An algorithmic approach based on generating trees for enumerating pattern-avoiding inversion sequences

TL;DR

This paper presents a new algorithmic approach using generating trees to enumerate pattern-avoiding inversion sequences, determines their generating trees, and solves six open classification cases of Wilf-equivalence among pattern pairs.

Contribution

Introduces an algorithm based on generating trees for enumerating pattern-avoiding inversion sequences and solves six open Wilf-equivalence classification cases.

Findings

Derived generating functions for several pattern-avoiding classes.

Established generating trees for multiple pattern-classes.

Solved six open Wilf-equivalence classification cases.

Abstract

We introduce an algorithmic approach based on generating tree method for enumerating the inversion sequences with various pattern-avoidance restrictions. For a given set of patterns, we propose an algorithm that outputs either an accurate description of the succession rules of the corresponding generating tree or an ansatz. By using this approach, we determine the generating trees for the pattern-classes $I_n(000, 021), I_n(100, 021)$, $I_n(110, 021), I_n(102, 021)$, $I_n(100,012)$, $I_n(011,201)$, $I_n(011,210)$ and $I_n(120,210)$. Then we use the kernel method, obtain generating functions of each class, and find enumerating formulas. Lin and Yan studied the classification of the Wilf-equivalences for inversion sequences avoiding pairs of length-three patterns and showed that there are 48 Wilf classes among 78 pairs. In this paper, we solve six open cases for such pattern classes.