this paper the R-transform of an 1-dimensional distribution is viewed as the particular case n = 1 of the Eqn.(1.3); we warn the reader that this differs by a factor of z from the notation used in [21].

Abstract not found

Abstract. In this paper, we present a reduction algorithm which transforms m-regular partitions of [n] = {1, 2,..., n} to (m−1)-regular partitions of [n − 1]. We show that this algorithm preserves the noncrossing property. This yields a simple explanation of an identity due to Simion-Ullman and Klazar in connection with enumeration problems on noncrossing partitions and RNA secondary structures. For ordinary noncrossing partitions, the reduction algorithm leads to a representation of noncrossing partitions in terms of independent arcs and loops, as well as an identity of Simion and Ullman which expresses the Narayana numbers in terms of the Catalan numbers.

. We derive a formula for expressing free cumulants whose entries are products of random variables in terms of the lattice structure of non-crossing partitions. We show the usefulness of that result by giving direct and conceptually simple proofs for a lot of results about R-diagonal elements. Our investigations do not assume the trace property for the considered linear functionals. Introduction Free probability theory, due to Voiculescu [17, 18], is a non-commutative probability theory where the classical concept of "independence" is replaced by a non-commutative analogue, called "freeness". Originally this theory was introduced in an operator-algebraic context for dealing with questions on special von Neumann algebras. However, since these beginnings free probability theory has evolved into a theory with a lot of links to quite different fields. In particular, there exists a combinatorial facet: main aspects of free probability theory can be considered as the combinatorics of non-cr...

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

Two combinatorial statistics, the pyramid weight and the number of exterior pairs, are investigated on the set of Dyck paths. Explicit formulae are given for the generating functions of Dyck paths of prescribed pyramid weight and prescribed number of exterior pairs. The proofs are combinatorial and rely on the method of q-grammars as well as on two new q-analogues of the Catalan numbers derived from statistics on noncrossing partitions. Connections with the combinatorics of Motzkin paths are pointed out.

Abstract. The aim of the paper is to calculate face numbers of simple generalized permutohedra, and study their f-, h- and γ-vectors. These polytopes include permutohedra, associahedra, graph-associahedra, simple graphic zonotopes, nestohedra, and other interesting polytopes. We give several explicit formulas for h-vectors and γ-vectors involving descent statistics. This includes a combinatorial interpretation for γ-vectors of a large class of generalized permutohedra which are flag simple polytopes, and confirms for them Gal’s conjecture on nonnegativity of γ-vectors. We calculate explicit generating functions and formulae for h-polynomials of various families of graph-associahedra, including those corresponding to all Dynkin diagrams of finite and affine types. We also discuss relations with Narayana numbers and with Simon Newcomb’s problem. We give (and conjecture) upper and lower bounds for f-, h-, and γ-vectors within several classes of generalized permutohedra. An appendix discusses the equivalence of various notions of deformations of simple polytopes.

A finite sequence u = a 1 a 2 : : : a p of some symbols is contained in another sequence v = b 1 b 2 : : : b q if there is a subsequence b i 1 b i 2 : : : b i p of v which can be identified, after an injective renaming of symbols, with u. We say that u = a 1 a 2 : : : a p is k-regular if i \Gamma j k whenever a i = a j ; i ? j. We denote further by juj the length p of u and by kuk the number of different symbols in u. In this expository paper we give a survey of combinatorial results concerning the containment relation. Many of them are from the author's PhD thesis with the same title. Extremal results concern the growth rate of the function Ex(u; n) = max jvj, the maximum is taken over all kuk-regular sequences v, kvk n, not containing u. This is a generalization of the case u = ababa : : : which leads to Davenport-Schinzel sequences. Enumerative results deal with the numbers of abab-free and abba-free sequences. We mention a well quasiordering result and a tree generalization of our extremal function from sequences (=colored paths) to colored trees.

We give a simple and natural proof of (an extension of) the identity P (k; l; n) = P 2 (k \Gamma 1; l \Gamma 1; n \Gamma 1). The number P (k; l; n) counts noncrossing partitions of f1; 2; : : : ; lg into n parts such that no part contains two numbers x and y, 0 ! y \Gamma x ! k. The lower index 2 indicates partitions with no part of size three or more. We use the identity to give quick proofs of the closed formulae for P (k; l; n) when k is 1, 2, or 3.

We show that the posets of shuffles introduced by Greene in 1988 are flag-symmetric, and we describe a permutation action of the symmetric group on the maximal chains which is local and yields a representation of the symmetric group whose character has Frobenius characteristic closely related to the flag symmetric function. A key tool is provided by a new labeling of the maximal chains of a poset of shuffles. This labeling and the structure of the orbits of maximal chains under the local action lead to combinatorial derivations of enumerative properties obtained originally by Greene. As a further consequence, a natural notion of type of shuffles emerges and the monoid of multiplicative functions on the poset of shuffles is described in terms of operations on power series. The main results concerning the flag symmetric function and the local action on the maximal chains of a poset of shuffles are obtained from new general results regarding chain labelings of posets.