Rodica Simion

On the structure of the lattice of noncrossing partitions

Abstract We show that the lattice of noncrossing (set) partitions is self-dual and that it admits a symmetric chain decomposition. The self-duality is proved via an order-reversing involution. Two proofs are given of the existence of the symmetric chain decomposition, one recursive and one constructive. Several identities involving Catalan numbers emerge from the construction of the symmetric chain decomposition.

Some q-analogues of the Schröder numbers arising from combinatorial statistics on lattice paths

Abstract We present enumerative results concerning plane lattice paths starting at the origin, with steps (1,0), (1,1) and (0,1). Such paths with a specified endpoint are counted by the Delannoy numbers, while those paths which in addition do not run above the line y=x are counted by the Schroder numbers. We develop q-analogues of the Delannoy and Schroder numbers derived from several combinatorial statistics: the number of diagonal steps, the area under the path, and the major index. We investigate the symmetry and unimodality of the resulting polynomials, and determine the asymptotic behavior of the expected number of diagonal steps and area under a path. Using the number of diagonal steps statistic, we describe the ƒ-vector of the associahedron in terms of lattice paths counted by the Schroder numbers.

Combinatorial statistics on non-crossing partitions

Abstract Four statistics, ls, rb, rs, and lb, previously studied on all partitions of {1, 2, …, n}, are applied to non-crossing partitions. We consider single and joint distributions of these statistics and prove equidistribution results. We obtain q- and p, q-analogues of Catalan and Narayana numbers which refine the rank symmetry and unimodality of the lattice of non-crossing partitions. Two unimodality conjectures, one of which pertains to Youngs lattice, are stated. We exhibit relations between statistics on non-crossing partitions and established permutation statistics applied to restricted permutations. All our proofs are combinatorial, relying on the construction of bijective correspondences and on structural properties of the lattice of non-crossing partitions.

A multiindexed sturm sequence of polynomials and unimodality of certain combinatorial sequences

Abstract With any multiset n we associate the numbers O (n, k) of compositions of n into exactly k parts. The polynomials kn(x) = Σk O(n, k)xk are shown to form a multiindexed Sturm sequence over (−1, 0). As consequences we obtain the unimodality of the sequence {O(n, k)}k for any n, of the generalized Eulerian numbers, and of the number of compositions of n with certain supplementary conditions imposed on the parts. The strong logarithmic concavity of the Stirling numbers of the second kind also follows as a corollary.

Chains in the lattice of noncrossing partitions

Abstract The lattice of noncrossing set partitions is known to admit an R -labeling. Under this labeling, maximal chains give rise to permutations. We discuss structural and enumerative properties of the lattice of noncrossing partitions, which pertain to a new permutation statistic, m (σ), defined as the number of maximal chains labeled by σ. Mobius inversion results and related facts about the lattice of unrestricted set partitions are also presented.

Specializations of generalized Laguerre polynomials

Three specialization of a set of orthogonal polynomials with “8 different q’s” are given. The polynomials are identified as q-analogues of Laguerre polynomials, and the combinatorial interpretation of the moments gives infinitely many new Mahonian statistics on permutations.

Two combinatorial statistics on Dyck paths

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 Dyek 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 non-crossing partitions. Connections with the combinatories of Motzkin paths are pointed out.

Solution to a problem of C. D. Godsil regarding bipartite graphs with unique perfect matching

We give the solution to the following question of C. D. Godsil[2]: Among the bipartite graphsG with a unique perfect matching and such that a bipartite graph obtains when the edges of the matching are contracted, characterize those having the property thatG+≅G, whereG+ is the bipartite multigraph whose adjacency matrix,B+, is diagonally similar to the inverse of the adjacency matrix ofG put in lower-triangular form. The characterization is thatG must be obtainable from a bipartite graph by adding, to each vertex, a neighbor of degree one. Our approach relies on the association of a directed graph to each pair (G, M) of a bipartite graphG and a perfect matchingM ofG.

Explicit Formulae for Some Kazhdan-Lusztig Polynomials

We consider the Kazhdan-Lusztig polynomials Pu,v(q) indexed by permutations u, v having particular forms with regard to their monotonicity patterns. The main results are the following. First we obtain a simplified recurrence relation satisfied by Pu,v(q) when the maximum value of v ∈ Sn occurs in position n − 2 or n − 1. As a corollary we obtain the explicit expression for Pe,3 4 ... n 1 2(q) (where e denotes the identity permutation), as a q-analogue of the Fibonacci number. This establishes a conjecture due to M. Haiman. Second, we obtain an explicit expression for Pe, 3 4 ... (n − 2) n (n − 1) 1 2(q). Our proofs rely on the recurrence relation satisfied by the Kazhdan-Lusztig polynomials when the indexing permutations are of the form under consideration, and on the fact that these classes of permutations lend themselves to the use of induction. We present several conjectures regarding the expression for Pu,v(q) under hypotheses similar to those of the main results.

Flag-symmetry of the poset of shuffles and a local action of the symmetric group

Abstract 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.