### 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}.

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}.

Original language | English |
---|---|

Pages (from-to) | 1624-1637 |

Number of pages | 14 |

Journal | Journal of Combinatorial Theory Series A |

Volume | 118 |

Issue number | 5 |

DOIs | |

Publication status | Published - Jul 2011 |

### Keywords

- partition matrix
- composition matrix
- ascent sequences
- inversion table
- permutation
- (2+2)-free poset

## Fingerprint Dive into the research topics of 'Partition and composition matrices'. Together they form a unique fingerprint.

## Cite this

Claesson, A., Dukes, M., & Kubitzke, M. (2011). Partition and composition matrices.

*Journal of Combinatorial Theory Series A*,*118*(5), 1624-1637 . https://doi.org/10.1016/j.jcta.2011.02.001