Rook Reciprocity Theorem

$$\sum_{k=0}^d r_k^B (d-k)!x^k=\sum_{k=0}^d (-1)^k r_k^{\bar B}(d-k)!x^k(x+1)^{d-k}.$$

References

Chow, T. Y. ``The Path-Cycle Symmetric Function of a Digraph.'' Adv. Math. 118, 71-98, 1996.

Chow, T. ``A Short Proof of the Rook Reciprocity Theorem.'' Electronic J. Combinatorics 3, R10 1-2, 1996. http://www.combinatorics.org/Volume_3/volume3.html#R10.

Goldman, J. R.; Joichi, J. T.; and White, D. E. ``Rook Theory I. Rook Equivalence of Ferrers Boards.'' Proc. Amer. Math. Soc. 52, 485-492, 1975.

Riordan, J. An Introduction to Combinatorial Analysis. New York: Wiley, 1958.