Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Partition and composition matrices   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}.

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The categories of this paper are: adap-org
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The categories of this paper are: cs
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The categories of this paper are: hep-lat
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The categories of this paper are: q-alg
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The categories of this paper are: astro-ph
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The categories of this paper are: math-ph
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The categories of this paper are: eess
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The categories of this paper are: physics
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The categories of this paper are: supr-con
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The categories of this paper are: nlin
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The categories of this paper are: econ
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The categories of this paper are: dg-ga
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The categories of this paper are: solv-int
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The categories of this paper are: cond-mat
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The categories of this paper are: comp-gas
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The categories of this paper are: chao-dyn