Set partitions and parity successions

By a (parity) succession within a sequence w = w₁w₂ …, we will mean an index i such that w_i ≡ w_{i + 1} (mod2). In this paper, we address the problem of counting successions in set partitions, represented sequentially as restricted growth functions. Among our results, we find explicit formulas for the relevant generating functions and for the number of parity-al­ternating set partitions, i.e., those having no successions. We also compute a formula for the total number of successions within all partitions of a fixed length and number of blocks, and a combinatorial proof of this result is provided. Finally, we consider the problem of counting successions in non-crossing partitions, i.e., those having no occurrence of the pattern 1212, and determine, with the aid of programming, formulas for the generating functions.