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Mathematics Colloquium


Date:  Tuesday, February 04, 2014

Title:  Factoring rook polynomials

Abstract:  A boardB is a subset of the squares of an nn chess board. Let rk(B) denote the number of ways of placing k nonattacking rooks on B which means that every row and column has at most one rook. Various specializations of rk(B)count permutations, derangements, and set partitions. One nice set of boards are the Ferrers boards B = (b1; : : : ; bn) where the bj are a weakly increasing sequence of nonnegative integers and the corresponding board is obtained by choosing the lowest bj squares in column j of the chess board for 1<=  j <= n. In a landmark paper, Goldman, Joichi, and White showed that if B is a Ferrers board then an appropriately chosen generating function for the rk(B) factors over the integers. They also gave various applications of this result, such as a new proof of a theorem of Foata and Schutzenberger. In the firsthalf of this lecture we will provide an introduction to these beautiful results. The second half will be devoted to recent research concerning a generalization of rook placements where the rows of a board are grouped into levels and one can have at most one rook in any level or any column. This part is joint work with Kenneth Barrese, Nicholas Loehr and Je rey Remmel.

Speaker:  Bruce Sagan
Institution:  Michigan State University


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