15 research outputs found

    A Bijection Between the Recurrent Configurations of a Hereditary Chip-Firing Model and Spanning Trees

    Full text link
    Hereditary chip-firing models generalize the Abelian sandpile model and the cluster firing model to an exponential family of games induced by covers of the vertex set. This generalization retains some desirable properties, e.g. stabilization is independent of firings chosen and each chip-firing equivalence class contains a unique recurrent configuration. In this paper we present an explicit bijection between the recurrent configurations of a hereditary chip-firing model on a graph and its spanning trees.Comment: 13 page

    Graph searching and a generalized parking function

    Get PDF
    Parking functions have been a focus of mathematical research since the mid-1970s. Various generalizations have been introduced since the mid-1990s and deep relationships between these and other areas of mathematics have been discovered. Here, we introduced a new generalization, the G-multiparking function, where G is a simple graph on a totally ordered vertex set {1, 2, . . . , n}. We give an algorithm that converts a G-multiparking function into a rooted spanning forest of G by using a graph searching technique to build a sequence F1, F2, . . . , Fn, where each Fi is a subforest of G and Fn is a spanning forest of G. We also give another algorithm that converts a rooted spanning forest of G to a G-multiparking function and prove that the resulting functions (between the sets of G-multiparking functions and rooted spanning forests of G) are bijections and inverses of each other. Each of these two algorithms relies on a choice function , which is a function from the set of pairs (F,W) (where F is a subforest of G and W is a set of some of the leaves of F) into W. There are many possible choice functions for a given graph, each giving formality to the concept of choosing how a graph searching algorithm should procede at each step (i.e. if the algorithm has reached Fi, then Fi+1 is some forest on the vertex set V (Fi)∪{(Fi,W)} for some W). We also define F-redundant edges, which are edges whose removal from G does not affect the execution of either algorithm mentioned above. This concept leads to a categorization of the edges of G, which in turn provides a new formula for the Tutte polynomial of G and other enumerative results

    Zonotopal algebra

    Get PDF
    A wealth of geometric and combinatorial properties of a given linear endomorphism XX of RN\R^N is captured in the study of its associated zonotope Z(X)Z(X), and, by duality, its associated hyperplane arrangement H(X){\cal H}(X). This well-known line of study is particularly interesting in case n\eqbd\rank X \ll N. We enhance this study to an algebraic level, and associate XX with three algebraic structures, referred herein as {\it external, central, and internal.} Each algebraic structure is given in terms of a pair of homogeneous polynomial ideals in nn variables that are dual to each other: one encodes properties of the arrangement H(X){\cal H}(X), while the other encodes by duality properties of the zonotope Z(X)Z(X). The algebraic structures are defined purely in terms of the combinatorial structure of XX, but are subsequently proved to be equally obtainable by applying suitable algebro-analytic operations to either of Z(X)Z(X) or H(X){\cal H}(X). The theory is universal in the sense that it requires no assumptions on the map XX (the only exception being that the algebro-analytic operations on Z(X)Z(X) yield sought-for results only in case XX is unimodular), and provides new tools that can be used in enumerative combinatorics, graph theory, representation theory, polytope geometry, and approximation theory.Comment: 44 pages; updated to reflect referees' remarks and the developments in the area since the article first appeared on the arXi
    corecore