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Palindromicity of multivariate Eulerian polynomials

Alejandro González Nevado

TL;DR

This work extends the palindromic property of univariate Eulerian polynomials to multivariate Eulerian polynomials by defining a reciprocal operation that respects a ghost variable and by proving that $A_n(\boldsymbol{x})$ is mirrorpalindromic, i.e., $rec(A_n)(\boldsymbol{x})=A_n(\mathrm{mirror}(\boldsymbol{x}))$. The authors construct a bijection on permutations with prescribed descent-top sets, using a lifting to $\mathfrak{S}_{n+2}$ and the index-reversing map $\tau(i)=n+3-i$, to equate coefficient structures and derive permutation-count identities and excedance-based reformulations. These results yield concrete combinatorial identities and reveal how descent-top and excedance statistics mirror under the multivariate framework. They also identify obstructions: even complete monomialmaximal multiaffine polynomials need not be mirrorpalindromic, indicating richer symmetry structures worth exploring further.

Abstract

We lift to the multivariate Eulerian polynomials the identity implying that univariate Eulerian polynomials are palindromic. As a consequence of this generalization, we obtain nice combinatorial identities that can be directly extracted from this polynomial relation and the bijection between permutations involved in the proof of the identity.

Palindromicity of multivariate Eulerian polynomials

TL;DR

This work extends the palindromic property of univariate Eulerian polynomials to multivariate Eulerian polynomials by defining a reciprocal operation that respects a ghost variable and by proving that is mirrorpalindromic, i.e., . The authors construct a bijection on permutations with prescribed descent-top sets, using a lifting to and the index-reversing map , to equate coefficient structures and derive permutation-count identities and excedance-based reformulations. These results yield concrete combinatorial identities and reveal how descent-top and excedance statistics mirror under the multivariate framework. They also identify obstructions: even complete monomialmaximal multiaffine polynomials need not be mirrorpalindromic, indicating richer symmetry structures worth exploring further.

Abstract

We lift to the multivariate Eulerian polynomials the identity implying that univariate Eulerian polynomials are palindromic. As a consequence of this generalization, we obtain nice combinatorial identities that can be directly extracted from this polynomial relation and the bijection between permutations involved in the proof of the identity.
Paper Structure (9 sections, 10 theorems, 51 equations)

This paper contains 9 sections, 10 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reciprocal notion for multivariate polynomials
  4. Fixing subsets of permutations
  5. Counting for clarity
  6. Multivariate Eulerian mirrorpalindromicity
  7. Mirrorpalindromicity from the permutation point of view
  8. Mirrorpalindromicity and excedances
  9. Conditions, obstructions and conclusion

Key Result

Theorem 2.3

Let $a_{n}$ be the $n$-th univariate Eulerian polynomial. Then $a_{n}=\mathop{\mathrm{rec}}\nolimits(a_{n}).$ This means that univariate Eulerian polynomials are palindromic.

Theorems & Definitions (34)

  • Definition 1.1
  • Definition 1.2: Tops sets
  • Definition 1.3
  • Definition 2.1: Reciprocal polynomial
  • Definition 2.2: Palindromicity as symmetry
  • Theorem 2.3
  • proof
  • Definition 3.1: Multivariate reciprocal
  • Definition 3.3: Monomialmaximality
  • Proposition 3.4: Eulerian monomialmaximality
  • ...and 24 more