Generalized Parking Functions, Tree Inversions, and Multicolored Graphs

Generalized Parking Functions, Tree Inversions and Multicolored Graphs

A generalized x-parking function associated to x = (a, b, b, . . . , b) ∈ Nn is a sequence (a1, a2, . . . , an) of positive integers whose non-decreasing rearrangement b1 ≤ b2 ≤ · · · ≤ bn satisfies bi ≤ a + (i − 1)b. The set of x-parking functions is equinumerate with the set of sequences of rooted b-forests on [n]. We construct a bijection between these two sets. We show that the sum enumerat...

Hyperplane Arrangements, Parking Functions and Tree Inversions

Parking Functions and Acyclic Orientations of Graphs

A. Given an undirected graph G = (V, E), and a designated vertex q ∈ V , the notion of a G-parking function (with respect to q) has recently been developed and studied by various authors. This notion generalizes the classical notion of a parking function associated with the complete graph. In this work, we study properties of certain maximum G-parking functions and relate them, in a bije...

On the Enumeration of Generalized Parking Functions

Let x = (x1, x2, . . . , xn) ∈ Nn . Define a x-parking function to be a sequence (a1, a2, . . . , an) of positive integers whose non-decreasing rearrangement b1 ≤ b2 ≤ · · · ≤ bn satisfies bi ≤ x1+ · · ·+xi. Let Pn(x) denote the number of x-parking functions. We discuss the enumerations of such generalized parking functions. In particular, We give the explicit formulas and present two proofs, o...

Deformation of Weyl Modules and Generalized Parking Functions

Local Weyl modules over two-dimensional currents with values in glr are deformed into spaces with bases related to parking functions. Using this construction we 1) reproof that dimension of the space of diagonal coinvariants is not less than the number of parking functions; 2) describe the limit of Weyl modules in terms of semi-infinite forms and find the limit of characters; 3) propose a lower...