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A note on p-Ascent Sequences

Kitaev, Sergey and Remmel, Jeffrey (2017) A note on p-Ascent Sequences. Journal of Combinatorics, 8 (3). pp. 487-506. ISSN 2156-3527

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Ascent sequences were introduced by Bousquet-M\'elou, Claesson, Dukes, and Kitaev in \cite{BCDK}, who showed that ascent sequences of length $n$ are in 1-to-1 correspondence with \tpt-free posets of size $n$. 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 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 \cite{KR} 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 in \cite{DKRS} 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.