A note on p-Ascent Sequences

Ascent sequences were introduced by Bousquet-Mélou, 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 \geq 1. A sequence $(a_1, \ldots, a_n)$ of non-negative integers is a p-ascent sequence if $a_0 =0$ and for all $i \geq 2$, $a_i$ is at most p plus the number of ascents in $(a_1, \ldots, a_{i-1})$. 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 Steingrímsson 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.