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    Weighted projections of alternating sign matrices: Latin-like squares and the ASM polytope

    Citation

    O'Brien, C. (2024) 'Weighted projections of alternating sign matrices: Latin-like squares and the ASM polytope', The Electronic Journal of Combinatorics, 31(1), available: https://doi.org/10.37236/11741.
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    O'Brien, C. (2024) Weighted projections of alternating sign matrices Latin-like squares and the ASM polytope.pdf (442.7Kb)
    Date
    2024-01-12
    Author
    O'Brien, Cian
    Peer Reviewed
    Yes
    Metadata
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    O'Brien, C. (2024) 'Weighted projections of alternating sign matrices: Latin-like squares and the ASM polytope', The Electronic Journal of Combinatorics, 31(1), available: https://doi.org/10.37236/11741.
    Abstract
    The weighted projection of an alternating sign matrix (ASM) was introduced by Brualdi and Dahl (2018) as a step towards characterising a generalisation of Latin squares they defined using alternating sign hypermatrices. Given row-vector z_n=(n,…,2,1), the weighted projection of an ASM A is equal to z_nA. Brualdi and Dahl proved that the weighted projection of an n×n ASM is majorized by the vector z_n, and conjectured that any positive integer vector majorized by z_n is the weighted projection of some ASM. The main result of this paper presents a proof of this conjecture, via monotone triangles. A relaxation of a monotone triangle, called a row-increasing triangle, is introduced. It is shown that for any row-increasing triangle T, there exists a monotone triangle M such that each entry of M occurs the same number of times as in T. A construction is also outlined for an ASM with given weighted projection. The relationship of the main result to existing results concerning the ASM polytope ASMn is examined, and a characterisation is given for the relationship between elements of ASMn corresponding to the same point in the regular n-permutohedron. Finally, the limitations of the main result for characterising alternating sign hypermatrix Latin-like squares are considered.
    Keywords
    Alternating sign matrix
    Latin square
    ASM polytope
    Language (ISO 639-3)
    eng
    Publisher
    Electronic Journal of Combinatorics
    Rights
    Open Access CC BY - ND license (International 4.0)
    License URI
    https://www.combinatorics.org/ojs/index.php/eljc/article/view/v31i1p10
    DOI
    10.37236/11741
    URI
    https://dspace.mic.ul.ie/handle/10395/3470
    ISSN
    1077-8926
    Collections
    • Mathematics and Computer Studies (Peer-reviewed publications)

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