# Partition and composition matrices

Claesson, Anders and Dukes, Mark and Kubitzke, Martina
(2011)
*Partition and composition matrices.*
Journal of Combinatorial Theory Series A, 118 (5).
pp. 1624-1637.

## 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}.

Item type: | Article |
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ID code: | 34507 |

Keywords: | partition matrix, composition matrix, ascent sequences, inversion table, permutation, (2+2)-free poset, Electronic computers. Computer science, Discrete Mathematics and Combinatorics, Computational Theory and Mathematics, Theoretical Computer Science |

Subjects: | Science > Mathematics > Electronic computers. Computer science |

Department: | Faculty of Science > Computer and Information Sciences |

Depositing user: | Pure Administrator |

Date Deposited: | 21 Oct 2011 08:50 |

Last modified: | 27 Apr 2016 17:33 |

Related URLs: | |

URI: | http://strathprints.strath.ac.uk/id/eprint/34507 |

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