Abstract
We consider the parameter rank introduced for graph configurations by M. Baker and S. Norine. We focus on complete graphs and obtain an efficient algorithm to determine the rank for these graphs. The analysis of this algorithm leads to the definition of a parameter on Dyck words, which we call prerank. We prove that the distribution of area and prerank on Dyck words of given length $2n$ leads to a polynomial with variables $q,t$ which is symmetric in these variables. This polynomial is different from the $q,t-$Catalan polynomial studied by A. Garsia, J. Haglund and M. Haiman.
Highlights
We consider the following solitary game on an undirected connected graph with no loops: at the beginning a configuration u is given, meaning that integer values ui are attributed to the n vertices x1, x2, . . . xn of the graph
Our aim here is to study the values of this parameter when G is the complete graph on n vertices, for these graphs it was noticed (see Proposition 2.8. in Cori and Rossin (2000)) that the recurrent configurations correspond to the parking functions which play a central role in combinatorics
In the case of complete graphs, we prove that our greedy algorithm to compute the rank has a linear complexity when assuming that arithmetic operations on the ui may be performed in constant time
Summary
In all this paper n denotes the number of vertices of the graph G and m the number of its edges. We will consider configurations on this graph, which are elements of the discrete lattice Zn. Each configuration u may be considered as assigning (positive or negative) tokens to the vertices. When there is no possibility of confusion the symbol xi will denote the configuration in which the value 1 is assigned to vertex xi is and the value 0 is assigned to all others. N i=1 ei,j is the degree of the vertex xi, play a central role througout this paper. The degree of the configuration u is the sum of the ui’s and is denoted deg(u). We denote by LG the subgroup of Zn generated by the ∆(i), and two configurations u and v will be said toppling equivalent if u − v ∈ LG, which will be written as u ∼LG v
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