Q-Eulerian polynomials: Excedance number and major index
By John Shareshian, Michelle L. Wachs. 2006
67 citations recorded by the source corpus.
We derive a new q-analog of a classical formula for the exponential generating function of the Eulerian polynomials. This arose in our work on poset topology and was presented as a conjecture at the second author's FPSAC 2006 lecture. We have since proved our conjecture and a symmetric function generalization of it. Our q-Eulerian polynomials are the enumerators for the joint distribution of the excedance statistic and the major index. There is a vast literature on q-Eulerian polynomials that involves other combinations of Eulerian and Mahonian permutation statistics, but this is the first result to address the combination of excedance number and major index. Our proof involves an intriguing new class of symmetric functions and a bijection of Gessel and Reutenauer, which can be viewed as a necklace analog of Stanley's theory of P-partitions. We also discuss connections with (1) the representation of the symmetric group on the homology of a poset introduced by Bjorner and Welker, (2) the representation of the symmetric group on the cohomology of the toric variety associated with the Coxeter complex of the symmetric group, studied by Procesi, Stanley, Stembridge, Dolgachev and Lunts, (3) the enumeration of words with no adjacent repeats studied by Carlitz, Scoville and Vaughan and by Dollhopf, Goulden and Greene, and (4) Stanley's chromatic symmetric functions.
Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/89691475d80dc38c47a4d9d60902aa4156cc7758
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In Algebra and Number Theory
In Discrete Mathematics and Combinatorics
In Geometry and Topology
Citations in the corpus
- Surveys in Combinatorics 2011: The cyclic sieving phenomenon: a survey (2010, background): “Sagan, Shareshian, and Wachs [67] used the Eulerian quasisymmetric functions as developed in the earlier papers by the last two authors, as well as a result of Désarménien [14] about evaluating principal specializations at roots of unity, to demonstrate the following cyclic sieving result.”