### Abstract

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.

Language | English |
---|---|

Pages | 487-506 |

Journal | Journal of Combinatorics |

Volume | 8 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2017 |

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### Keywords

- ascent sequences
- p-ascent sequences

### Cite this

*Journal of Combinatorics*,

*8*(3), 487-506. https://doi.org/10.4310/JOC.2017.v8.n3.a5

}

*Journal of Combinatorics*, vol. 8, no. 3, pp. 487-506. https://doi.org/10.4310/JOC.2017.v8.n3.a5

**A note on p-Ascent Sequences.** / Kitaev, Sergey; Remmel, Jeffrey.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A note on p-Ascent Sequences

AU - Kitaev, Sergey

AU - Remmel, Jeffrey

PY - 2017

Y1 - 2017

N2 - 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.

AB - 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.

KW - ascent sequences

KW - p-ascent sequences

U2 - 10.4310/JOC.2017.v8.n3.a5

DO - 10.4310/JOC.2017.v8.n3.a5

M3 - Article

VL - 8

SP - 487

EP - 506

JO - Journal of Combinatorics

T2 - Journal of Combinatorics

JF - Journal of Combinatorics

SN - 2156-3527

IS - 3

ER -