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Flag Descents and Eulerian Polynomials for Wreath Product Quotients
Combinatorics · Submitted 18 Nov 2016
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We prove that the flag descent polynomial is palindromic over this same quotient by giving a combinatorial proof that the flag descent statistic is symmetrically distributed over the collection of colored permutations with fixed last color by way of a new combinatorial tool, the colored winding number of a colored permutation.
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The abstract, sentence by sentence
We investigate the $\alpha$-colored Eulerian polynomials and a notion of descents introduced in a recent paper of Hedmark and show that such polynomials can be computed as a polynomial encoding descents computed over a quotient of the wreath product $\mathbb{Z}_\alpha\wr\mathfrak{S}_n$. Moreover, we consider the flag descent statistic computed over this same quotient and find that the flag Eulerian polynomial remains palindromic. We prove that the flag descent polynomial is palindromic over this same quotient by giving a combinatorial proof that the flag descent statistic is symmetrically distributed over the collection of colored permutations with fixed last color by way of a new combinatorial tool, the colored winding number of a colored permutation. We conclude with some conjectures, observations, and open questions.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We investigate the $\alpha$-colored Eulerian polynomials and a notion of descents introduced in a recent paper of Hedmark and show that such polynomials can be computed as a polynomial encoding descents computed over a quotient of the wreath product $\mathbb{Z}_\alpha\wr\mathfrak{S}_n$. | contribution 0.90 | 0.90 | 0.55 | asserted 1.00 |
| s1 | Moreover, we consider the flag descent statistic computed over this same quotient and find that the flag Eulerian polynomial remains palindromic. | result 0.89 | 0.95 | 0.22 | asserted 1.00 |
| s2 | We prove that the flag descent polynomial is palindromic over this same quotient by giving a combinatorial proof that the flag descent statistic is symmetrically distributed over the collection of colored permutations with fixed last color by way of a new combinatorial tool, the colored winding number of a colored permutation. | result 0.58 | 0.96 | 0.43 | asserted 1.00 |
| s3 | We conclude with some conjectures, observations, and open questions. | future work 1.00 | 0.56 | 0.06 | unclear 0.85 |
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Original abstract
We investigate the $\alpha$-colored Eulerian polynomials and a notion of descents introduced in a recent paper of Hedmark and show that such polynomials can be computed as a polynomial encoding descents computed over a quotient of the wreath product $\mathbb{Z}_\alpha\wr\mathfrak{S}_n$. Moreover, we consider the flag descent statistic computed over this same quotient and find that the flag Eulerian polynomial remains palindromic. We prove that the flag descent polynomial is palindromic over this same quotient by giving a combinatorial proof that the flag descent statistic is symmetrically distributed over the collection of colored permutations with fixed last color by way of a new combinatorial tool, the colored winding number of a colored permutation. We conclude with some conjectures, observations, and open questions.
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In this paper, we extend the work of [ref] by providing a new algebraic interpretation of $A_n^α$.
Colored Eulerian Polynomials and the Colored Permutohedron · Introduction · 1.00
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