Permutations by number of rises and successions

Abstract

such that Ij,j (j= l(1)n) and the latter asks for the number of permutations for which there are a fixed number of j such that Ij <ij+,. Two notational conventions will prove useful in what follows. First, it shall be convenient to write the permutation (1.1) as the one line array (11, 12, * , In) and, secondly, when enumerating permutations by number of rises, we shall attach an initial rise. Thus, for example, the permutations