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This is essentially a reference request/name inquiry. Is there a name for the set $M_k$ formed by $k$ by by $k$ matrices with non-negative integer entries and positive values on the diagonal? Related, are there any papers about these sets, or at least for $k=2?$

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    $\begingroup$ You might find something by searching for "symmetric contingency table". $\endgroup$
    – Brendan McKay
    Commented Nov 6, 2021 at 3:50

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