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 (https://doi.org/10.4310/JOC.2017.v8.n3.a5)

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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.