A note on $p$-ascent sequences

Abstract

A sequence (a 1 , . . . , a n ) of nonnegative integers is an ascent sequence if a 0 = 0 and for all i ≥ 2, a i is at most 1 plus the number of ascents in (a 1 , . . . , a i−1 ). Ascent sequences were introduced by Bousquet-Mélou, Claesson, Dukes, and Kitaev in [1], who showed that these sequences of length n are in 1-to-1 correspondence with (2 + 2)-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 p-ascent sequences, where p ≥ 1. A sequence (a 1 , . . . , a n ) of nonnegative integers is a p-ascent sequence if a 0 = 0 and for all i ≥ 2, a i is at most p plus the number of ascents in (a 1 , . . . , a i−1 ). Thus, in our terminology, ascent sequences are 1-ascent sequences. We generalize a result of the authors in [15] 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.

Further Combinatorics and Applications of Two-Toned Tilings

“…We pause here to identify a number of connections between the numbers a s (r, n) and other results in the literature. It appears that the rows of Table 3 are related to p-ascent sequences as defined in [10]. Namely, {a 1 (2, n)} n≥0 appears to coincide with 3-ascent sequences; see OEIS sequence A049611.…”

The combinatorics of Jeff Remmel

“…Jeff studied the bivincular pattern related to the interval orders and ascent sequences encoding them, as well as to several other remarkable combinatorial objects [16,40,45]. An occurrence of in a permutation is an occurrence of the pattern 2 3 1 in which the first and second elements are next to each other, and the first element is one more than the last element.…”

Asymptotics and statistics on Fishburn matrices and their generalizations

“…Stoimenow [39] found a recursive formula for the numbers of regular linearized chord diagram with a given length of the leftmost chord. Subsequently, it was discovered [6,21,30,34,35,43] that these numbers are equivalent to the following ones:…”