Ballots and rooks

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.