Consecutive permutation patterns, point processes, and moment sequences

We present a point of view on consecutive permutation patterns that interprets these in terms of (1) natural generalizations of the descent set of a permutation, (2) paths of a $k$-dependent point process, (3) refined clusters in the cluster method, and, surprisingly, (4) as conjectured moments of probability measures on the real line. At the heart of this paper is a recursive enumeration formula that allows us to get a grip on the aforementioned quantities and further enables us to formulate and numerically verify the conjecture (4), which provides a new unifying perspective on moment sequences arising from the study of permutation patterns