[Paper Review] Edgewise subdivisions, local $h$-polynomials and excedances in the wreath product $\ZZ_r \wr \mathfrak{S}_n$

Christos A. Athanasiadis|arXiv (Cornell University)|Oct 1, 2013

Advanced Combinatorial Mathematics23 references5 citations

TL;DR

This paper generalizes the combinatorial interpretation of local h-polynomials in simplicial subdivisions by linking the r-th edgewise subdivision of the barycentric subdivision of the simplex to flag excedances in the wreath product ℤ_r ≀ 𝔖_n. It proves that the local h-polynomial is γ-nonnegative and provides a combinatorial interpretation of the γ-coefficients via balanced derangements and flag excedance statistics, extending classical results on derangements and excedances in symmetric groups.

ABSTRACT

The coefficients of the local $h$-polynomial of the barycentric subdivision of the simplex with $n$ vertices are known to count derangements in the symmetric group $\mathfrak{S}_n$ by the number of excedances. A generalization of this interpretation is given for the local $h$-polynomial of the $r$th edgewise subdivision of the barycentric subdivision of the simplex. This polynomial is shown to be $γ$-nonnegative and a combinatorial interpretation to the corresponding $γ$-coefficients is provided. The new combinatorial interpretations involve the notions of flag excedance and descent in the wreath product $\ZZ_r \wr \mathfrak{S}_n$. A related result on the derangement polynomial for $\ZZ_r \wr \mathfrak{S}_n$, studied by Chow and Mansour, is also derived from results of Linusson, Shareshian and Wachs on the homology of Rees products of posets.

Motivation & Objective

• To extend the classical interpretation of local h-polynomials in barycentric subdivisions—where coefficients count derangements by excedance—to higher-order edgewise subdivisions.
• To establish a combinatorial interpretation of the local h-polynomial of the r-th edgewise subdivision of the barycentric subdivision of the (n−1)-simplex.
• To prove that this local h-polynomial is γ-nonnegative and to provide a combinatorial interpretation of the γ-coefficients using flag excedance and balanced elements in the wreath product ℤ_r ≀ 𝔖_n.
• To connect the derangement polynomial d_n^r(x) in ℤ_r ≀ 𝔖_n to the homology of Rees products of posets, recovering results from Chow and Mansour via Linusson, Shareshian, and Wachs.

Proposed method

• The paper uses the definition of the local h-polynomial via inclusion-exclusion over face restrictions: ℓ_V(Γ,x) = ∑_{F⊆V} (−1)^{n−|F|} h(Γ_F,x).
• It applies results from Rees products of posets (Linusson, Shareshian, Wachs) to derive the generating function for the h-polynomial of the r-th edgewise subdivision.
• It introduces the flag excedance statistic fexc(w) on ℤ_r ≀ 𝔖_n as r·exc_A(w) + csum(w), where exc_A counts excedances at zero-colored positions and csum is the sum of colors.
• It decomposes the derangement polynomial d_n^r(x) into symmetric and antisymmetric parts f_n^r,+(x) and f_n^r,-(x), which are shown to satisfy functional equations reflecting symmetry.
• It proves γ-nonnegativity by expressing the local h-polynomial in the γ-basis: ℓ_V(Γ,x) = ∑_{i=0}^{⌊n/2⌋} ξ_{n,r,i} x^i (1+x)^{n−2i}, with ξ_{n,r,i} counting permutations with i descending runs and no singleton runs.
• It uses the principle of inclusion-exclusion and generating function manipulation to relate the h-polynomial of the r-th edgewise subdivision to the flag excedance generating function over balanced elements in ℤ_r ≀ 𝔖_k.

Experimental results

Research questions

• RQ1 How can the local h-polynomial of the r-th edgewise subdivision of the barycentric subdivision of the simplex be interpreted combinatorially?
• RQ2 What is the combinatorial meaning of the γ-coefficients in the γ-expansion of this local h-polynomial?
• RQ3 How do flag excedances in the wreath product ℤ_r ≀ 𝔖_n relate to the structure of edgewise subdivisions and local h-polynomials?
• RQ4 Can the derangement polynomial d_n^r(x) in ℤ_r ≀ 𝔖_n be interpreted via poset homology and what does this imply for its unimodality and real-rootedness?
• RQ5 Is the local h-polynomial of the r-th edgewise subdivision real-rooted, and does it satisfy γ-nonnegativity?

Key findings

• The local h-polynomial of the r-th edgewise subdivision of the barycentric subdivision of the (n−1)-simplex is given by ℓ_V(Γ,x) = ∑_{w∈(𝒟_n^r)^b} x^{fexc(w)/r}, where (𝒟_n^r)^b is the set of balanced derangements in ℤ_r ≀ 𝔖_n.
• The local h-polynomial is γ-nonnegative, with coefficients ξ_{n,r,i} counting permutations in ℤ_r ≀ 𝔖_n with i descending runs and no singleton runs.
• The γ-expansion ℓ_V(Γ,x) = ∑_{i=0}^{⌊n/2⌋} ξ_{n,r,i} x^i (1+x)^{n−2i} provides a combinatorial interpretation of the γ-coefficients via flag excedance and descent statistics.
• The derangement polynomial d_n^r(x) is shown to be unimodal with peak at ⌊(n+1)/2⌋, and its real-rootedness is confirmed via the decomposition f_n^r,+(x) + f_n^r,-(x) and symmetry properties.
• The generating function for the h-polynomial of the r-th edgewise subdivision is h(sd(2^F)^⟨r⟩,x) = E_r(∑_{w∈ℤ_r≀𝔖_k} x^{fexc(w)}) = ∑_{w∈(ℤ_r≀𝔖_k)^b} x^{fexc(w)/r}, linking it to balanced elements.
• The paper confirms that d_n^r(x) is real-rooted and unimodal, and provides a new transparent proof of unimodality via the γ-expansion and combinatorial interpretation of ξ_{n,r,i}.