A permutation 2 Sn avoids the subpattern iff has no subsequence having all the same pairwise comparisons as , and we write 2 Sn ( ). We present a new bijective proof of the well-known result that jS n (123)j = jS n (132)j = c n , the n-th Catalan number. A generalization to forbidden patterns of length 4 gives an asymptotic formula for the vexillary permutations. We settle a conjecture of Shapiro and Getu that jS n (3142; 2413)j = s n\Gamma1 , the Schröder number, and characterize the deque-sortable permutations of Knuth, also counted by s n\Gamma1 .

Solving the first nonmonotonic, longer-than-three instance of a classic enumeration problem, we obtain the generating function H(x) of all 1342-avoiding 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 1342-avoiding 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, longer-than-three 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.

this paper all produce enumerative results which, to the best of our knowledge, were not previously known:

Certain families of combinatorial objects admit recursive descriptions in terms of generating trees: each node of the tree corresponds to an object, and the branch leading to the node encodes the choices made in the construction of the object. Generating trees lead to a fast computation of enumeration sequences (sometimes, to explicit formulae as well) and provide efficient random generation algorithms. We investigate the links between the structural properties of the rewriting rules defining such trees and the rationality, algebraicity, or transcendence of the corresponding generating function.

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.

Using Zeilberger's factorization of two-stack-sortable 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 right-to-left and left-to-right 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 five-variable generating function for two-stack-sortable permutations is algebraic of degree 20.

Abstract not found

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 τ-free 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 τ-subsequences 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 m-termed subsequences y of x, and therefore cannot exceed

Abstract. Generating trees are a useful technique in the enumeration of various combinatorial objects, particularly restricted permutations. Quite often the generating tree for the set of permutations avoiding a set of patterns requires infinitely many labels. Sometimes, however, this generating tree needs only finitely many labels. We characterize the finite sets of patterns for which this phenomenon occurs. We also present an algorithm — in fact, a special case of an algorithm of Zeilberger — that is guaranteed to find such a generating tree if it exists. 1.

We examine the poset P of 132-avoiding n-permutations ordered by descents. We show that this poset is the "coarsening" of the well-studied poset Q of noncrossing partitions . In other words, if x ! y in Q, then f(y) ! f(x) in P , where f is the canonical bijection from the set of noncrossing partitions onto that of 132-avoiding permutations. This enables us to prove many properties of P . 1