn. In this paper we study the generating function for the number of alternating permutations on n letters that avoid or contain exactly once 132 and also avoid or contain exactly once an arbitrary pattern on k letters. In several interesting cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind.
RESTRICTED 132 PERMUTATIONS AND GENERALIZED PATTERNS
, 2001
"... Recently, Babson and Steingrimsson (see [BS]) introduced generalized permutations patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We study generating functions for the number of permutations on n letters avoiding 132 (or containing 1 ..."
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Cited by 14 (6 self)
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Recently, Babson and Steingrimsson (see [BS]) introduced generalized permutations patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We study generating functions for the number of permutations on n letters avoiding 132 (or containing 132 exactly once) and an arbitrary generalized pattern τ on k letters, or containing τ exactly once. In several cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind, and generating function of Motzkin numbers. 1
Continued fractions, statistics, and generalized patterns
, 2001
"... Recently, Babson and Steingrimsson (see [BS]) introduced generalized permutations patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Following [BCS], let ekπ (respectively; fkπ) be the number of the occurrences of the generalized pattern 1 ..."
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Cited by 11 (7 self)
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Recently, Babson and Steingrimsson (see [BS]) introduced generalized permutations patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Following [BCS], let ekπ (respectively; fkπ) be the number of the occurrences of the generalized pattern 123...k (respectively; 213...k) in π. In the present note, we study the distribution of the statistics ekπ and fkπ in a permutation avoiding the classical pattern 132. Also we present an applications, which relates the Narayana numbers, Catalan numbers, and increasing subsequences, to permutations avoiding the classical pattern 132 according to a given statistics on ekπ, or on fkπ.
Restricted Motzkin permutations, Motzkin paths, continued fractions, and Chebyshev polynomials
 DISC. MATH
, 2005
"... We say that a permutation π is a Motzkin permutation if it avoids 132 and there do not exist a < b such that πa < πb < πb+1. We study the distribution of several statistics in Motzkin permutations, including the length of the longest increasing and decreasing subsequences and the number of ..."
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Cited by 10 (5 self)
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We say that a permutation π is a Motzkin permutation if it avoids 132 and there do not exist a < b such that πa < πb < πb+1. We study the distribution of several statistics in Motzkin permutations, including the length of the longest increasing and decreasing subsequences and the number of rises and descents. We also enumerate Motzkin permutations with additional restrictions, and study the distribution of occurrences of fairly general patterns in this class of permutations.
Permutations restricted by two distinct patterns of length three
 Adv. Appl. Math
"... Define Sn(R; T) to be the number of permutations on n letters which avoid all patterns in the set R and contain each pattern in the multiset T exactly once. In this paper we enumerate Sn({α}; {β}) and Sn(∅; {α, β}) for all α ̸ = β ∈ S3. We show that there are five Wilflike classes associated with e ..."
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Cited by 9 (2 self)
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Define Sn(R; T) to be the number of permutations on n letters which avoid all patterns in the set R and contain each pattern in the multiset T exactly once. In this paper we enumerate Sn({α}; {β}) and Sn(∅; {α, β}) for all α ̸ = β ∈ S3. We show that there are five Wilflike classes associated with each of Sn({α}; {β}) and Sn(∅; {α, β}) for all α ̸ = β ∈ S3. 1.