Cycle lemma, parking functions and related multigraphs

For positive integers a and b, an (a, b̄)-parking function of length n is a sequence (p₁, . . ., p_n) of nonnegative integers whose weakly increasing order q₁ ≤ . . . ≤ q_n satisfies the condition q_i < a + (i - 1)b. In this paper, we give a new proof of the enumeration formula for (a, b̄)-parking functions by using of the cycle lemma for words, which leads to some enumerative results for the (a, b̄)-parking functions with some restrictions such as symmetric property and periodic property. Based on a bijection between (a, b̄)-parking functions and rooted forests, we enumerate combinatorially the (a, b̄)-parking functions with identical initial terms and symmetric (a, b̄)-parking functions with respect to the middle term. Moreover, we derive the critical group of a multigraph that is closely related to (a, b̄)-parking functions.