A Combinatorial Bijection on k-Noncrossing Partitions

Abstract

For any integer k ≥ 2, we prove combinatorially the following Euler (binomial) transformation identity $${\rm{NC}}_{n + 1}^{(k)}(t) = t\sum\limits_{i = 0}^n {\left({\matrix{n \cr i \cr}} \right)} {\rm{NW}}_i^{(k)}(t),$$ where NC (k)m (t) (resp. NW (k)m (t)) is the sum of weights, tnumber of blocks, of partitions of {1,…,m} without k-crossings (resp. enhanced k-crossings). The special k = 2 and t = 1 case, asserting the Euler transformation of Motzkin numbers are Catalan numbers, was discovered by Donaghey 1977. The result for k = 3 and t = 1, arising naturally in a recent study of pattern avoidance in ascent sequences and inversion sequences, was proved only analytically.