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34
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.
The Enumeration of Permutations with a Prescribed Number of "Forbidden" Patterns
 Adv. in Appl. Math
, 1998
"... We initiate a general approach for the fast enumeration of permutations with a prescribed number of occurrences of `forbidden' patterns, that seems to indicate that the enumerating sequence is always Precursive. We illustrate the method completely in terms of the patterns `abc',`cab' and `abcd'. 0. ..."
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Cited by 44 (1 self)
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We initiate a general approach for the fast enumeration of permutations with a prescribed number of occurrences of `forbidden' patterns, that seems to indicate that the enumerating sequence is always Precursive. We illustrate the method completely in terms of the patterns `abc',`cab' and `abcd'. 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.
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 23 (1 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.
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.
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 20 (9 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
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.
Continued fractions and generalized patterns
, 2001
"... In [BS] Babson and Steingrimsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Let fτ;r(n) be the number of 132avoiding permutations on n letters that contain exactly r occurrences of τ, where τ a ..."
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Cited by 13 (6 self)
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In [BS] Babson and Steingrimsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Let fτ;r(n) be the number of 132avoiding permutations on n letters that contain exactly r occurrences of τ, where τ a generalized pattern on k letters. Let Fτ;r(x) and Fτ(x, y) be the generating functions defined by Fτ;r(x) = ∑ n n≥0 fτ;r(n)x and Fτ(x, y) = ∑ r≥0 Fτ;r(x)y r. We find an explicit expression for Fτ(x, y) in the form of a continued fraction for where τ given as a generalized pattern; τ = 123...k, τ = 213...k, τ = 123...k, or τ = k...321. In particularly, we find Fτ(x, y) for any τ generalized pattern of length 3. This allows us to express Fτ;r(x) via Chebyshev polynomials of the second kind, and continued fractions.