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  1. PO
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    <p>Are you needing this to calculate <a href="http://en.wikipedia.org/wiki/Fisher%27s_exact_test" rel="nofollow noreferrer">Fisher's exact test</a>? Because that requires what you're doing, and based on that page, it seems there will in general be a vast number of solutions, so you probably can't do better than a brute force recursive enumeration if you want every solution. OTOH it seems Monte Carlo approximations are successfully used by some software instead of full-blown enumerations.</p> <p>I asked a <a href="https://stackoverflow.com/questions/5086872/is-it-possible-to-shuffle-a-2d-matrix-while-preserving-row-and-column-frequencie">similar question</a>, which might be helpful. Although that question deals with preserving frequencies of letters in each row and column rather than sums, some results can be translated across. E.g. if you find any submatrix (pair of not-necessarily-adjacent rows and pair of not-necessarily-adjacent columns) with numbers</p> <pre><code>xy yx </code></pre> <p>Then you can rearrange these to</p> <pre><code>yx xy </code></pre> <p>without changing any row or column sums. However:</p> <ul> <li><a href="https://stackoverflow.com/a/5087786/47984">mhum's answer</a> proves that there will in general be valid matrices that cannot be reached by any sequence of such 2x2 swaps. This can be seen by taking his 3x3 matrices and mapping <code>A -&gt; 1</code>, <code>B -&gt; 2</code>, <code>C -&gt; 4</code> and noticing that, because no element appears more than once in a row or column, frequency preservation in the original matrix is equivalent to sum preservation in the new matrix. <em>However...</em></li> <li><a href="https://stackoverflow.com/a/5091273/47984">someone's answer</a> links to a mathematical proof that it actually will work for matrices whose entries are just 0 or 1.</li> </ul> <p>More generally, if you have any submatrix</p> <pre><code>ab cd </code></pre> <p>where the (not necessarily unique) minimum is d, then you can replace this with any of the d+1 matrices</p> <pre><code>ef gh </code></pre> <p>where h = d-i, g = c+i, f = b+i and e = a-i, for any integer 0 &lt;= i &lt;= d.</p>
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    1. POEnumerate matrix combinations with fixed row and column sums
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    1. COIndeed, I need this algorithm to apply Fisher's Exact test on large data samples (but relatively low cell values, hence excluding any asymptotic test such as the Chi² test). I also need the algorithm for another problem, more practical, involving many sellers (each with a set of available items in variable quantity) and a buyer trying to find the best configuration given a set of items to buy in different quantities (lowest cost, including fixed and variable fees for each seller and each type of item...) - but I'll take a look at Monte Carlo tests anyway.
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      1. USMaxime Rossini
    2. COIf the row and column sums are low, it might be feasible to enumerate all possibilities by deciding for each row (say) in turn how to distribute indistinguishable "marbles" to "bins", where each cell in the row is a bin and the row sum tells you how many marbles you have. There are `(n+r-1) choose (r-1)` ways of distributing r marbles across n bins for a single row; multiply the result you get for each row together to get a loose upper bound on the number of configurations (loose because some distributions will be outlawed by column sums).
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