Abstract
The paper by M. Baker and S. Norine in 2007 introduced a new parameter on configurations of graphs and gave a new result in the theory of graphs which has an algebraic geometry flavor. This result was called Riemann-Roch formula for graphs since it defines a combinatorial version of divisors and their ranks in terms of configurations on graphs. The so called chip firing game on graphs and the sandpile model in physics play a central role in this theory. In this paper we present an algorithm for the determination of the rank of configurations for the complete graph $K_n$. This algorithm has linear arithmetic complexity. The analysis of number of iterations in a less optimized version of this algorithm leads to an apparently new parameter which we call the prerank. This parameter and the parameter dinv provide an alternative description to some well known $q,t$-Catalan numbers. Restricted to a natural subset of configurations, the two natural statistics degree and rank lead to a distribution which is described by a generating function which, up to a change of variables and a rescaling, is a symmetric fraction involving two copies of Carlitz $q$-analogue of the Catalan numbers. In annex, we give an alternative presentation of the theorem of Baker and Norine in purely combinatorial terms.
Highlights
On a graph, a configuration is a map from its vertices to the integers of Z
In this paper we present an algorithm for the determination of the rank of configurations for the complete graph Kn
The complete graph Kn is the non-oriented graph with n vertices and exactly one edge between any pair of distinct vertices
Summary
A configuration is a map from its vertices to the integers of Z. We use the numerous symmetries of the complete graphs to obtain a greedy algorithm computing the rank The complexity of this version is not clear but the aim of Section 3 is to optimize this algorithm. A word v on the two-letter alphabet {a, b} is a Dyck word of size n if |v|a = |v|b = n and for any prefix p of v, |p|a |p|b where |v|c denotes the number of occurrences of letter c in the word v This data is embedded in a cut skew cylinder to make use of a cyclic symmetry. Our optimized algorithm on the configuration u = (ui)i may be interpreted as a spiral traversal of un + 1 cells on a related cut skew cylinder counting the occurrences of cells in one of the two components defined by the cut. We prove the correctness of our first version of a greedy algorithm computing the rank for a configuration in a complete graph
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