We determine all permutation classes defined by pattern avoidance which are equinumerous to the class of permutations whose Schubert variety is smooth. We also provide a lattice path interpretation for the numbers of such permutations. 1 Introduction Let q =(q 1 ,q 2 ,...,q k ) # S k be a permutation, and let k # n. We say that the permutation p =(p 1 ,p 2 ,,p n )#S n contains a subsequence (or pattern) of type q if there is a set of indices 1 # i q1 <i q 2 < <i q k #nsuch that p(i 1 ) <p(i 2 )< <p(i k ). Otherwise we say that p is q-avoiding. For example, a permutation is 132-avoiding if it doesn't contain three (not necessarily consecutive) elements among which the leftmost is the smallest and the middle one is the largest. The enumeration of permutations of length n (or, in what follows, n-permutations) avoiding one given pattern q is a di#cult problem and has recently generated a fairly extensive research. See [2] [3] for an overview of these results. 1 the electronic jour...

Abstract Recently, Babson and Steingr'imsson introduced generalised permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We investigate simultaneous avoidance of two or more 3-patterns without internal dashes, that is, where the pattern corresponds to a contiguous subword in a permutation.

Let σ ∈ Sk and τ ∈ Sn be permutations. We say τ contains σ if there exist 1 ≤ x1 < x2 <... < xk ≤ n such that τ(xi) < τ(xj) if and only if σ(i) < σ(j). If τ does not contain σ we say τ avoids σ. Let F (n, σ) = |{τ ∈ Sn | τ avoids σ}|. Stanley and Wilf conjectured that for any σ ∈ Sk there exists a constant c = c(σ) such that F (n, σ) ≤ c n for all n. Here we prove the following weaker statement: For every fixed σ ∈ Sk, F (n, σ) ≤ c nγ ∗ (n) , where c = c(σ) and γ ∗ (n) is an extremely slow growing function, related to the Ackermann hierarchy. 1

A simple permutation is one that does not map any non-trivial interval onto an interval. It is shown that, if the number of simple permutations in a pattern restricted class of permutations is finite, the class has an algebraic generating function and is defined by a finite set of restrictions. Some partial results on classes with an infinite number of simple permutations are given. Examples of results obtainable by the same techniques are given; in particular it is shown that every pattern restricted class properly contained in the 132-avoiding permutations has a rational generating function.

Many families of pattern-avoiding permutations can be described by a generating tree in which each node carries one integer label, computed recursively via a rewriting rule. A typical example is that of 123-avoiding permutations. The rewriting rule automatically gives a functional equation satis ed by the bivariate generating function that counts the permutations by their length and the label of the corresponding node of the tree. These equations are now well understood, and their solutions are always algebraic series.

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.

A partition u of [k] = f1; 2; : : : ; kg is contained in another partition v of [l] if [l] has a k-subset on which v induces u. We are interested in counting partitions v not containing a given partition u or a given set of partitions R. This concept is related to that of forbidden permutations. A strengthening of Stanley--Wilf conjecture is proposed.

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.

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...

A class of permutations is called closed if 2 implies 2 , where the relation is the natural containment of permutations. Let n be the set of all permutations of 1; 2; : : : ; n belonging to . We investigate the counting functions n 7! j n j of closed classes. Our main result says that if j n j < 2 for at least one n 1, then there is a unique k 1 such that F n;k j n j F n;k n holds for all n 1 with a constant c > 0. Here F n;k are the generalized Fibonacci numbers which grow like powers of the largest positive root of x 1. We characterize also the constant and the polynomial growth of closed permutation classes and give two more results on these.