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Rook polynomial

The polynomial whose coefficient of x to the k counts the ways to place k non-attacking rooks on a board of cells. Its coefficients turn inclusion–exclusion into a count of the permutations that avoid the board, and it multiplies over blocks that share no row or column.

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

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 5 couples seated so that no one sits beside a partner. A round table with 10 seats alternating women and men, labelled by couple, arranged so that no man sits next to his partner, with the number of such arrangements.

A round table with no couple together

Seat n couples round a table, men and women alternating, so that nobody sits beside their partner. Once the women are placed the men face a board of forbidden cells that bends round a corner — and that corner is the whole difficulty. The forbidden cells form a cycle, a count of non-adjacent points on a cycle finishes the problem, and the chance of a good seating creeps towards e^(−2) far more slowly than the hat problem reaches 1/e.

probability · Inclusion exclusion

Named alongside it

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

Counting two waysDerangemente, the numberInclusion exclusionPermutationGenerating functionLatin squareLimitPermanent

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