A combinatorial bijection on $k$-noncrossing partitions (Representation Theory and its Combinatorial Aspects)

Abstract

For any integer k ≥ 2, we prove combinatorially the following Euler (binomial) transformation identity NC(k)(t)n₊1=tΣn(n i), i=0NW(k)(t)i where NC(k)(t)m(resp. NW(k)(t)m) is the enumerative polynomial on part1t1ons of {1, ... , m } avoiding k-crossings (resp. enhanced k-crossings) by number of blocks. 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. It is based on the preprint (arXiv:1905.10526) with Zhicong Lin.