Rook polynomial
Named by 3 essays across one field — each of them below, with the objects they name alongside it.
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.
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.
A card game almost nobody wins
Shuffle a deck and deal it face up, calling ace, two, three, up to king and round again, four times through. If a card ever matches the rank being called, the game is lost, and it is won about one time in sixty-two. The chance is 0.016232727…, exact, because the forbidden pairings form thirteen separate four-by-four squares and inclusion–exclusion turns their rook numbers into a sum — a sum whose terms climb past ten before they cancel to a sixtieth, and which is also the integral of a power of a Laguerre polynomial. The obvious guess, e to the minus four, is eleven per cent too high, and two hundred thousand deals can tell.
Named alongside it
The objects these essays reach for when they reach for this one.
DerangementInclusion exclusionCounting two wayse, the numberGenerating functionPermutationExhaustive searchLatin squareLimitPermanentPoisson approximation