A combinatorial bijection on $k$-noncrossing partitions

Abstract

For any integer k ≥ 2, we prove combinatorially the following Euler (binomial) transformation identitywhere) is the sum of weights, t number 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.