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17
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.
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 27 (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.
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.
MultiStatistic Enumeration of TwoStack Sortable Permutations
, 1998
"... Using Zeilberger's factorization of twostacksortable permutations, we write a functional equation of a strange sort that defines their generating function according to five statistics: length, number of descents, number of righttoleft and lefttoright maxima, and a fifth statistic that is close ..."
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Cited by 16 (2 self)
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Using Zeilberger's factorization of twostacksortable permutations, we write a functional equation of a strange sort that defines their generating function according to five statistics: length, number of descents, number of righttoleft and lefttoright maxima, and a fifth statistic that is closely linked to the factorization. Then, we show how one can translate this functional equation into a polynomial one. We thus prove that our fivevariable generating function for twostacksortable permutations is algebraic of degree 20.
The patterns of permutations
 Discrete Math
, 2002
"... To Dan Kleitman, on his birthday, with all good wishes. Let n, k be positive integers, with k ≤ n, and let τ be a fixed permutation of {1,...,k}. 1 We will call τ the pattern. We will look for the pattern τ in permutations σ of n letters. A pattern τ is said to occur in a permutation σ if there are ..."
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Cited by 15 (0 self)
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To Dan Kleitman, on his birthday, with all good wishes. Let n, k be positive integers, with k ≤ n, and let τ be a fixed permutation of {1,...,k}. 1 We will call τ the pattern. We will look for the pattern τ in permutations σ of n letters. A pattern τ is said to occur in a permutation σ if there are integers 1 ≤ i1 <i2 <...<ik ≤ n such that for all 1 ≤ r<s ≤ k we have τ(r) <τ(s) if and only if σ(ir) <σ(is). Example: Suppose τ = (132). Then this pattern of k = 3 letters occurs several times in the following permutation σ, ofn = 14 letters (one such occurrence is underlined): σ =(529414 10 13615811 7 13 12) 1 Some areas of research and recent results Among the active areas of research are the following. 1. For a given pattern τ, let f(n, τ) be the number of τfree permutations of n letters. Describe the equivalence classes of patterns that have the same f. 2. What can be said about the asymptotics of f(n, τ) for n →∞and fixed τ? 3. For fixed τ what is the maximum number of occurrences of τ in a permutation of n letters? Call this g(τ,n). Which permutation has the maximum? 1 We will often refer to {1, 2,...,k} as the letters on which the permutation acts, however their numerical sizes will be very relevant. 1 2 Packing density First, as regards the question of stuffing in as many τ’s as possible, Fred Galvin (unpublished) has shown the following. Theorem 1 (Galvin) For fixed τ ∈ Sk, is decreasing, and thus exists. � � ∞ g(τ,n) �n � k lim n→∞ n=k g(τ,n) Galvin’s proof is reproduced here with his permission: Let τ be a fixed pattern of length k. If x is any sequence of distinct numbers, of length ≥ k, let g(x) be the number of τsubsequences of x, and let h(x) =g(x) / � x  � k.Forn≥k, let f(n, τ) H(n) = max {h(x):x  = n} = �. Suppose k ≤ m<n, and let x be a permutation of length n with h(x) =H(n). Note that h(x) is equal to the average of h(y) over all mtermed subsequences y of x, and therefore cannot exceed
ON LINEAR TRANSFORMATIONS PRESERVING THE PÓLYA FREQUENCY PROPERTY
"... We prove that certain linear operators preserve the Pólya frequency property and realrootedness, and apply our results to settle some conjectures and open problems in combinatorics proposed by Bóna, Brenti and ReinerWelker. ..."
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Cited by 14 (4 self)
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We prove that certain linear operators preserve the Pólya frequency property and realrootedness, and apply our results to settle some conjectures and open problems in combinatorics proposed by Bóna, Brenti and ReinerWelker.
Description Trees for Some Families of Planar Maps.
 Proceedings of the 9th Conference on Formal Power Series and Algebraic Combinatorics
, 1997
"... In this paper, we introduce description trees, to give a general framework for the recursive decompositions of several families of planar maps studied by W.T. Tutte. These trees reflect the combinatorial structure of the decompositions and carry out various combinatorial parameters. We also introduc ..."
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Cited by 13 (1 self)
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In this paper, we introduce description trees, to give a general framework for the recursive decompositions of several families of planar maps studied by W.T. Tutte. These trees reflect the combinatorial structure of the decompositions and carry out various combinatorial parameters. We also introduce left regular trees as canonical representants of some new conjugacy classes on planted plane trees. We give an enumeration formula for these trees. In several cases the combination of these two ingredients yields a purely combinatorial proof of some elegant formulae of W.T. Tutte and gives uniform random generation algorithms for the corresponding planar maps. Rsum Dans cet article, nous introduisons des arbres de description pour coder la dcomposition rcursive de plusieurs familles de cartes planaires tudies par W.T. Tutte. Ces arbres refltent la structure combinatoire de la dcomposition et portent diffrents paramtres combinatoires. Nous introduisons aussi les arbres rguliers gauches qu...
CLASSIFICATION OF BIJECTIONS BETWEEN 321 AND 132AVOIDING PERMUTATIONS
 SÉMINAIRE LOTHARINGIEN DE COMBINATOIRE 60 (2008), ARTICLE B60D
, 2008
"... It is wellknown, and was first established by Knuth in 1969, that the number of 321avoiding permutations is equal to that of 132avoiding permutations. In the literature one can find many subsequent bijective proofs of this fact. It turns out that some of the published bijections can easily be ob ..."
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Cited by 10 (4 self)
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It is wellknown, and was first established by Knuth in 1969, that the number of 321avoiding permutations is equal to that of 132avoiding permutations. In the literature one can find many subsequent bijective proofs of this fact. It turns out that some of the published bijections can easily be obtained from others. In this paper we describe all bijections we were able to find in the literature and show how they are related to each other via “trivial” bijections. We classify the bijections according to statistics preserved (from a fixed, but large, set of statistics), obtaining substantial extensions of known results. Thus, we give a comprehensive survey and a systematic analysis of these bijections. We also give a recursive description of the algorithmic bijection given by Richards in 1988 (combined with a bijection by Knuth from 1969). This bijection is equivalent to the celebrated bijection of Simion and Schmidt (1985), as well as to the bijection given by Krattenthaler in 2001, and it respects 11 statistics — the largest number of statistics any of the bijections respects.
Sorted and/or Sortable Permutations
"... In his Ph.D. thesis [21], Julian West studied in depth a map \Pi that acts on permutations of the symmetric group Sn by partially sorting them through a stack. The main motivation of this paper is to characterize and count the permutations of \Pi(Sn ), which we call sorted permutations. This is equi ..."
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Cited by 5 (0 self)
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In his Ph.D. thesis [21], Julian West studied in depth a map \Pi that acts on permutations of the symmetric group Sn by partially sorting them through a stack. The main motivation of this paper is to characterize and count the permutations of \Pi(Sn ), which we call sorted permutations. This is equivalent to counting preorders of increasing binary trees. We rst nd a local characterization of sorted permutations. Then, using an extension of Zeilberger's factorisation of twostack sortable permutations [23], we obtain for the generating function of sorted permutations an unusual functional equation. Out of curiosity, we apply the same treatment to four other families of permutations (general permutations, onestack sortable permutations, twostack sortable permutations, sorted and sortable permutations) and compare the functional equations we obtain. All of them have similar features, involving a divided dioeerence. Moreover, most of them have interesting qanalogs obtained by counting i...
On (Some) Functional Equations Arising In Enumerative Combinatorics
"... I shall try to convince the audience not to be shy with functional equation approaches in enumerative combinatorics. They not only solve (some) problems, but they often teach us a lot too. The proofs they provide for a specific problem might be less nice than more combinatorial proofs. But functiona ..."
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Cited by 4 (0 self)
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I shall try to convince the audience not to be shy with functional equation approaches in enumerative combinatorics. They not only solve (some) problems, but they often teach us a lot too. The proofs they provide for a specific problem might be less nice than more combinatorial proofs. But functional equation approaches sometimes give a unified description of apparently distinct problems, and the efforts we make to solve one specific functional equation often teach us what to do in a more generic case. The talk will be based on recent examples dealing with very classical combinatorial objects: lattice paths, maps, permutations.