On (102,000)-avoiding inversion sequences

TL;DR

This paper introduces a new combinatorial framework for counting (102,000)-avoiding inversion sequences with fixed parameters, deriving explicit formulas and revealing connections to Fuss-Catalan numbers.

Contribution

It develops a novel method using simple H-paths to obtain generating functions and explicit formulas for these inversion sequences, linking them to well-known combinatorial numbers.

Findings

Explicit formulas for counting (102,000)-avoiding inversion sequences.

Connection established between these sequences and 3-Fuss-Catalan numbers.

Derived generating functions with respect to length, distinct elements, and rank.

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.