Cycle Lemma, Parking Functions and Related Multigraphs

Abstract

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