Question: An ascent in a permutation is a consecutive pair of the form i,i + 1. The ascents in the following permutation are underlined: 12534.
(a) Design a chessboard to represent a permutation of 1, 2, 3, 4, 5 so that a check in entry (i, j) means that i is followed immediately by j in the permutation. Darken entries so as to exclude all ascents.
(b) Find the rook polynomial for the darkened squares in part (a).
(c) Find the number of permutations of 1, 2, 3, 4, 5 with k ascents.