Partition and composition matrices

TL;DR

This paper introduces matrix analogues for set partitions, establishing bijections with inversion tables, posets, parking functions, and permutations, and enumerates specific classes like bidiagonal matrices.

Contribution

It defines and explores partition and composition matrices, linking them to various combinatorial structures and providing enumeration results for special cases.

Findings

Partition matrices correspond to inversion tables.

Composition matrices relate to (2+2)-free posets and parking functions.

Bidiagonal partition matrices are counted by permutations sortable by two parallel pop-stacks.

Abstract

This paper introduces two matrix analogues for set partitions. A composition matrix on a finite set X is an upper triangular matrix whose entries partition X, and for which there are no rows or columns containing only empty sets. A partition matrix is a composition matrix in which an order is placed on where entries may appear relative to one-another. We show that partition matrices are in one-to-one correspondence with inversion tables. Non-decreasing inversion tables are shown to correspond to partition matrices with a row ordering relation. Partition matrices which are s-diagonal are classified in terms of inversion tables. Bidiagonal partition matrices are enumerated using the transfer-matrix method and are equinumerous with permutations which are sortable by two pop-stacks in parallel. We show that composition matrices on X are in one-to-one correspondence with (2+2)-free posets on X. Also, composition matrices whose rows satisfy a column-ordering relation are shown to be in one-to-one correspondence with parking functions. Finally, we show that pairs of ascent sequences and permutations are in one-to-one correspondence with (2+2)-free posets whose elements are the cycles of a permutation, and use this relation to give an expression for the number of (2+2)-free posets on {1,...,n}.