### 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 |
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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 |

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### 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