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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 5 essays across 4 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 The cyclic square of order 6: no transversal, and one of 5 cells. A Latin square of order 6 with a partial transversal of 5 cells shaded — one cell in each row and column but one, each with a different symbol. No full transversal exists.

One cell short of a transversal

A transversal of a Latin square picks one cell in every row and every column with every symbol different. The cyclic squares of even order have none, and that was settled by a parity argument centuries old. Whether every square of odd order has one is a conjecture from 1967 that nobody has proved; whether every square comes within one cell of having one was settled only in 2023, and only for squares large enough.

computation · Latin squares Permutations that avoid 8 forbidden cells. A 5 by 5 grid with 8 forbidden cells shaded and one permutation that avoids them marked, beside the numbers of ways to place non-attacking rooks on the forbidden cells and the count of avoiding permutations they give.

The cells a permutation must miss

A derangement is a permutation that misses the diagonal of a square grid. Forbid any other set of cells instead and inclusion–exclusion still counts what is left — driven entirely by one list of numbers, the ways to place non-attacking rooks on the forbidden cells. Boards that look nothing alike can share that list, and rooks on a staircase turn out to count the ways to split a set.

probability · Inclusion exclusion Thirty-six domino tilings, cancelling in pairs until the signs are added. The 36 domino tilings of a 4×4 board with their determinant terms: unsigned, 18 positive and 18 negative (determinant 0); with Kasteleyn signs, all 36 positive.

Signs that make a determinant count

Hall's theorem says whether a board can be covered by dominoes. How many ways it can be covered is a different question, and for a general graph a hopeless one — the count is a permanent, and nobody expects an efficient way to compute permanents. On a flat board it is a determinant. Put a minus sign on the vertical pairs in every second column and the thirty-six coverings of a four-by-four board stop cancelling and add up; the chessboard has 12,988,816. A hole in the board breaks the rule in a way a cut repairs, and on K₃,₃, which cannot be drawn flat, no choice among its 512 signings works at all.

discrete · Halls theorem

Named alongside it

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

MatchingLatin squareCounting argumentDeterminantExhaustive searchPermutationAsymptoticsBijectionBipartite graphBoundsCancellationCounterexample

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