4,592 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
The Tutte Polynomial of a Morphism of Matroids 6. A Multi-Faceted Counting Formula for Hyperplane Regions and Acyclic Orientations
We show that the 4-variable generating function of certain orientation
related parameters of an ordered oriented matroid is the evaluation at (x + u,
y+v) of its Tutte polynomial. This evaluation contains as special cases the
counting of regions in hyperplane arrangements and of acyclic orientations in
graphs. Several new 2-variable expansions of the Tutte polynomial of an
oriented matroid follow as corollaries.
This result hold more generally for oriented matroid perspectives, with
specific special cases the counting of bounded regions in hyperplane
arrangements or of bipolar acyclic orientations in graphs.
In corollary, we obtain expressions for the partial derivatives of the Tutte
polynomial as generating functions of the same orientation parameters.Comment: 23 pages, 2 figures, 3 table
A family of bijections between G-parking functions and spanning trees
For a directed graph G on vertices {0,1,...,n}, a G-parking function is an
n-tuple (b_1,...,b_n) of non-negative integers such that, for every non-empty
subset U of {1,...,n}, there exists a vertex j in U for which there are more
than b_j edges going from j to G-U. We construct a family of bijective maps
between the set P_G of G-parking functions and the set T_G of spanning trees of
G rooted at 0, thus providing a combinatorial proof of |P_G| = |T_G|.Comment: 11 pages, 4 figures; a family of bijections containing the two
original bijections is presented; submitted to J. Combinatorial Theory,
Series
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