p-Ascent Sequences

A sequence $(a_1, \ldots, a_n)$ of nonnegative integers is an {\em ascent sequence} if $a_0 =0$ and for all $i \geq 2$, $a_i$ is at most 1 plus the number of ascents in $(a_1, \ldots, a_{i-1})$. Ascent sequences were introduced by Bousquet-M\'elou, Claesson, Dukes, and Kitaev, who showed that these sequences of length $n$ are in 1-to-1 correspondence with \tpt-free posets of size $n$, which, in turn, are in 1-to-1 correspondence with interval orders of size $n$. Ascent sequences are also in bijection with several other classes of combinatorial objects including the set of upper triangular matrices with nonnegative integer entries such that no row or column contains all zeros, permutations that avoid a certain mesh pattern, and the set of Stoimenow matchings. In this paper, we introduce a generalization of ascent sequences, which we call {\em $p$-ascent sequences}, where $p \geq 1$. A sequence $(a_1, \ldots, a_n)$ of nonnegative 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 by enumerating $p$-ascent sequences with respect to the number of $0$s. We also generalize a result of Dukes, Kitaev, Remmel, and Steingr\'{\i}msson 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.