Difference ascent sequences and related combinatorial structures

TL;DR

The paper develops bijective correspondences for generalized $d$-ascent sequences by introducing difference $d$ permutations and difference $d$ posets, each in bijection with $\,\mathcal{A}^d_n$, thereby extending known results for ascent and weak ascent sequences. It provides a recursive construction $oldsymbol{φ}$ that maps difference $d$ permutations to $d$-ascent sequences with $ ext{act}(π) = \text{dasc}(\boldsymbol{φ}(π)) + 1$, and proves that $\,\mathcal{S}^d_n$ corresponds to a bivincular pattern family $\Sigma_{d+3}$, yielding a pattern-avoidance interpretation. A parallel bijection $oldsymbol{ψ}$ links difference $d$ posets to $d$-ascent sequences and aligns $ ext{Act}(P)$ with $\text{dAsc}(\boldsymbol{ψ}(P))$, generalizing the weak- and ordinary-ascent connections to the $d$-setting. Finally, the paper constructs a direct, weight-preserving bijection $ heta$ between matrices with a column restriction and Fishburn matrices, resolving a $D$ukes–Sagan problem and connecting to the existing $\text{mx}$ mappings. Collectively, these results complete the bijective landscape for $d$-ascent sequences across permutations, posets, and matrices, and answer several open questions in the field.

Abstract

Ascent sequences were introduced by Bousquet-Mélou, Claesson, Dukes and Kitaev, and are in bijection with unlabeled $(2+2)$-free posets, Fishburn matrices, permutations avoiding a bivincular pattern of length $3$, and Stoimenow matchings. Analogous results for weak ascent sequences have been obtained by Bényi, Claesson and Dukes. Recently, Dukes and Sagan introduced a more general class of sequences which are called $d$-ascent sequences. They showed that some maps from the weak case can be extended to bijections for general $d$ while the extensions of others continue to be injective but not surjective. The main objective of this paper is to restore these injections to bijections. To be specific, we introduce a class of permutations which we call difference $d$ permutations and a class of factorial posets which we call difference $d$ posets, both of which are shown to be in bijection with $d$-ascent sequences. Moreover, we also give a direct bijection between a class of matrices with a certain column restriction and Fishburn matrices. Our results give answers to several questions posed by Dukes and Sagan.

Figures (4)

• Figure 1: Two factorial posets.
• Figure 2: A poset in $\mathcal{P}^0_5$ but not in $\mathcal{P}_5(P_{3})$.
• Figure 3: An example of the bijection $\psi$ between $\mathcal{P}^d_n$ and $\mathcal{A}^d_n$.
• Figure 4: A matrix of $\mathcal{M}'_{19}$ with its $\mathrm{rmax}$ and $\mathrm{rmin}$ values.

Theorems & Definitions (21)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.4
• Lemma 2.5
• Example 2.6
• Theorem 2.7
• Theorem 3.1
• Theorem 3.3
• Theorem 3.4
• Lemma 3.5