Abstract. We present results on the enumeration of crossings and nestings for matchings and set partitions. Using a bijection between partitions and vacillating tableaux, we show that if we fix the sets of minimal block elements and maximal block elements, the crossing number and the nesting number of partitions have a symmetric joint distribution. It follows that the crossing numbers and the nesting numbers are distributed symmetrically over all partitions of [n], as well as over all matchings on [2n]. As a corollary, the number of knoncrossing partitions is equal to the number of k-nonnesting partitions. The same is also true for matchings. An application is given to the enumeration of matchings with no k-crossing (or with no k-nesting).

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Trees are combinatorial structures that arise naturally in diverse applications. They occur in branching decision structures, taxonomy, computer languages, combinatorial optimization, parsing of sentences, and cluster expansions of statistical mechanics. Intuitively, a tree is a collection of branches connected at nodes. Formally, it can be defmed as a connected graph without cycles. Schroder trees, introduced in this paper, are a class of trees for which the set of subtrees at any vertex is endowed with the structure of ordered partitions. An ordered partition is a partition of a set in which the blocks are linearly ordered. Labeled rooted trees and labeled planed trees are both special classes of Schrdder trees. The main result gives a biection between Schrider trees and forests of small trees-namely, rooted trees of height one. Using this bijection, it is easy to encode a Schroder tree by a sequence of integers. Several classical algorithms for trees, including a combinatorial proof of the Lagrange inversion formula, are immediate consequences of this bijection.

Let B k,i (n) be the number of partitions of n with certain difference condition and let A k,i (n) be the number of partitions of n with certain congruence condition. The Rogers-Ramanujan-Gordon theorem states that B k,i (n) = A k,i (n). Lovejoy obtained an overpartition analogue of the Rogers-Ramanujan-Gordon theorem for the cases i = 1 and i = k. We find an overpartition analogue of the Rogers-Ramanujan-Gordon theorem in the general case. Let D k,i (n) be the number of overpartitions of n satisfying certain difference condition and C k,i (n) be the number of overpartitions of n whose non-overlined parts satisfy certain congruence condition. We show that C k,i (n) = D k,i (n). By using a function introduced by Andrews, we obtain a recurrence relation that implies that the generating function of D k,i (n) equals the generating function of C k,i (n). By introducing the Gordon marking of an overpartition, we find a generating function formula for D k,i (n) that can be considered an overpartition analogue of an identity of Andrews for ordinary partitions.

The Turán inequalities and the higher order Turán inequalities arise in the study of Maclaurin coefficients of an entire function in the Laguerre-Pólya class. A real sequence {a n } is said to satisfy the Turán inequalities if for n ≥ 1, a 2 n − a n−1 a n+1 ≥ 0. It is said to satisfy the higher order Turán inequalities if for n ≥ 1, 4(a 2 n − a n−1 a n+1 )(a 2 n+1 − a n a n+2 ) − (a n a n+1 − a n−1 a n+2 ) 2 ≥ 0. A sequence satisfying the Turán inequalities is also called log-concave. For the partition function p(n), DeSalvo and Pak showed that for n > 25, the sequence {p(n)} n>25 is log-concave, that is, p(n) 2 − p(n − 1)p(n + 1) > 0 for n > 25. It was conjectured by Chen that p(n) satisfies the higher order Turán inequalities for n ≥ 95. In this paper, we prove this conjecture by using the Hardy-Ramanujan-Rademacher formula to derive an upper bound and a lower bound for p(n + 1)p(n − 1)/p(n) 2 . Consequently, for n ≥ 95, the Jensen polynomials g 3,n−1 (x) = p(n − 1) + 3p(n)x + 3p(n + 1)x 2 + p(n + 2)x 3 have only real zeros. We conjecture that for any positive integer m ≥ 4 there exists an integer N (m) such that for n ≥ N (m), the polynomials m k=0 m k p(n + k)x k have only real zeros. This conjecture was independently posed by Ono.

Abstract. In this paper, we introduce the notion of a grammatical labeling to describe a recursive process of generating combinatorial objects based on a context-free grammar. For example, by labeling the ascents and descents of a Stirling permutation, we obtain a grammar for the second-order Eulerian polynomials. By using the grammar for 0-1-2 increasing trees given by Dumont, we obtain a grammatical derivation of the generating function of the André polynomials obtained by Foata and Schützenberger, without solving a differential equation. We also find a grammar for the number T (n, k) of permutations of [n] = {1, 2, . . . , n} with k exterior peaks, which was independently discovered by Ma. We demonstrate that Gessel's formula for the generating function of T (n, k) can be deduced from this grammar. Moreover, by using grammars we show that the number of the permutations of [n] with k exterior peaks equals the number of increasing trees on [n] with 2k + 1 vertices of even degree. A combinatorial proof of this fact is also presented.

A tangled-diagram over [n] = {1, . . . , n} is a graph of degree less than two whose vertices 1, . . . , n are arranged in a horizontal line and whose arcs are drawn in the upper halfplane with a particular notion of crossings and nestings. Generalizing the construction of Chen et.al.we prove a bijection between generalized vacillating tableaux with less than k rows and knoncrossing tangled-diagrams and study their crossings and nestings. We show that the number of k-noncrossing and k-nonnesting tangled-diagrams are equal and enumerate tangled-diagrams.