On (102,000)-avoiding inversion sequences

TL;DR

The paper addresses the enumeration of $(102,000)$-avoiding inversion sequences with constraints on distinct elements by mapping to weighted $H$-walks and simple $H$-paths, yielding a trivariate generating function in length, dist, and rank. The authors develop a robust combinatorial framework built on labeled $F$-paths and related bijections, derive algebraic generating functions, and extract explicit counting formulas, including Fuss–Catalan connections. Key results include closed forms for sequences with fixed dist and rank, as well as total counts by dist, and a demonstration that certain subsets are counted by the $3$-Fuss-Catalan numbers. The methods provide exact enumeration and reveal structural links between inversion sequences and generalized Catalan objects, with potential implications for pattern-avoidance combinatorics and related path families.

Abstract

In this article, we study (102,000)-avoiding inversion sequences with a fixed number of distinct elements. By introducing simple H-paths, we derive the trivariate generating function for these inversion sequences with respect to their length, number of distinct elements, and rank. As consequences, we obtain an explicit formula for the number of (102,000)-avoiding inversion sequences with fixed length and number of distinct elements and we also provide a formula for those with fixed number of distinct elements and rank. In particular, we show that both the number of (102,000)-avoiding inversion sequences with a fixed number of distinct elements whose maximum element occurs exactly once and the number of those whose rank is zero are given by the 3-Fuss-Catalan numbers.

Figures (1)

• Figure 1: A labeled $F$-path $Q$ of semilength $26$ and height $4$, with the labels $(a;b)$ are omitted for brevity.

Theorems & Definitions (19)

• Definition 2.1
• Example 2.2
• Theorem 2.3
• Lemma 3.1
• proof
• Definition 3.2
• Lemma 3.3
• Lemma 3.4
• proof
• Theorem 3.5