Pattern avoidance in "flattened" partitions

TL;DR

This paper investigates pattern avoidance in flattened set partitions, revealing five distinct counting sequences depending on the pattern and partition ordering, thus connecting combinatorial structures with well-known sequences.

Contribution

It introduces a novel analysis of pattern avoidance in flattened partitions, identifying specific counting sequences associated with different pattern and ordering scenarios.

Findings

Five distinct counting sequences identified

Connections established between pattern avoidance and classical sequences

Provides enumeration results for flattened partitions avoiding patterns

Abstract

To flatten a set partition (with apologies to Mathematica) means to form a permutation by erasing the dividers between its blocks. Of course, the result depends on how the blocks are listed. For the usual listing--increasing entries in each block and blocks arranged in increasing order of their first entries--we count the partitions of [n] whose flattening avoids a single 3-letter pattern. Five counting sequences arise: a null sequence, the powers of 2, the Fibonacci numbers, the Catalan numbers, and the binomial transform of the Catalan numbers.