Generalized Parking Functions, Tree Inversions, and Multicolored Graphs

  • Authors:

  • Catherine H. Yan

  • Affiliations:

  • Department of Mathematics, Texas A&M University, College Station, Texas, 77843-3368, f1catherine.yan@math.tamu.eduf1

  • Venue:
  • Advances in Applied Mathematics
  • Year:
  • 2001

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Abstract

A generalized x-parking function associated to a positive integer vector of the form (a,b,b,...,b) is a sequence (a"1,a"2,...,a"n) of positive integers whose nondecreasing rearrangement b"1@?b"2@?...@?b"n satisfies b"i@?a+(i-1)b. The set of x-parking functions has the same cardinality as the set of sequences of rooted b-forests on [n]. We construct a bijection between these two sets. We show that the sum enumerator of complements of x-parking functions is identical to the inversion enumerator of sequences of rooted b-forests by generating function analysis. Combinatorial correspondences between the sequences of rooted forests and x-parking functions are also given in terms of depth-first search and breadth-first search on multicolored graphs.