A Combinatorial Bijection on k-Noncrossing Partitions

For any integer k ≥ 2, we prove combinatorially the following Euler (binomial) transformation identity NC n + 1 ( k ) ( t ) = t ∑ i = 0 n ( n i ) NW i ( k ) ( t ) , where NC m ( k ) ( t ) (resp. NW m ( k ) ( t )) 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.