On the Number of Possible Row and Column Sums of 0,1-Matrices

  • Daniel Goldstein
  • Richard Stong


For $n$ a positive integer, we show that the number of 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 Benton, Snow, and Wallach.

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.

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