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 Young's 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.