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

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 × 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×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.