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Suppose that we are looking for an $m\times n$ matrix that contains only zeros and ones and that we desire certain numbers of ones for each row and each column: $x_i$ ones for row $i$ and $y_j$ ones for column $j$.

How many matrices are there that respect those conditions ? Optionally, how can one efficiently find them ?

This seems connected to Gale–Ryser theorem and bipartite realization problem (https://en.wikipedia.org/wiki/Bipartite_realization_problem) as user antkam pointed out.

Related question: Number of $(0,1)$ $m \times n$ matrices with exactly $k$ ones and no row or column contains only zeros

Related article: https://www.sciencedirect.com/science/article/pii/S0012365X08003981

W. Volante
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  • This sounds like a fun enumerative problem. What have you done so far? – The Count Apr 25 '19 at 16:09
  • For now I only have that the number of matrices is maximal when all $x_i$ and $y_i$ are set to 1 and minimal when all $x_i$ are set to $n$ and all $y_i$ are set to $m$. But no idea about how many matrices for a given $m$,$n$ and a set of conditions. – W. Volante Apr 25 '19 at 16:49
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    The question of "does there exist at least one such matrix" is solved back in 1957 and I wouldn't be surprised if there is literature on your related problem of "how many such matrices". – antkam Apr 25 '19 at 20:19
  • Also, I'd think the number is not maximal when all $x_i= y_j = 1$. In that case (and assuming $m=n$) the number equals the number of permutations, $n!$. But if every $x_i=y_j=2$ (and $m=n$), the number equals the number of distinct pairs of "non-overlapping" permutations, i.e. $\pi, \sigma$ s.t. $\pi(k) \neq \sigma(k) \forall k$. This is clearly way more than the number of permutations. My guess would be the maximal happens when every $x_i=n/2$ and every $y_j =m/2$. – antkam Apr 25 '19 at 20:22
  • Real world example of this: I had a professor who graded assignments as follows. There were 12 reading summaries due during the semester, but he only graded 6 for each student (separate random selection of 6 for each student). Say there were 20 students in the class, and the professor wanted to grade the same number of each reading summary. He'd need a 20 × 12 matrix where each row had six 1s and each column had ten 1s. Alas I'm sorry to see no answers have been given to your question. – DreiCleaner Feb 22 '20 at 16:38

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