The sets of flattened partitions with forbidden patterns

TL;DR

This paper investigates pattern avoidance in flattened partitions, counting such structures for single and pairs of patterns, and explores related combinatorial statistics, revealing connections to well-known sequences.

Contribution

It introduces new enumeration results for flattened partitions avoiding patterns and links these to classical combinatorial sequences and statistics.

Findings

Catalan, Fibonacci, Motzkin numbers appear in counts

Bijections established for runs and inversions

Enumeration for single and pairs of pattern avoidance

Abstract

The study of pattern avoidance in permutations, and specifically in flattened partitions is an active area of current research. In this paper, we count the number of distinct flattened partitions over [n] avoiding a single pattern, as well as a pair of two patterns. Several counting sequences, namely Catalan numbers, powers of two, Fibonacci numbers and Motzkin numbers arise. We also consider other combinatorial statistics, namely runs and inversions, and establish some bijections in situations where the statistics coincide.