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Partition and composition matrices : two matrix analogues of set partitions

Claesson, Anders and Dukes, Mark and Kubitzke, Martina (2011) Partition and composition matrices : two matrix analogues of set partitions. In: DMTCS Proceedings. Discrete Mathematics & Theoretical Computer Science, Nancy, France, pp. 221-232.

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This paper introduces two matrix analogues for set partitions; partition and composition matrices. These two analogues are the natural result of lifting the mapping between ascent sequences and integer matrices given in Dukes & Parviainen (2010). We prove 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 the set X are in one-to-one correspondence with (2+2)-free posets on X. 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}.