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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.

##### MSC:
 05A15 Exact enumeration problems, generating functions 05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
##### Keywords:
divisibility result
Zbl 1076.05015
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