Cycle lemma, parking functions and related multigraphs

Sen Peng Eu; Tung Shan Fu; Chun Ju Lai

doi:10.1007/s00373-010-0921-1

Cycle lemma, parking functions and related multigraphs

Sen Peng Eu, Tung Shan Fu, Chun Ju Lai

Department of Mathematics

Research output: Contribution to journal › Article

Abstract

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.

Original language	English
Pages (from-to)	345-360
Number of pages	16
Journal	Graphs and Combinatorics
Volume	26
Issue number	3
DOIs	https://doi.org/10.1007/s00373-010-0921-1
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

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Eu, S. P., Fu, T. S., & Lai, C. J. (2010). Cycle lemma, parking functions and related multigraphs. Graphs and Combinatorics, 26(3), 345-360. https://doi.org/10.1007/s00373-010-0921-1

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abstract = "For positive integers a and b, an (a, {\=b})-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, {\=b})-parking functions by using of the cycle lemma for words, which leads to some enumerative results for the (a, {\=b})-parking functions with some restrictions such as symmetric property and periodic property. Based on a bijection between (a, {\=b})-parking functions and rooted forests, we enumerate combinatorially the (a, {\=b})-parking functions with identical initial terms and symmetric (a, {\=b})-parking functions with respect to the middle term. Moreover, we derive the critical group of a multigraph that is closely related to (a, {\=b})-parking functions.",

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N2 - 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.

AB - 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.

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