On conjugates for set partitions and integer compositions

TL;DR

This paper introduces a new conjugate operation for set partitions that swaps the number of singletons and adjacencies, extending the concept of conjugation beyond integer partitions and compositions.

Contribution

It proposes a novel conjugate for set partitions, linking it to classical conjugates and revealing its effects on specific statistics like singletons and adjacencies.

Findings

Conjugate for set partitions interchanges singletons and adjacencies.

Restriction of the conjugate to noncrossing partitions relates to Kreweras' work.

Identifies pairs of statistics interchanged by the conjugate.

Abstract

There is a familiar conjugate for integer partitions: transpose the Ferrers diagram, and a conjugate for integer compositions: transpose a Ferrers-like diagram. Here we propose a conjugate for set partitions and show that it interchanges # singletons and # adjacencies. Its restriction to noncrossing partitions cropped up in a 1972 paper of Kreweras. We also exhibit an analogous pairs of statistics interchanged by the composition conjugate.