A Note On P-Ascent Sequences

Ascent sequences were introduced by Bousquet-Melou, Claesson, Dukes, and Kitaev in [1], who showed that ascent sequences of length n are in 1-to-1 correspondence with (2 + 2)-free posets of size n. In this paper, we introduce a generalization of ascent sequences, which we call p-ascent sequences, where p >= 1. A sequence (a(1),..., a(n)) of non-negative integers is a p-ascent sequence if a(0) - 0 and for all i >= 2, a(i) is at most p plus the number of ascents in (a(1),..., a(i-1)). Thus, in our terminology, ascent sequences are 1-ascent sequences. We generalize a result of the authors in [9] by enumerating p-ascent sequences with respect to the number of 0s. We also generalize a result of Dukes, Kitaev, Remmel, and Steingriimsson in [4] by finding the generating function for the number of p-ascent sequences which have no consecutive repeated elements. Finally, we initiate the study of pattern-avoiding p-ascent sequences.