There is a reciprocity law relating rook numbers for a given board and the complementary board, the Rook Reciprocity Theorem. Is there a similar reciprocity law for non-rook chess pieces?
"Reciprocity law" for non-rook chess pieces
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combinatorics
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0@example: A *board* is a subset $B$ of $[n]\times[n]$ for some $n\in\Bbb Z^+$; the *complementary board* is $([n]\times[n])\setminus B$. The *rook numbers* $r_k(B)$ give the numbers of ways of placing $k$ mutually non-attacking rooks on $B$, i.e., the number of $k$-element subsets of $B$ such that no two elements have the same first or the same second coordinate. – 2012-04-15