Adrien Boussicault ; Valentin Féray

Application of graph combinatorics to rational identities of type $A^\ast$
dmtcs:2722 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2009,
DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009)

https://doi.org/10.46298/dmtcs.2722
Application of graph combinatorics to rational identities of type $A^\ast$Article
To a word $w$, we associate the rational function $\Psi_w = \prod (x_{w_i}  x_{w_{i+1}})^{1}$. The main object, introduced by C. Greene to generalize identities linked to MurnaghanNakayama rule, is a sum of its images by certain permutations of the variables. The sets of permutations that we consider are the linear extensions of oriented graphs. We explain how to compute this rational function, using the combinatorics of the graph $G$. We also establish a link between an algebraic property of the rational function (the factorization of the numerator) and a combinatorial property of the graph (the existence of a disconnecting chain).