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The sum over all permutations of the same products a determinant takes, with the minus signs deleted. It counts perfect matchings and the ways of extending a Latin rectangle, and having no cancellation in it, it cannot be evaluated by elimination.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

How many Latin squares there are, orders 1 to 8. The number of Latin squares of each small order, the ones up to six counted by exhaustive search and the larger ones quoted, on a logarithmic scale.

Nine thousand four hundred and eight

There are four Latin squares of order four once the first row and column are fixed, fifty-six of order five, and nine thousand four hundred and eight of order six. The exact answer is known for eleven orders and for no more — and yet a half-finished square can always be finished.

computation · latin squares A determinant of −5 and a permanent of 23 from the same six products. The six products of a three-by-three matrix listed once, added with signs to give the determinant and without signs to give the permanent, with a row operation applied to both.

The same sum without its minus signs

Delete the signs from the determinant's sum over permutations and what is left counts things directly rather than by cancellation. It is a better count and a far worse object — because the cancellation was what made the determinant computable.

algebra · determinant

Named alongside it

The objects these essays reach for when they reach for this one.

Latin squareMatchingAsymptoticsBijectionBoundsCancellationCountingCounting argumentDeterminantExhaustive searchFactorialHalls condition

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