15 research outputs found
A Bijection Between the Recurrent Configurations of a Hereditary Chip-Firing Model and Spanning Trees
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
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
A wealth of geometric and combinatorial properties of a given linear
endomorphism of is captured in the study of its associated zonotope
, and, by duality, its associated hyperplane arrangement .
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 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 variables that are dual to each other: one encodes
properties of the arrangement , while the other encodes by duality
properties of the zonotope . The algebraic structures are defined purely
in terms of the combinatorial structure of , but are subsequently proved to
be equally obtainable by applying suitable algebro-analytic operations to
either of or . The theory is universal in the sense that it
requires no assumptions on the map (the only exception being that the
algebro-analytic operations on yield sought-for results only in case
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