Gamma-positive polynomials

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#Gamma positivity

See the survey by C. Athanasiadis [Ath18].

A polynomial $p(x)$ of degree $n$ with non-negative coefficients is palindromic (or symmetric) if $x^n p(x^{-1}) = p(x).$ A palindromic polynomial $p(x)$ is said to be gamma-positive if it can be expressed as \[p(x) = \sum_{i=0}^{\lfloor n/2 \rfloor} \gamma_i x^i (1+x)^{n-2i}, \qquad \gamma_i \geq 0.\] The gamma-polynomial of $p(x)$ is $\gamma_p(x) \coloneqq \sum_j \gamma_j x^j.$ If $p(x)$ is palindromic with center of symmetry $n/2,$ the expansion above is unique.

Example

The Eulerian polynomial \[A_4(x)=1+11x+11x^2+x^3\] has degree $3,$ and its gamma expansion is \[A_4(x)=(1+x)^3+8x(1+x).\] Thus $\gamma_{A_4}(x)=1+8x,$ so $A_4(x)$ is gamma-positive.

Gamma-positivity immediately implies unimodality: writing $p(x) = \sum_k a_k x^k$ with $a_k = a_{n-k}$ and expanding the binomial coefficients, each $\gamma_i$ contributes a unimodal symmetric summand, so $p(x)$ is unimodal.

A corrected form of Observation 4.2 in T. K. Petersen’s book [Observation 4.2, Pet15] is the following criterion. If $p(x)$ is palindromic with non-negative coefficients, then $p(x)$ is real-rooted if and only if $\gamma_p(x)$ has only real non-positive zeros. Equivalently, if $p(x)$ is gamma-positive, then $p(x)$ is real-rooted if and only if $\gamma_p(x)$ is real-rooted. The root-location condition cannot be omitted: the polynomial \[p(x)=1+x+x^2=(1+x)^2-x\] has non-negative coefficients and $\gamma_p(x)=1-x$ is real-rooted, but $p(x)$ is not real-rooted. See also [Lemma 4.1, Br04] and [Remark 3.1.1, Gal05] for earlier related forms of the criterion.
�. Gal conjectured [Gal05] that the $h$-polynomial of any flag simplicial sphere is gamma-positive. This is still open in general but known for several families.
Suppose $p(x)$ is palindromic with center of symmetry $n/2.$ If $\gamma_p(x)$ is ultra-log-concave of order $\lfloor n/2 \rfloor$ with no internal zeros, then $p(x)$ is also ultra-log-concave of order $n$ with no internal zeros [BFJ24]. The converse does not hold.
Suppose $p(x)$ is palindromic. If $\gamma_p(x)$ is log-concave with no internal zeros, then $p(x)$ is also log-concave with no internal zeros [FPV25]. The proof uses a binomial-coefficient inequality obtained through a path-counting argument. The converse does not hold.

Theorem (Eulerian transformation, [Ath25]).

Let $\mathcal{P}_n[x]$ be the cone spanned by the polynomials $x^{n-k}(1+x)^k,$ for $0\leq k\leq n,$ with non-negative coefficients. Define the Eulerian transformation $A^\circ:\setR[x]\to\setR[x]$ by \[A^\circ(1)=1,\qquad A^\circ(x^m)=xA_m(x) \quad \text{for } m\geq 1.\] For every $p(x)\in \mathcal{P}_n[x],$ the polynomial $A^\circ(p(x))$ is real-rooted. Moreover, it fits into an interlacing chain between $xA_n(x)$ and $A^\circ((1+x)^n),$ and has a real-rooted interlacing symmetric decomposition. In particular, $A^\circ(p(x))$ is unimodal and gamma-positive.

#Alternating gamma-positivity

A polynomial is alternatingly gamma-positive if it can be written as \[f(x)=\sum_k \gamma_k(-x)^k(1+x)^{n-2k}, \qquad \gamma_k\geq 0.\] This is useful when a polynomial becomes naturally expressed in powers of $x^2$ or through Hermite–Biehler decompositions. For example, S.-M. Ma, H. Qi, J. Yeh and Y.-N. Yeh prove [MQYY22] that the product of two alternatingly gamma-positive polynomials is alternatingly gamma-positive, and that if $f(x)$ is gamma-positive, then $f(x^{2m})$ is alternatingly gamma-positive for every $m\geq 1.$ J. Shareshian and M. L. Wachs prove gamma-positivity for several variations of Eulerian polynomials [SW20]. Their results fit naturally with the peak-set and valley-hopping viewpoint on Eulerian gamma expansions.

They also show that $A_n(x^2)$ and the type $B$ Eulerian polynomial $B_n(x^2)$ are alternatingly gamma-positive, and that \[N(B_n,x^2)+(n+1)xN(A_{n-1},x^2)\] is alternatingly gamma-positive and Hurwitz stable. A general result from [MQYY22] is that every gamma-positive polynomial is alternatingly semi-gamma-positive.

#Examples of gamma-positivity

Theorem (Foata–Strehl action on Eulerian polynomials).

The Eulerian polynomials $A_n(x)$ are palindromic and gamma-positive. The gamma-coefficients $\gamma_{n,k}$ count permutations $\sigma \in \symS_n$ with $k$ peaks (positions $i$ with $\sigma_{i-1} \lt{} \sigma_i \gt{} \sigma_{i+1}$). Thus \[A_n(x) = \sum_{k=0}^{\lfloor (n-1)/2 \rfloor} \gamma_{n,k}\, x^k(1+x)^{n-1-2k}.\] The proof uses the Foata–Strehl action (or valley-hopping action) on $\symS_n:$ for each $i \in \{2,\dotsc,n-1\},$ define an involution $\varphi_i$ that swaps the relative position of $i$ with its neighbors when $i$ is a local minimum. The $2^{n-2}$ orbits of the abelian group generated by $\{\varphi_i\}$ are exactly the gamma-classes; orbits of size $2^k$ contribute $\gamma_{n,k}.$ See [Ath18] and [Br08] for generalization to sign-graded posets. We also get gamma-positivity for the peak polynomials by the same method.

Theorem (Narayana polynomials, [Br06]).

The Narayana polynomials \[N_n(x) = \frac{1}{n}\sum_{k=1}^n \binom{n}{k}\binom{n}{k-1} x^k\] are palindromic and gamma-positive. The gamma-coefficients are the Narayana numbers of the second kind. Since $N_n(x)$ is also real-rooted (see [Br06]), the gamma-positivity follows from the real-rooted criterion above.

Theorem (Descent polynomials of standard Young tableaux).

Let $W_\lambda(t) = \sum_{T \in \SYT(\lambda)} t^{\des(T)}.$ For any shape $\lambda,$ $W_\lambda(t)$ is palindromic and gamma-positive. This follows because $W_\lambda(t)$ is real-rooted (see [Bre89] and the discussion in tableaux examples), combined with the criterion that palindromic real-rooted polynomials are gamma-positive.

Theorem (Zig-zag poset, [PZ24]).

Let $Z_m(t)$ be the $P$-Eulerian polynomial associated with the zig-zag poset on $m$ vertices. Then $Z_m(t)$ is gamma-positive: \[Z_m(t) = t \cdot \sum_{0 \leq 2j \leq m-2} \gamma_{m,j}\,t^j (1+t)^{m-2-2j}, \qquad \gamma_{m,j} \geq 0.\] Moreover, \[Z_m(t) = t \cdot \sum_{\pi \in U_m} t^{\ret_1(\pi)}\] where the sum is over up-down permutations, and $\ret_1(\pi)$ counts instances where $i+1$ appears to the left of $i$ but not adjacent to it. The stronger assertion that all $Z_m(t)$ are real-rooted remains open; it is not the same problem as the Delannoy row polynomial case for regular snake posets. It is an open problem to give a combinatorial explanation for the symmetry and unimodality of the $\gamma_{m,j}.$

Y. Kahane uses Ehrhart theory of the zig-zag poset to give another proof of the Watanabe–Yoshida conjecture on Hilbert–Kunz multiplicity [Kah25]. The proof gives an explicit combinatorial formula for the coefficients of the shifted Ehrhart polynomial and expresses its generating function through a Hadamard product involving Euler numbers.

M. Beck and D. Deligeorgaki interpret canon permutations using labeled posets [BD24]. If $P$ is a naturally labeled poset, their canon polynomial satisfies $C_n^{P,\omega}(x)=A_n(x)h^*_{P\times[n]}(x).$ This explains the palindromicity of canon descent polynomials and gives gamma-positivity together with a combinatorial interpretation of the gamma-coefficients in the classical canon-permutation case.

Theorem (Brändén, [Br04]).

For any graded poset $P$ with a $\hat{0}$ and $\hat{1},$ if the $cd$-index of $P$ has non-negative coefficients, then the $h$-polynomial of the order complex of the proper part $\overline{P}$ is gamma-positive. In particular, this applies to face posets of polytopes, Boolean algebras, and the partition lattice.

Alessio D'Ali and A. Higashitani prove an equivariant analogue of Brändén’s gamma-positivity theorem for graded posets [DH25]. For any graded poset $P$ and any subgroup $G\leq \operatorname{Aut}(P),$ the order polytope $\mathcal{O}(P)$ is gamma-effective: the coefficients of its equivariant gamma-polynomial are actual $G$-characters. Their proof develops equivariant Ehrhart theory for order polytopes of sign-graded posets.

H.-C. Liao proves equivariant gamma-positivity for the Chow ring and augmented Chow ring of a matroid under the action of any automorphism group [Lia24]. The proof gives an explicit combinatorial interpretation of the equivariant gamma-coefficients, confirms a conjecture of Angarone–Nathanson–Reiner, and for uniform matroids extends the Schur-gamma-positivity results of J. Shareshian and M. Wachs for Eulerian and binomial Eulerian quasisymmetric functions.

L. Ferroni disproves the Ohsugi–Tsuchiya conjecture that the Ehrhart $h^*$-polynomials of all symmetric edge polytopes are gamma-positive [Fer26]. The counterexamples form an infinite family of series-parallel graphs obtained by joining two vertices with internally disjoint paths. The smallest example detected there is the graph with five paths of length $8$; its symmetric edge polytope has dimension $36,$ and the corresponding gamma-vector has $\gamma_{16}=-9799680.$

#Methods for proving gamma-positivity

There are several standard approaches:

Real-rootedness. If $p(x)$ is palindromic and real-rooted with non-negative coefficients, then it is automatically gamma-positive. This gives gamma-positivity for Eulerian, Narayana, and SYT descent polynomials.
Group action. Exhibit an involution (or group action) whose orbits correspond to the gamma-classes. The Foata–Strehl action is the canonical example.
Symmetric decomposition. Write $p(x) = a(x)(1+x) + x^c b(x)$ where $a,b$ are palindromic; if $a$ and $b$ are gamma-positive, so is $p.$ See [BS19] for this approach via the symmetric decomposition of the $h^*$-polynomial.
Algebraic / representation-theoretic. Show that $p(x)$ is the Hilbert series of a representation where the palindromy and gamma-positivity are forced by the structure, e.g. via the coinvariant algebra or $\mathfrak{sl}_2$-module structure.

#Open problems

In [CDM23], the authors consider polynomials related to Hurwitz numbers. These are conjectured to be real-rooted and interlace in a certain fashion [Conj. 3.14, CDM23]. The authors prove this statement for one-point Hurwitz numbers.

Gal’s conjecture [Gal05] says that $h$-polynomials of flag simplicial spheres are gamma-positive. It remains open in general. S. Park studies how PL homeomorphism constraints, especially the link condition for edge contractions, interact with gamma-vector growth for simplicial spheres [Par26]. For high-dimensional spheres with non-negative gamma-vector, the averaged constraints induced by the link condition are often already forced by the general $M$-vector inequalities, including when $g_1$ grows linearly with the dimension. In regimes where the constraints are nontrivial, they give lower bounds on ratios of adjacent $g$-vector components and hence on top gamma-vector components. The paper also interprets gamma-vector components through monomer–dimer covers and extends the positivity framework using orthogonal polynomials and lattice paths.

I. Gessel and Y. Zhuang [GZ24] give symmetric-function formulas for two-sided permutation statistics and pose several real-rootedness and gamma-positivity conjectures. In particular, they conjecture that the inverse peak and inverse left-peak distributions over permutations with a fixed value of the corresponding peak statistic are real-rooted, and that the polynomial \[\widehat{A}_{n,k}(t) \coloneqq \sum_{\substack{\pi\in\symS_n\\ \operatorname{pk}(\pi)=k}} t^{\operatorname{ides}(\pi)+1}\] is gamma-positive with center of symmetry $(n+1)/2.$ They also propose a stronger refinement by fixed pinnacle set.

#Symmetric decomposition

See [BS19] for the symmetric decomposition and its relationship with real-rootedness.

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