TY - JOUR

T1 - Cycle lemma, parking functions and related multigraphs

AU - Eu, Sen Peng

AU - Fu, Tung Shan

AU - Lai, Chun Ju

N1 - Funding 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.

PY - 2010

Y1 - 2010

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

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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 -