Abstract
For positive integers a and b, an (a, b̄)-parking function of length n is a sequence (p1, . . ., pn) of nonnegative integers whose weakly increasing order q1 ≤ . . . ≤ qn satisfies the condition qi < 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.
Original language | English |
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Pages (from-to) | 345-360 |
Number of pages | 16 |
Journal | Graphs and Combinatorics |
Volume | 26 |
Issue number | 3 |
DOIs | |
Publication status | Published - 2010 Mar 8 |
Keywords
- Critical group
- Cycle lemma
- Labeled rooted forest
- Parking function
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics