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Cycle Lemma, Parking Functions and Related Multigraphs

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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.

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Authors and Affiliations

  1. Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung, 811, Taiwan, ROC

    Sen-Peng Eu

  2. Mathematics Faculty, National Pingtung Institute of Commerce, Pingtung, 900, Taiwan, ROC

    Tung-Shan Fu

  3. Department of Electrical Engineering, National Taiwan University, Taipei, 106, Taiwan, ROC

    Chun-Ju Lai

Authors
  1. Sen-Peng Eu
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  2. Tung-Shan Fu
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  3. Chun-Ju Lai
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Correspondence to Sen-Peng Eu.

Additional information

S.-P. Eu is partially supported by National Science Council (NSC), Taiwan under grant 98-2115-M-390-002-MY3. T.-S. Fu is partially supported by NSC under grant 97-2115-M-251-001-MY2.

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Eu, SP., Fu, TS. & Lai, CJ. Cycle Lemma, Parking Functions and Related Multigraphs. Graphs and Combinatorics 26, 345–360 (2010). https://doi.org/10.1007/s00373-010-0921-1

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