Eulerian Polynomials: Structure & Extensions

• Eulerian polynomials are generating functions that encode descent and excedance statistics on permutations and related Coxeter-theoretic objects.
• Their structural properties—such as palindromicity, real-rootedness, unimodality, and γ-positivity—yield strong combinatorial and analytic implications.
• Recent extensions include multivariate, q-analog, and geometric generalizations, connecting Eulerian theory to poset refinements, Stirling permutations, and orthogonal polynomials.

Eulerian polynomials are generating polynomials for descent- and excedance-type statistics on permutations and, more broadly, on a wide range of Coxeter-theoretic and combinatorial objects. In the classical case one writes

$$A_n(t)=\sum_{\sigma\in \mathfrak S_n} t^{\operatorname{des}(\sigma)} =\sum_{\sigma\in \mathfrak S_n} t^{\operatorname{exc}(\sigma)},$$

and equivalently

$$\sum_{k\ge 0}(k+1)^n t^k=\frac{A_n(t)}{(1-t)^{n+1}}.$$

Modern usage extends the term to multivariate, $q$-, type $B$, $P$-, second-order, and other Eulerian-type families arising from Stirling permutations, posets, signed permutations, Weyl groups, multiset permutations, and related structures (Shareshian et al., 2017, Ma et al., 19 Jun 2025).

1. Classical definitions and normalizations

The classical Eulerian numbers $A(n,k)$ count permutations of $[n]$ with exactly $k$ descents, and the classical Eulerian polynomial is

$$A_n(t)=\sum_{k=0}^{n-1}A(n,k)t^k =\sum_{\sigma\in\mathfrak S_n} t^{\operatorname{des}(\sigma)}.$$

MacMahon’s excedance interpretation gives the equivalent form

$$A_n(t)=\sum_{\sigma\in\mathfrak S_n} t^{\operatorname{exc}(\sigma)},$$

and the exponential generating function is

$$\sum_{k\ge 0}(k+1)^n t^k=\frac{A_n(t)}{(1-t)^{n+1}}.$$0

A standard recurrence is

$$\sum_{k\ge 0}(k+1)^n t^k=\frac{A_n(t)}{(1-t)^{n+1}}.$$1

and a classical Worpitzky identity is

$$\sum_{k\ge 0}(k+1)^n t^k=\frac{A_n(t)}{(1-t)^{n+1}}.$$2

These formulas place Eulerian polynomials simultaneously in enumerative combinatorics, generating-function theory, and finite-difference calculus (Shareshian et al., 2017, Iijima et al., 2016).

The literature uses several normalizations. One common convention is $$\sum_{k\ge 0}(k+1)^n t^k=\frac{A_n(t)}{(1-t)^{n+1}}.$$3; another shifts the exponent and writes $$\sum_{k\ge 0}(k+1)^n t^k=\frac{A_n(t)}{(1-t)^{n+1}}.$$4. The same coefficient triangle also appears in “descending power” form

$$\sum_{k\ge 0}(k+1)^n t^k=\frac{A_n(t)}{(1-t)^{n+1}}.$$5

and some analytic treatments keep the generating function as primary and consequently write objects denoted $$\sum_{k\ge 0}(k+1)^n t^k=\frac{A_n(t)}{(1-t)^{n+1}}.$$6 that are rational rather than polynomial under that normalization. For technical work, the normalization must therefore be checked explicitly rather than inferred from notation alone (Brändén et al., 2016, Barry, 2011, Kim et al., 2016).

2. Structural properties

Eulerian polynomials are palindromic in the standard shifted sense, and Frobenius proved that all zeros of $$\sum_{k\ge 0}(k+1)^n t^k=\frac{A_n(t)}{(1-t)^{n+1}}.$$7 are real. Real-rootedness implies unimodality and log-concavity of the coefficient sequence, and it remains one of the central analytic signatures of the classical family (Brändén et al., 2016).

A stronger structural statement is $$\sum_{k\ge 0}(k+1)^n t^k=\frac{A_n(t)}{(1-t)^{n+1}}.$$8-positivity. For classical Eulerian polynomials one has the Foata–Schützenberger expansion

$$\sum_{k\ge 0}(k+1)^n t^k=\frac{A_n(t)}{(1-t)^{n+1}}.$$9

where $q$0 counts permutations with no double descents, no final descent, and $q$1. The same paper develops binomial–Eulerian polynomials $q$2, proves analogous $q$3-positive expansions for them, and extends both constructions to $q$4-analogs $q$5 and $q$6, yielding $q$7-$q$8-positivity and hence $q$9-unimodality (Shareshian et al., 2017).

A further refinement is bi-$B$0-positivity and ratio monotonicity for Eulerian-type recurrences

$B$1

If $B$2, then $B$3 is bi-$B$4-positive and the reciprocal polynomial $B$5 is ratio monotone. This framework covers several Eulerian-type families, including $B$6 $B$7-Eulerian polynomials, $B$8-Eulerian polynomials, type $B$9 $P$0-Eulerian polynomials, generalized Carlitz–Scoville Eulerian polynomials, and $P$1-colored Eulerian polynomials (Liu et al., 3 Sep 2025).

3. Posets, multivariate refinements, and stability

Stanley’s $P$2-Eulerian polynomial generalizes $P$3 from permutations to linear extensions of a labeled poset $P$4: $P$5 A multivariate refinement records ascent and descent bottoms through separate variables,

$P$6

Brändén and Leander show that for suitable poset classes these multivariate $P$7-Eulerian polynomials are stable, meaning nonvanishing whenever all variables lie in the open upper half-plane. Their proofs use the Malvenuto–Reutenauer algebra, stability-preserving linear operators, and an isomorphic algebra on Dyck paths. Stability is preserved under disjoint unions, interleaved disjoint unions, certain ordinal sums, and holds in particular for naturally labeled decreasing forests and their duals. The paper also emphasizes that the Neggers–Stanley conjecture is false in general, even though many important classes retain real-rootedness or stability (Brändén et al., 2016).

This multivariate viewpoint turns Eulerian polynomials into a model case for the broader theory of stable polynomials. It makes visible interactions among descent statistics that are invisible after univariate specialization, and it explains why real-rootedness, unimodality, and related inequalities persist under a number of algebraic constructions (Brändén et al., 2016).

4. Second-order Eulerian polynomials and Stirling permutations

A major extension replaces permutations by classical Stirling permutations $P$8 of the multiset $P$9. The second-order Eulerian polynomials $A(n,k)$0 are defined analytically by

$A(n,k)$1

and combinatorially by

$A(n,k)$2

Thus ascents, descents, and plateaux are equidistributed on $A(n,k)$3. The trivariate refinement

$A(n,k)$4

satisfies Dumont’s differential recursion and is symmetric in $A(n,k)$5 (Ma et al., 19 Jun 2025).

Recent work develops this second-order theory much further. It gives a simplified convolution formula for $A(n,k)$6, derives a lower Hessenberg determinantal expression, studies trivariate and six-variable refinements, and introduces the statistics proper ascent-plateau, improper ascent-plateau, and trace on restricted Stirling permutations. A six-variable Eulerian-type polynomial on restricted Stirling permutations is shown to coincide with a six-variable Eulerian-type polynomial on signed permutations, and specializations provide unified Stirling-permutation models for $A(n,k)$7-Eulerian polynomials and derangement polynomials of types $A(n,k)$8 and $A(n,k)$9. The same work also presents a box sorting algorithm leading to a bijection between the terms in the expansion of $[n]$0 and ordered weak set partitions, and then, via standard Young tableaux and grammars, gives three interpretations of the second-order Eulerian polynomials (Ma et al., 19 Jun 2025).

A complementary development introduces Stirling permutations of the second kind. If $[n]$1 denotes these cycle-structured objects, then the second-order Eulerian polynomial also appears as

$[n]$2

Moreover, cyclic Stirling permutations of the second kind counted by cycle ascent plateaux recover the classical Eulerian polynomial through

$[n]$3

The same paper links Eulerian polynomials of types $[n]$4 and $[n]$5 to perfect matching polynomials and to inversion-sequence polynomials $[n]$6 (Ma et al., 2016).

5. Algebraic, geometric, and arithmetic avatars

Eulerian polynomials admit several non-combinatorial realizations. Using exponential Riordan arrays, the “descending power” Eulerian polynomials $[n]$7 and the shifted sequence $[n]$8 are shown to be moment sequences for explicit families of monic orthogonal polynomials. This yields three-term recurrences, Jacobi continued fractions for the ordinary generating functions, and closed product formulas for the corresponding Hankel transforms (Barry, 2011).

From the generating function

$[n]$9

one can derive nonlinear differential equations whose repeated differentiation expresses $k$0 as a polynomial in $k$1. Comparing this with the Taylor expansion of $k$2 gives explicit identities connecting shifted Eulerian polynomials $k$3 with higher-order Eulerian polynomials $k$4, defined by

$k$5

These identities also translate into formulas for weighted power sums $k$6 (Kim et al., 2016).

Eulerian polynomials also admit rigidity statements in algebraic form. One such result is the congruence

$k$7

valid for $k$8. More strongly, among monic degree-$k$9 polynomials this congruence characterizes the Eulerian polynomial: if a monic degree-$$A_n(t)=\sum_{k=0}^{n-1}A(n,k)t^k =\sum_{\sigma\in\mathfrak S_n} t^{\operatorname{des}(\sigma)}.$$0 polynomial satisfies it for some $$A_n(t)=\sum_{k=0}^{n-1}A(n,k)t^k =\sum_{\sigma\in\mathfrak S_n} t^{\operatorname{des}(\sigma)}.$$1, then it must equal $$A_n(t)=\sum_{k=0}^{n-1}A(n,k)t^k =\sum_{\sigma\in\mathfrak S_n} t^{\operatorname{des}(\sigma)}.$$2 (Iijima et al., 2016).

In Lie-theoretic and geometric settings, Eulerian polynomials arise through Weyl arrangements. For a Weyl subarrangement $$A_n(t)=\sum_{k=0}^{n-1}A(n,k)t^k =\sum_{\sigma\in\mathfrak S_n} t^{\operatorname{des}(\sigma)}.$$3, an $$A_n(t)=\sum_{k=0}^{n-1}A(n,k)t^k =\sum_{\sigma\in\mathfrak S_n} t^{\operatorname{des}(\sigma)}.$$4-Eulerian polynomial is defined by

$$A_n(t)=\sum_{k=0}^{n-1}A(n,k)t^k =\sum_{\sigma\in\mathfrak S_n} t^{\operatorname{des}(\sigma)}.$$5

and, for compatible $$A_n(t)=\sum_{k=0}^{n-1}A(n,k)t^k =\sum_{\sigma\in\mathfrak S_n} t^{\operatorname{des}(\sigma)}.$$6, the characteristic quasi-polynomial satisfies

$$A_n(t)=\sum_{k=0}^{n-1}A(n,k)t^k =\sum_{\sigma\in\mathfrak S_n} t^{\operatorname{des}(\sigma)}.$$7

where $$A_n(t)=\sum_{k=0}^{n-1}A(n,k)t^k =\sum_{\sigma\in\mathfrak S_n} t^{\operatorname{des}(\sigma)}.$$8 is the shift operator and $$A_n(t)=\sum_{k=0}^{n-1}A(n,k)t^k =\sum_{\sigma\in\mathfrak S_n} t^{\operatorname{des}(\sigma)}.$$9 is the Ehrhart quasi-polynomial of the closed fundamental alcove. When $$A_n(t)=\sum_{\sigma\in\mathfrak S_n} t^{\operatorname{exc}(\sigma)},$$0, $$A_n(t)=\sum_{\sigma\in\mathfrak S_n} t^{\operatorname{exc}(\sigma)},$$1 is the Lam–Postnikov generalized Eulerian polynomial $$A_n(t)=\sum_{\sigma\in\mathfrak S_n} t^{\operatorname{exc}(\sigma)},$$2; in type $$A_n(t)=\sum_{\sigma\in\mathfrak S_n} t^{\operatorname{exc}(\sigma)},$$3, $$A_n(t)=\sum_{\sigma\in\mathfrak S_n} t^{\operatorname{exc}(\sigma)},$$4 is the classical Eulerian polynomial $$A_n(t)=\sum_{\sigma\in\mathfrak S_n} t^{\operatorname{exc}(\sigma)},$$5 (Ashraf et al., 2019).

6. Generalized families and recent extensions

The scope of Eulerian theory now includes many additional families. For multipermutations of $$A_n(t)=\sum_{\sigma\in\mathfrak S_n} t^{\operatorname{exc}(\sigma)},$$6 and their signed analogues, the polynomials $$A_n(t)=\sum_{\sigma\in\mathfrak S_n} t^{\operatorname{exc}(\sigma)},$$7, $$A_n(t)=\sum_{\sigma\in\mathfrak S_n} t^{\operatorname{exc}(\sigma)},$$8, $$A_n(t)=\sum_{\sigma\in\mathfrak S_n} t^{\operatorname{exc}(\sigma)},$$9, and $$\sum_{k\ge 0}(k+1)^n t^k=\frac{A_n(t)}{(1-t)^{n+1}}.$$00 are real-rooted; moreover $$\sum_{k\ge 0}(k+1)^n t^k=\frac{A_n(t)}{(1-t)^{n+1}}.$$01, $$\sum_{k\ge 0}(k+1)^n t^k=\frac{A_n(t)}{(1-t)^{n+1}}.$$02, and $$\sum_{k\ge 0}(k+1)^n t^k=\frac{A_n(t)}{(1-t)^{n+1}}.$$03 are bi-$$\sum_{k\ge 0}(k+1)^n t^k=\frac{A_n(t)}{(1-t)^{n+1}}.$$04-positive and therefore unimodal with modes in the middle. Interlacing relations connect these families and unify their proofs of real-rootedness and unimodality (Ma et al., 2019).

Segmented permutations produce another two-parameter extension. If $$\sum_{k\ge 0}(k+1)^n t^k=\frac{A_n(t)}{(1-t)^{n+1}}.$$05 records descents and segmentations, then

$$\sum_{k\ge 0}(k+1)^n t^k=\frac{A_n(t)}{(1-t)^{n+1}}.$$06

where $$\sum_{k\ge 0}(k+1)^n t^k=\frac{A_n(t)}{(1-t)^{n+1}}.$$07 denotes ordered Bell polynomials, and the exponential generating function is

$$\sum_{k\ge 0}(k+1)^n t^k=\frac{A_n(t)}{(1-t)^{n+1}}.$$08

This family comes with a noncommutative lift in the algebra of segmented compositions and satisfies a Worpitzky-type relation (Nunge, 2018).

Other generalizations are defined by modifying the ambient combinatorial or analytic data. General Eulerian numbers $$\sum_{k\ge 0}(k+1)^n t^k=\frac{A_n(t)}{(1-t)^{n+1}}.$$09 and polynomials $$\sum_{k\ge 0}(k+1)^n t^k=\frac{A_n(t)}{(1-t)^{n+1}}.$$10 are attached to arithmetic progressions and extend Worpitzky-type identities, finite power-sum formulas, and exponential generating functions (Xiong et al., 2012). The $$\sum_{k\ge 0}(k+1)^n t^k=\frac{A_n(t)}{(1-t)^{n+1}}.$$11-Eulerian polynomials

$$\sum_{k\ge 0}(k+1)^n t^k=\frac{A_n(t)}{(1-t)^{n+1}}.$$12

refine Eulerian theory by fixed points and major index, admit symmetric $$\sum_{k\ge 0}(k+1)^n t^k=\frac{A_n(t)}{(1-t)^{n+1}}.$$13-Eulerian identities, and support a new recurrence formula (Lin, 2012). Two-parameter and Dirichlet-type constructions connect Eulerian polynomials to Bernstein polynomials, polylogarithms, Bernoulli and Euler numbers, $$\sum_{k\ge 0}(k+1)^n t^k=\frac{A_n(t)}{(1-t)^{n+1}}.$$14-adic $$\sum_{k\ge 0}(k+1)^n t^k=\frac{A_n(t)}{(1-t)^{n+1}}.$$15-integrals, and Eulerian $$\sum_{k\ge 0}(k+1)^n t^k=\frac{A_n(t)}{(1-t)^{n+1}}.$$16-functions (Araci et al., 2012, Araci et al., 2012).

Recent work continues this expansion. Colored multiset Eulerian polynomials form a common generalization of MacMahon’s multiset Eulerian polynomials and colored Eulerian polynomials; the symmetric cases are characterized, and sufficient conditions are given for self-interlacing, which in turn implies real-rootedness, log-concavity, unimodality, the alternatingly increasing property, and bi-$$\sum_{k\ge 0}(k+1)^n t^k=\frac{A_n(t)}{(1-t)^{n+1}}.$$17-positivity (Deligeorgaki et al., 2024). In parallel, remixed Eulerian numbers supply a $$\sum_{k\ge 0}(k+1)^n t^k=\frac{A_n(t)}{(1-t)^{n+1}}.$$18-deformation of Postnikov’s mixed Eulerian numbers, recover $$\sum_{k\ge 0}(k+1)^n t^k=\frac{A_n(t)}{(1-t)^{n+1}}.$$19-binomial coefficients, Carlitz $$\sum_{k\ge 0}(k+1)^n t^k=\frac{A_n(t)}{(1-t)^{n+1}}.$$20-Eulerian polynomials, and Garsia–Remmel $$\sum_{k\ge 0}(k+1)^n t^k=\frac{A_n(t)}{(1-t)^{n+1}}.$$21-hit numbers as special cases, and are shown to be symmetric and unimodal as polynomials in $$\sum_{k\ge 0}(k+1)^n t^k=\frac{A_n(t)}{(1-t)^{n+1}}.$$22 (Nadeau et al., 2022).

Taken together, these developments show that Eulerian polynomials are no longer a single sequence but a large and interconnected class of generating functions. Classical descent enumeration remains the prototype, but the contemporary theory encompasses stability, $$\sum_{k\ge 0}(k+1)^n t^k=\frac{A_n(t)}{(1-t)^{n+1}}.$$23-positivity, orthogonality, hyperplane arrangements, Stirling permutations, multiset and colored analogues, and several analytic and arithmetic deformations, all organized around the same basic principle: a polynomial encoding the distribution of ascent-, descent-, or excedance-like statistics on a structured combinatorial family.