q-Eulerian Polynomials: Excedance Number and Major index

John Shareshian; Michelle L. Wachs

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q-Eulerian Polynomials: Excedance Number and Major index

John Shareshian, Michelle L. Wachs

Combinatorics · Submitted 11 Aug 2006

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Our q-Eulerian polynomials are the enumerators for the joint distribution of the excedance statistic and the major index.

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What the abstract itself states, as probabilities: claims first 0.82 · states limitation 0.14 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.21 · compares with prior results 0.79 · mentions code or data availability 0.01.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

In this research announcement we present a new q-analog of a classical formula for the exponential generating function of the Eulerian polynomials. The Eulerian polynomials enumerate permutations according to their number of descents or their number of excedances. 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 which involve other combinations of Mahonian and Eulerian permutation statistics, but the combination of major index and excedance number seems to have been completely overlooked until now. We use symmetric function theory to prove our formula. In particular, we prove a symmetric function version of our formula, which involves an intriguing new class of symmetric functions. We also present connections with representations of the symmetric group on the homology of a poset recently introduced by Bj\"orner and Welker and on the cohomology of the toric variety associated with the Coxeter complex of the symmetric group, studied by Procesi, Stanley, Stembridge, Dolgachev and Lunts.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In this research announcement we present a new q-analog of a classical formula for the exponential generating function of the Eulerian polynomials.	contribution 0.99	0.92	0.38	asserted 1.00
s1	The Eulerian polynomials enumerate permutations according to their number of descents or their number of excedances.	background 1.00	0.05	0.05	asserted 1.00
s2	Our q-Eulerian polynomials are the enumerators for the joint distribution of the excedance statistic and the major index.	contribution 0.64	0.89	0.65	asserted 1.00
s3	There is a vast literature on q-Eulerian polynomials which involve other combinations of Mahonian and Eulerian permutation statistics, but the combination of major index and excedance number seems to have been completely overlooked until now.	gap 0.99	0.17	0.09	asserted 0.97
s4	We use symmetric function theory to prove our formula.	method 0.99	0.95	0.05	asserted 1.00
s5	In particular, we prove a symmetric function version of our formula, which involves an intriguing new class of symmetric functions.	contribution 0.73	0.96	0.11	asserted 1.00
s6	We also present connections with representations of the symmetric group on the homology of a poset recently introduced by Bj\"orner and Welker and on the cohomology of the toric variety associated with the Coxeter complex of the symmetric group, studied by Procesi, Stanley, Stembridge, Dolgachev and Lunts.	contribution 0.83	0.81	0.07	asserted 1.00

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Original abstract

In this research announcement we present a new q-analog of a classical formula for the exponential generating function of the Eulerian polynomials. The Eulerian polynomials enumerate permutations according to their number of descents or their number of excedances. 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 which involve other combinations of Mahonian and Eulerian permutation statistics, but the combination of major index and excedance number seems to have been completely overlooked until now. We use symmetric function theory to prove our formula. In particular, we prove a symmetric function version of our formula, which involves an intriguing new class of symmetric functions. We also present connections with representations of the symmetric group on the homology of a poset recently introduced by Bj\"orner and Welker and on the cohomology of the toric variety associated with the Coxeter complex of the symmetric group, studied by Procesi, Stanley, Stembridge, Dolgachev and Lunts.

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