The distribution of descents and length in a Coxeter group
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We give a method for computing the $q$-Eulerian distribution $$ W(t,q)=\sum_{w \in W} t^{{\rm des}(w)} q^{l(w)} $$ as a rational function in $t$ and $q$, where $(W,S)$ is an arbitrary Coxeter system, $l(w)$ is the length function in $W$, and ${\rm des}(w)$ is the number of simple reflections $s \in S$ for which $l(ws) < l(w)$. Using this we compute generating functions encompassing the $q$-Eulerian distributions of the classical infinite families of finite and affine Weyl groups.
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2012 ◽
Vol DMTCS Proceedings vol. AR,...
(Proceedings)
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2014 ◽
Vol 41
(4)
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pp. 911-948
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1985 ◽
Vol 55
(2)
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pp. 103-130
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