Ballots and rooks

Abstract

The level code representation of the simplest ballot problem (weak lead lattice paths from (0, 0) to ( n, n) is the set of sequences ( b 1,…, b n ) defined by b 1 = 1, b i −1 ≤ b i ≤ i, 2 ≤ i ≤ n. Each sequence is monotone non-decreasing, has a specification ( c 1, c 2,…, c n ) with c i the number of sequence elements equal to i (hence c 1 + c 2…+ c n = n), and may be permuted in n! c 1!…c n! ways. The set of permuted sequences, as noted in [4], is the set of parking functions, introduced by Konheim and Weiss in [1]. To count parking functions by number of fixed points, associate the rook polynomial for matching a deck of cards of specification ( c 1,…, c n ), c i cards marked i, with a deck of n distinct cards. The hit polynomial H n ( x) corresponding to the sum of such rook polynomials over all sequences (I am using the terminology of [2]) is the required enumerator and turns out to be simply (n+1) 2H n(x)=(x+n) n+1−(x−1) n+1 .