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Alternating Sign Matrices: Extensions, König-Properties, and Primary Sum-Sequences

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Abstract

This paper is concerned with properties of permutation matrices and alternating sign matrices (ASMs). An ASM is a square \((0,\pm 1)\)-matrix such that, ignoring 0’s, the 1’s and \(-1\)’s in each row and column alternate, beginning and ending with a 1. We study extensions of permutation matrices into ASMs by changing some zeros to \(+1\) or \(-1\). Furthermore, several properties concerning the term rank and line covering of ASMs are shown. An ASM A is determined by a sum-matrix \(\varSigma (A)\) whose entries are the sums of the entries of its leading submatrices (so determined by the entries of A). We show that those sums corresponding to the nonzero entries of a permutation matrix determine all the entries of the sum-matrix and investigate some of the properties of the resulting sequence of numbers. Finally, we investigate the lattice-properties of the set of ASMs (of order n), where the partial order comes from the Bruhat order for permutation matrices.

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Acknowledgements

The authors thank a referee for several useful comments.

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Authors and Affiliations

  1. Department of Mathematics, University of Wisconsin, Madison, WI, 53706, USA

    Richard A. Brualdi

  2. Department of Mathematics, University of Oslo, Oslo, Norway

    Geir Dahl

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  1. Richard A. Brualdi
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  2. Geir Dahl
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Correspondence to Geir Dahl.

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Brualdi, R.A., Dahl, G. Alternating Sign Matrices: Extensions, König-Properties, and Primary Sum-Sequences. Graphs and Combinatorics 36, 63–92 (2020). https://doi.org/10.1007/s00373-019-02119-x

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