Geodesic
On the ranks of configurations on the complete graph
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013) (2013).

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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. 
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     author = {Cori, Robert and Borgne, Yvan Le},
     title = {On the ranks of configurations on the complete graph},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013)},
     year = {2013},
     doi = {10.46298/dmtcs.2332},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2332/}
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Cori, Robert; Borgne, Yvan Le. On the ranks of configurations on the complete graph. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013) (2013). doi : 10.46298/dmtcs.2332. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2332/

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