### Abstract

For positive integers a and b, an (a, b̄)-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, 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 | |

Publication status | Published - 2010 Mar 8 |

### Fingerprint

### Keywords

- Critical group
- Cycle lemma
- Labeled rooted forest
- Parking function

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

### Cite this

*Graphs and Combinatorics*,

*26*(3), 345-360. https://doi.org/10.1007/s00373-010-0921-1

**Cycle lemma, parking functions and related multigraphs.** / Eu, Sen-Peng; Fu, Tung Shan; Lai, Chun Ju.

Research output: Contribution to journal › Article

*Graphs and Combinatorics*, vol. 26, no. 3, pp. 345-360. https://doi.org/10.1007/s00373-010-0921-1

}

TY - JOUR

T1 - Cycle lemma, parking functions and related multigraphs

AU - Eu, Sen-Peng

AU - Fu, Tung Shan

AU - Lai, Chun Ju

PY - 2010/3/8

Y1 - 2010/3/8

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.

KW - Critical group

KW - Cycle lemma

KW - Labeled rooted forest

KW - Parking function

UR - http://www.scopus.com/inward/record.url?scp=77952323085&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77952323085&partnerID=8YFLogxK

U2 - 10.1007/s00373-010-0921-1

DO - 10.1007/s00373-010-0921-1

M3 - Article

AN - SCOPUS:77952323085

VL - 26

SP - 345

EP - 360

JO - Graphs and Combinatorics

JF - Graphs and Combinatorics

SN - 0911-0119

IS - 3

ER -