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Application of graph combinatorics to rational identities of type $A^\ast$
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009) (2009).

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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 Murnaghan-Nakayama 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). 
@article{DMTCS_2009_special_256_a44,
     author = {Boussicault, Adrien and F\'eray, Valentin},
     title = {Application of graph combinatorics to rational identities of type $A^\ast$},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009)},
     year = {2009},
     doi = {10.46298/dmtcs.2722},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2722/}
}
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%J Discrete mathematics & theoretical computer science
%D 2009
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Boussicault, Adrien; Féray, Valentin. Application of graph combinatorics to rational identities of type $A^\ast$. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009) (2009). doi : 10.46298/dmtcs.2722. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2722/

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