Enumeration of (p, q)-parking functions

Parking functions are central in many aspects of combinatorics. We define in this communication a generalization of parking functions which we call (p1,..., pk)-parking functions. We give a characterization of them in terms of parking functions and we show that they can be interpreted as recurrent configurations in the sandpile model for some graphs. We also establish a correspondence with a Lukasiewicz language, which enables to enumerate (p1,..., pk)-parking functions as well as increasing ones.

[1]  Julian D. Gilbey,et al.  Parking functions, valet functions and priority queues , 1999 .

[2]  A. Joyal,et al.  Une th6orie combinatoire des s6ries formelles , 1981 .

[3]  A. Joyal.  Une théorie combinatoire des séries formelles , 1981 .

[4]  Julian D. Gilbey,et al.  Parking functions, valet functions and priority queues , 1999, Discret. Math..

[5]  John Riordan,et al.  Mappings of acyclic and parking functions , 1974 .

[6]  Aryeh Dvoretzky,et al.  The asymptotic density of certain sets of real numbers , 1947 .

[7]  Richard P. Stanley,et al.  Hyperplane Arrangements, Parking Functions and Tree Inversions , 1998 .

[8]  A. Konheim,et al.  An Occupancy Discipline and Applications , 1966 .

[9]  M. Lothaire,et al.  Combinatorics on words: Frontmatter , 1997 .

[10]  Germain Kreweras,et al.  Sur les partitions non croisees d'un cycle , 1972, Discret. Math..

[11]  Satya N. Majumdar,et al.  Equivalence between the Abelian sandpile model and the q→0 limit of the Potts model , 1992 .

[12]  Robert Cori,et al.  On the Sandpile Group of Dual Graphs , 2000, Eur. J. Comb..

[13]  Laurent Chottin,et al.  Énumération d'arbres et formules d'inversion de séries formellles , 1981, J. Comb. Theory, Ser. B.

[14]  Gilbert Labelle,et al.  Une nouvelle démonstration combinatoire des formules d'inversion de Lagrange , 1981 .

[15]  Norman Biggs.  The Tutte Polynomial as a Growth Function , 1999 .

[16]  Richard P. Stanley,et al.  Parking functions and noncrossing partitions , 1996, Electron. J. Comb..

[17]  Dhar,et al.  Self-organized critical state of sandpile automaton models. , 1990, Physical review letters.

[18]  Mark Haiman,et al.  Conjectures on the Quotient Ring by Diagonal Invariants , 1994 .

[19]  Rodica Simion,et al.  Chains in the lattice of noncrossing partitions , 1994, Discret. Math..