A contingency table is a non-negative integer matrix with specified row and column sums.
A contingency table is a non-negative integer matrix with specified row and column sums, so named by Karl Pearson in developing statistical tests of significance. Observations are counted in a table with appropriate row and column labels, whereby statistical tests may be done on the entries to determine how likely the results would arise if the row and column outcomes were independent events.
Given specified row and column sums, counting the number of possible contingency tables can be a hard problem. Indeed even the case of $2$ rows and $n$ columns is known to be #P-complete.
However existence of solutions, unless otherwise constrained, is easy: it is necessary and sufficient that the row sums and column sums give equal totals for the entries of the entire matrix (balance condition).
An example of a further constraint would be requiring 0/1 entries, called binary contingency tables. Necessary and sufficient criteria for these restricted solutions were given by Gale and Ryser (independently) in 1957.
