Multivariate Eulerian Polynomials

• Multivariate Eulerian polynomials are refined symmetric functions that generalize classical Eulerian polynomials by encoding detailed descent, ascent, and segmentation statistics.
• They leverage algebraic, combinatorial, and probabilistic techniques, including stability theory and context-free grammars, to reveal relationships between root geometry, log-concavity, and convexity properties.
• These polynomials have applications in algebraic combinatorics and complex systems, providing insights into exclusion processes, poset extensions, and operad structures.

A multivariate Eulerian polynomial is a refined symmetric or multiaffine polynomial generalizing the classical Eulerian polynomial, which encodes refined enumerative statistics of permutations, colored permutations, generalized Stirling permutations, poset extensions, segmented permutations, and several other combinatorial structures. Recent advances leverage algebraic, probabilistic, and combinatorial tools—particularly the theory of real-stable polynomials and context-free grammars—to establish deep relationships among root-locus geometry, log-concavity, γ-positivity, and rigidity properties of associated convex bodies. Multivariate Eulerian polynomials serve as a central object in algebraic combinatorics, with rigorous links to exclusion processes, spectrahedral relaxations, and operad structures.

1. Classical and Multivariate Eulerian Polynomials

The classical Eulerian polynomial $A_n(x)$ for the symmetric group $S_n$ is defined as

$A_n(x) = \sum_{\pi\in S_n} x^{\mathrm{des}(\pi)}$

where $\mathrm{des}(\pi)$ is the descent number of $\pi$. $A_n(x)$ is real-rooted—a property underpinned by its connection to stability theory.

Multivariate refinements encode further information beyond descent count, such as types and positions of descents, ascents, and values associated to these statistics. Notable multivariate forms include:

• Descent-top/ascend-top encoding: For $\sigma\in S_{n+1}$, define the descent-top set $$\mathcal{DT}(\sigma)=\{\sigma_i\mid \sigma_i > \sigma_{i+1}\}$$ and ascent-top set $$\mathcal{AT}(\sigma)=\{\sigma_{i+1}\mid \sigma_i<\sigma_{i+1}\}$$. The polynomial

$$A_n(\mathbf x,\mathbf y) = \sum_{\sigma\in S_{n+1}} \prod_{i\in\mathcal{DT}(\sigma)} x_i \prod_{j\in\mathcal{AT}(\sigma)} y_j$$

generalizes both descent and ascent tracking. The specialization $y_j=1$ yields a multivariate "real-zero" lifting $p_n(\mathbf x):=A_n(\mathbf x, \mathbf{1})$ (Nevado, 4 Jul 2025).

• Segmentation refinement: Segmented permutations introduce additional variables for descent/ascent tops adjacent to segmentation bars, producing a four-fold multivariate Eulerian family (Zhang et al., 2018).

For various Coxeter and wreath products, work of Brändén, Visontai, Williams, and others introduces multivariate Eulerian polynomials tracking generalized descent and top statistics, including in type $B$, colored and affine settings (Visontai et al., 2012, Brändén et al., 2014, Deligeorgaki et al., 2024).

2. Stability, Real-Rootedness, and Spectral Geometry

Stability is a multivariate polynomial property generalizing real-rootedness: a real-coefficient polynomial $f(z_1,\dots,z_m)$ is stable if $f(z_1,\dots,z_m)\neq 0$ whenever all $\mathrm{Im}(z_i)>0$. For $n=1$ this coincides with real-rootedness.

Key facts:

• All classical and multivariate Eulerian polynomials, including those for colored, signed, or multiset permutations, can be constructed or specialized to stable polynomials (Visontai et al., 2012, Brändén et al., 2014, Deligeorgaki et al., 2024).
• Context-free grammar methods provide recurrence operators (by differentiation and multiplication) that preserve stability; this is pivotal for inductive or recursive proofs of multivariate stability and for establishing univariate real-rootedness as a corollary (Chen et al., 2012, Nevado, 21 Jan 2026).
• Spectrahedral relaxations: Stable multivariate Eulerian polynomials admit interpretations as defining rigidly convex sets (RCS), which can be approximated by low-degree monic symmetric linear matrix pencils (MSLMPs). Accuracy can be assessed along the "diagonal" (setting $x_2=\cdots=x_{n+1}$), where one recovers the univariate Eulerian polynomials and their root bounds, sometimes improving known extremal estimates (Nevado, 4 Jul 2025).

3. Combinatorial and Algebraic Constructions

Several algebraic and combinatorial frameworks underpin the construction and interpretation of multivariate Eulerian polynomials:

• Context-free grammars: For each multivariate family, e.g., second-order or higher Eulerian, Legendre-Stirling, or segmented Eulerian polynomials, explicit grammars generate the multiaffine structure tracking each relevant statistic. Labels in permutations or generalized objects (such as Stirling or segmented permutations) correspond to variables in the polynomial via insertion rules, and stability is preserved under the associated differential operators (Chen et al., 2012, Zhang et al., 2018).
• Symmetric function expansions: k-th order Eulerian polynomials, particularly those connected to $(k+1)$-ary increasing trees, admit $e$-positive symmetric function expansions, with explicit combinatorial interpretation of coefficients in terms of degree profile of increasing plane trees (Ma et al., 2021).
• Dyck-path and algebraic models: The descent–ascent-bottom algebra quotient of the Malvenuto–Reutenauer algebra gives a new Dyck-path algebra whose product encodes the same multiplication as shuffle products of multivariate Eulerian polynomials, revealing deep links between permutations and lattice path combinatorics (Brändén et al., 2016).

4. Extensions: Colored, Segmental, and Poset Multivariate Eulerian Polynomials

• Colored and multiset Eulerian: Let $M_m$ be a multiset and $S_{M_m^r}$ the set of r-colored multiset permutations. The corresponding colored multiset Eulerian polynomial $A_{M_m^r}(x)$ encodes descent distributions, generalizing both MacMahon's and Brenti's classical variants. Symmetry, palindromicity, and self-interlacing properties yield bi-$\gamma$-positivity, log-concavity, unimodality, and real-rootedness (Deligeorgaki et al., 2024).
• Segmented permutations: Nunge’s generalization of Eulerian statistics to segmented permutations introduces bar statistics, leading to multivariate refinements with explicit four-variable structure, all stable. The interlacing and $q$-analog properties follow by action of stability-preserving differential operators on the classical multivariate base (Zhang et al., 2018).
• $P$-Eulerian (Poset) polynomials: For a finite labeled poset $P$, the multivariate $P$-Eulerian polynomial enumerates linear extensions by descent/ascent patterns, with two distinguished alphabets. For wide classes (including antichains, chains, and forestlike posets), these are stable, and hence all univariate specializations are real-rooted (Brändén et al., 2016).

5. Symmetric, Palindromic, and $\gamma$-Positivity Structures

• Palindromicity: The univariate palindromicity $x^nA_n(1/x) = A_n(x)$ lifts to multivariate settings via a mirror operator: the multivariate reciprocal of $A_n(x_2,\dots,x_{n+1})$ matches its image under variable reversal, yielding explicit combinatorial bijections among descent-top sets (Nevado, 21 Jan 2026).
• $\gamma$-positivity: Expansions of Eulerian polynomials in the basis $x^k(1+x)^{n-2k}$ imply nonnegativity of $\gamma$-coefficients, which can be interpreted as enumerating permutations by forbidden double descents, and for multivariate/trivariate polynomials, by joint statistics such as succession, fixed points, and excedances (Ma et al., 2020, Ma et al., 2021, Deligeorgaki et al., 2024).
• Extended polynomials: Higher order Eulerian polynomials and their multivariate expansions are $e$-positive, with explicit basis expansion in elementary symmetric functions and combinatorial tree interpretation of coefficients (Ma et al., 2021).

6. Applications and Connections

• Exclusion processes: The stationary distributions of partially asymmetric exclusion processes (ASEP) with various boundary conditions naturally interpolate multivariate Eulerian partition functions. The correspondence extends to stable families for colored permutations and their statistic refinements (Brändén et al., 2014).
• Negative dependence and Rayleigh properties: Multivariate stability implies that the associated measures are strongly Rayleigh, yielding negative dependence and correlation inequalities for systems modeled by Eulerian statistics (Brändén et al., 2014).
• Spectrahedral/certified convexity: Multivariate stable Eulerian polynomials define "ovaloids" via their zero-loci, and their associated RCSs can be efficiently approximated for use in convex geometry, with accuracy certified by the multiaffine structure and explicit MSLMP representations (Nevado, 4 Jul 2025).

7. Open Problems and Research Directions

• Type $D$ and affine multivariate Eulerian polynomials: Despite univariate real-rootedness results for types $A$, $B$, and $C$, the existence of genuinely stable multivariate refinements for type $D$ and affine types $\widetilde{B}$, $\widetilde{D}$ remains unresolved (Visontai et al., 2012).
• Uniform combinatorial interpretations: While grammar-based and symmetric function expansions exist for many families, direct combinatorial interpretations of certain $\gamma$-coefficients in colored or multiset cases are incomplete or only partially resolved (Deligeorgaki et al., 2024).
• Extension to new combinatorial structures: Connections between Eulerian statistics on generalized objects, such as Stirling, segmented, or $P$-partitions, and their stable, multivariate refinements continue to generate new research in algebraic, geometric, and probabilistic combinatorics (Chen et al., 2012, Zhang et al., 2018, Ma et al., 2021).

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Key References:

• W.Y.C. Chen, R.X.J. Hao & H.R.L. Yang: "Context-free Grammars and Multivariate Stable Polynomials over Stirling Permutations" (Chen et al., 2012)
• P. Brändén & M. Visontai: "Stable multivariate $W$-Eulerian polynomials" (Visontai et al., 2012)
• M. Leander & M. Visontai: "Multivariate Eulerian polynomials and exclusion processes" (Brändén et al., 2014)
• Z. Zhang & S. Zhang: "Multivariate Stable Eulerian Polynomials on Segmented Permutations" (Zhang et al., 2018)
• S. Ma, T. Ma, A. Yeh, H. Yeh: "Eulerian polynomials, Stirling permutations and increasing trees" (Ma et al., 2021)
• K. Kohnert: "Colored Multiset Eulerian Polynomials" (Deligeorgaki et al., 2024)
• P. Brändén & M. Leander: "Multivariate P-Eulerian polynomials" (Brändén et al., 2016)
• H. S. Eisenberg, F. Brändén: "Spectrahedral relaxations of Eulerian rigidly convex sets" (Nevado, 4 Jul 2025)
• R. Brändén, L. Nunge, "Palindromicity of multivariate Eulerian polynomials" (Nevado, 21 Jan 2026)