Parity successions in set partitions

Suppose that the elements within each block of a partition π of [n]={1,2,…,n} are written in ascending order. By a parity succession, we will mean a pair of adjacent elements x and y within some block of π such that x≡y(mod2). Here, we consider the problem of counting the partitions of [n] according to the number of successions, extending recent results concerning successions on subsets and permutations. Using linear algebra, we determine a formula for the generating function which counts partitions having a fixed number of blocks according to size and number of successions. Furthermore, a special case of our formula yields an explicit recurrence for the generating function which counts the parity-alternating partitions of [n], i.e., those that contain no successions.