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On the number of possible row and column sums of \(0,1\)-matrices. (English) Zbl 1098.05007
Summary: For \(n\) a positive integer, we show that the number of \(2n\)-tuples of integers that are the row and column sums of some \(n\times n\) matrix with entries in \(\{0,1\}\) is evenly divisible by \(n+1\). This confirms a conjecture of J. Benton, R. Snow, and N. Wallach [Linear Algebra Appl. 412, 30–38 (2006; Zbl 1076.05015)]. We also consider a \(q\)-analogue for \(m\times n\) matrices. We give an efficient recursion formula for this analogue. We prove a divisibility result in this context that implies the \(n+1\) divisibility result.

05A15 Exact enumeration problems, generating functions
05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
Zbl 1076.05015
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