Stirling Polynomials: Definitions & Applications

• Stirling polynomials are families of polynomials defined via Stirling numbers, permutations, and their generalizations, playing a key role in combinatorial transformations and basis changes.
• They are constructed using generating functions, recurrence relations, and inversion formulas that connect classical enumerative techniques with modern analytic methods.
• Their applications span refining Eulerian-type statistics, Bell polynomial identities, Lagrange inversion, and grammatical calculus in contemporary mathematical research.

Stirling polynomials are polynomial families built from Stirling numbers, Stirling permutations, and their higher-order or multivariate analogues. The label does not designate a single canonical sequence: in current usage it includes the second-order Eulerian polynomials on Stirling permutations, polynomial-valued extensions of Stirling numbers of the first and second kind, diagonal Jacobi- and Legendre-Stirling objects, and multivariate first- and second-kind Stirling polynomials tied to Bell polynomials, Faà di Bruno theory, and Lagrange inversion (Ma et al., 19 Jun 2025, Kim et al., 2017, Gessel et al., 2012, Schreiber, 2013, Adell et al., 2024).

1. Terminological scope and basic definitions

The literature uses the term “Stirling polynomials” in several adjacent senses. Some authors reserve it for the second-order Eulerian polynomials $C_n(x)$ arising from Stirling permutations; others use it for polynomial refinements such as $S_2(n,k\mid x)$, $S_{2,r}(n,k\mid x)$, $s(n,m,Y)$, $S(n,m,Y)$, or for multivariate families $S_{n,k}$ and $B_{n,k}$ that specialize to the classical Stirling numbers of the first and second kind (Ishii et al., 18 Aug 2025, Kim et al., 2017, Schreiber, 2013).

Family	Defining relation	Source
Second-order Eulerian polynomials $C_n(x)$	$$C_n(x)=\sum_{\sigma\in Q_n}x^{\operatorname{asc}(\sigma)}=\sum_{\sigma\in Q_n}x^{\operatorname{des}(\sigma)}=\sum_{\sigma\in Q_n}x^{\operatorname{plat}(\sigma)}$$	(Ma et al., 19 Jun 2025)
Stirling polynomials of the second kind $S_2(n,k\mid x)$	$S_2(n,k\mid x)$0	(Kim et al., 2017)
Extended Stirling polynomials $S_2(n,k\mid x)$1	$S_2(n,k\mid x)$2	(Kim et al., 2017)
Stirling polynomials $S_2(n,k\mid x)$3, $S_2(n,k\mid x)$4	$S_2(n,k\mid x)$5, $S_2(n,k\mid x)$6	(Ishii et al., 18 Aug 2025)
Diagonal Jacobi-Stirling polynomials $S_2(n,k\mid x)$7	$S_2(n,k\mid x)$8	(Gessel et al., 2012)

A persistent source of confusion is that the same notation may denote different variable orderings or different specializations. The data show, for example, both univariate Stirling-permutation enumerators and multivariate Bell-polynomial-type objects being called Stirling polynomials. The common thread is not a single formula but the role of Stirling structures as coefficient arrays, basis-change operators, or descent-generating functions.

2. Stirling permutations and second-order Eulerian polynomials

One central meaning of Stirling polynomial is the second-order Eulerian polynomial attached to classical Stirling permutations. A Stirling permutation is a permutation of the multiset $S_2(n,k\mid x)$9 in which, for each $S_{2,r}(n,k\mid x)$0, all entries between the two $S_{2,r}(n,k\mid x)$1’s are at least $S_{2,r}(n,k\mid x)$2. On this set, ascents, descents, and plateaux are equidistributed, and their common enumerator is

$S_{2,r}(n,k\mid x)$3

These polynomials satisfy

$S_{2,r}(n,k\mid x)$4

and admit a convolution formula

$S_{2,r}(n,k\mid x)$5

together with a lower Hessenberg determinantal expression (Ma et al., 19 Jun 2025).

Trivariate and higher refinements mark the joint distribution of ascents, descents, and plateaux. One such refinement satisfies Dumont’s symmetric derivative recurrence

$S_{2,r}(n,k\mid x)$6

and later work develops eight-variable and seventeen-variable expansions generalizing the trivariate second-order Eulerian and ascent-plateau polynomials (Ma et al., 19 Jun 2025, Ma et al., 2024).

Recent work on Stirling permutation codes converts these polynomial questions into code-theoretic and tree-theoretic ones. The code formalism yields equidistribution results such as up-down-pair with ascent-plateau and exterior up-down-pair with left ascent-plateau, proves that six bivariable set-valued statistics are equidistributed on the set of Stirling permutations, and gives $S_{2,r}(n,k\mid x)$7-positive expansions for enumerators by left ascent-plateaux, exterior up-down-pairs, and right plateau-descents. The sequel establishes a strong connection between signed permutations in the hyperoctahedral group and Stirling permutations, derives expansion formulas for the joint distribution of type $S_{2,r}(n,k\mid x)$8 and type $S_{2,r}(n,k\mid x)$9 descent statistics, and proves an interlacing property for coefficient polynomials (Ma et al., 2022, Ma et al., 2024).

The second-order Eulerian interpretation has also expanded to refined Eulerian-type objects. By introducing the statistics proper ascent-plateau, improper ascent-plateau, and trace, a six-variable Eulerian-type polynomial over a class of restricted Stirling permutations is shown to equal a six-variable Eulerian-type polynomial over signed permutations; suitable parametrizations then produce unified interpretations of $s(n,m,Y)$0-Eulerian polynomials and derangement polynomials of types $s(n,m,Y)$1 and $s(n,m,Y)$2 (Ma et al., 19 Jun 2025).

3. Polynomial extensions of Stirling numbers

A second major strand studies polynomial families whose coefficients or basis expansions are Stirling numbers. The classical Stirling polynomials of the second kind are defined by

$s(n,m,Y)$3

and the extended Stirling polynomials of the second kind by

$s(n,m,Y)$4

When $s(n,m,Y)$5, $s(n,m,Y)$6 reduces to the classical polynomial $s(n,m,Y)$7; when $s(n,m,Y)$8, $s(n,m,Y)$9 is the extended Stirling number of the second kind. These extended polynomials satisfy

$S(n,m,Y)$0

and are linked to the extended Bell polynomials by

$S(n,m,Y)$1

They also satisfy the Poisson-moment identity

$S(n,m,Y)$2

for a Poisson random variable $S(n,m,Y)$3 with parameter $S(n,m,Y)$4 (Kim et al., 2017).

A broader basis-change theory is obtained by attaching Stirling numbers to an arbitrary polynomial sequence $S(n,m,Y)$5 with $S(n,m,Y)$6. The associated Stirling numbers of the second kind are defined by

$S(n,m,Y)$7

and those of the first kind by

$S(n,m,Y)$8

Umbral calculus yields generating functions, inverse relations, and orthogonality identities, recovering classical Stirling numbers, Lah numbers, Gould-Hopper numbers, central factorial numbers, Bell, Bernoulli, Euler, Mittag-Leffler, and Laguerre examples within one framework (Kim et al., 2022).

A still more general construction is provided by the $S(n,m,Y)$9-Stirling numbers associated to potential polynomials. If

$S_{n,k}$0

then the coefficients in the monomial and falling-factorial expansions

$S_{n,k}$1

define the $S_{n,k}$2-Stirling numbers of the first and second kind. Their generating functions are

$S_{n,k}$3

and the two kinds are related through the classical Stirling numbers by

$S_{n,k}$4

This framework includes partial and complete Bell polynomials, degenerate and probabilistic Stirling numbers, $S_{n,k}$5-restricted Stirling numbers, and Lah numbers (Adell et al., 2024).

4. Multivariate, grammatical, and inversion-theoretic frameworks

A distinct multivariate theory replaces scalar Stirling numbers by homogeneous and isobaric polynomials. In this setting, the partial exponential Bell polynomials $S_{n,k}$6 serve as multivariate Stirling polynomials of the second kind, while a new family $S_{n,k}$7 specializes to the signed Stirling numbers of the first kind. The central inversion law is

$S_{n,k}$8

The families satisfy recurrences paralleling the classical ones and are connected by a Schlömilch-type formula expressing $S_{n,k}$9 in terms of Bell polynomials and conversely. Substituting $B_{n,k}$0 for the indeterminates $B_{n,k}$1 turns both families into differential polynomials depending on $B_{n,k}$2 and on its inverse $B_{n,k}$3, thereby connecting them to Comtet’s treatment of Lagrange inversion and to the Faà di Bruno Hopf algebra discussed by Haiman and Schmitt (Schreiber, 2013).

Context-free grammars provide another multivariate mechanism. For Legendre-Stirling permutations and marked Stirling permutations, explicit grammars generate multivariate refinements

$B_{n,k}$4

and

$B_{n,k}$5

These multivariate polynomials are stable, and the proof proceeds by showing that the differential operators induced by the grammars preserve stability for multiaffine polynomials, using the Borcea–Brändén theory. As a consequence, univariate specializations such as $B_{n,k}$6 and $B_{n,k}$7 are real-rooted. The construction resolves two problems posed by Haglund and Visontai about stable multivariate refinements (Chen et al., 2012).

Grammar calculus also enters the binomial-Stirling-Eulerian theory. The polynomials

$B_{n,k}$8

incorporate two Stirling statistics, left-to-right minima and right-to-left minima, together with Eulerian statistics. They admit an expansion

$B_{n,k}$9

and their $C_n(x)$0-positivity is proved both by Chen’s grammatical calculus and by a new group action on permutations, extending work of Shareshian and Wachs, Chung–Graham–Knuth, and Postnikov–Reiner–Williams (Ji et al., 2023).

5. Specialized combinatorial variants

Jacobi-Stirling polynomials arise from the diagonal Jacobi-Stirling numbers of the second kind. Writing

$C_n(x)$1

one obtains coefficient polynomials $C_n(x)$2 of degree $C_n(x)$3. Their diagonal generating functions have the form

$C_n(x)$4

where $C_n(x)$5 has nonnegative integer coefficients. Stanley’s $C_n(x)$6-partition theory identifies $C_n(x)$7 with the number of Jacobi-Stirling permutations having $C_n(x)$8 descents, and $C_n(x)$9 yields the Legendre-Stirling case. The associated numerator polynomials are conjectured, and in some cases proved, to be real-rooted (Gessel et al., 2012).

Stirling permutations on multisets lead to another notion of Stirling polynomial. For a multiset type $$C_n(x)=\sum_{\sigma\in Q_n}x^{\operatorname{asc}(\sigma)}=\sum_{\sigma\in Q_n}x^{\operatorname{des}(\sigma)}=\sum_{\sigma\in Q_n}x^{\operatorname{plat}(\sigma)}$$0 with $$C_n(x)=\sum_{\sigma\in Q_n}x^{\operatorname{asc}(\sigma)}=\sum_{\sigma\in Q_n}x^{\operatorname{des}(\sigma)}=\sum_{\sigma\in Q_n}x^{\operatorname{plat}(\sigma)}$$1, if $$C_n(x)=\sum_{\sigma\in Q_n}x^{\operatorname{asc}(\sigma)}=\sum_{\sigma\in Q_n}x^{\operatorname{des}(\sigma)}=\sum_{\sigma\in Q_n}x^{\operatorname{plat}(\sigma)}$$2 counts Stirling permutations with $$C_n(x)=\sum_{\sigma\in Q_n}x^{\operatorname{asc}(\sigma)}=\sum_{\sigma\in Q_n}x^{\operatorname{des}(\sigma)}=\sum_{\sigma\in Q_n}x^{\operatorname{plat}(\sigma)}$$3 descents, then the rational generating functions

$$C_n(x)=\sum_{\sigma\in Q_n}x^{\operatorname{asc}(\sigma)}=\sum_{\sigma\in Q_n}x^{\operatorname{des}(\sigma)}=\sum_{\sigma\in Q_n}x^{\operatorname{plat}(\sigma)}$$4

$$C_n(x)=\sum_{\sigma\in Q_n}x^{\operatorname{asc}(\sigma)}=\sum_{\sigma\in Q_n}x^{\operatorname{des}(\sigma)}=\sum_{\sigma\in Q_n}x^{\operatorname{plat}(\sigma)}$$5

define Stirling polynomials of the second and first kind. When all multiplicities equal $$C_n(x)=\sum_{\sigma\in Q_n}x^{\operatorname{asc}(\sigma)}=\sum_{\sigma\in Q_n}x^{\operatorname{des}(\sigma)}=\sum_{\sigma\in Q_n}x^{\operatorname{plat}(\sigma)}$$6, these reduce to classical Stirling numbers: $$C_n(x)=\sum_{\sigma\in Q_n}x^{\operatorname{asc}(\sigma)}=\sum_{\sigma\in Q_n}x^{\operatorname{des}(\sigma)}=\sum_{\sigma\in Q_n}x^{\operatorname{plat}(\sigma)}$$7 and $$C_n(x)=\sum_{\sigma\in Q_n}x^{\operatorname{asc}(\sigma)}=\sum_{\sigma\in Q_n}x^{\operatorname{des}(\sigma)}=\sum_{\sigma\in Q_n}x^{\operatorname{plat}(\sigma)}$$8. The authors construct $$C_n(x)=\sum_{\sigma\in Q_n}x^{\operatorname{asc}(\sigma)}=\sum_{\sigma\in Q_n}x^{\operatorname{des}(\sigma)}=\sum_{\sigma\in Q_n}x^{\operatorname{plat}(\sigma)}$$9-Stirling posets with

$S_2(n,k\mid x)$0

thereby extending the $S_2(n,k\mid x)$1-partition interpretation, and they also introduce Stirling numbers of odd type and generalizations of the central factorial numbers (Dzhumadil'daev et al., 2013).

A more arithmetic specialization occurs for the numbers $S_2(n,k\mid x)$2. A question posed in 1960 by D.S. Mitrinović and R.S. Mitrinović asked whether these numbers could always be expressed באמצעות specific Stirling polynomials. The answer is affirmative: for every $S_2(n,k\mid x)$3 there exist an integer $S_2(n,k\mid x)$4 and a primitive polynomial $S_2(n,k\mid x)$5 such that $S_2(n,k\mid x)$6 is given for all $S_2(n,k\mid x)$7 by an explicit formula involving $S_2(n,k\mid x)$8, $S_2(n,k\mid x)$9, a parity factor $S_2(n,k\mid x)$00, and $S_2(n,k\mid x)$01; moreover $S_2(n,k\mid x)$02 for all $S_2(n,k\mid x)$03 (Bencherif et al., 2014).

6. Analytic, arithmetic, and algebraic applications

Stirling polynomials interact strongly with Bernoulli, Euler, and related special polynomials. Using Faà di Bruno’s formula and Bell polynomials, higher-order Bernoulli and Euler polynomials admit closed formulas and determinant expressions in terms of Stirling numbers of the second kind. For example,

$S_2(n,k\mid x)$04

and

$S_2(n,k\mid x)$05

In this setting, the identity $S_2(n,k\mid x)$06 is the bridge between Bell polynomials and Stirling numbers (Dağlı, 2020).

Related formulas express Bernoulli, poly-Bernoulli, and Cauchy polynomials directly through Stirling and $S_2(n,k\mid x)$07-Stirling numbers. Examples include

$S_2(n,k\mid x)$08

together with poly-Bernoulli and Cauchy analogues, while high-order Bernoulli polynomials of both kinds at integer arguments are given by explicit finite sums involving $S_2(n,k\mid x)$09-Stirling numbers and binomial coefficients (Boyadzhiev, 2016, Mihoubi et al., 2014).

The Stirling transform organizes many additional identities. If $S_2(n,k\mid x)$10, then $S_2(n,k\mid x)$11, and this mechanism yields formulas connecting Stirling numbers with hyperharmonic numbers, derangement numbers, Bernoulli and Euler polynomials, powers, and factorials. In particular, the geometric polynomials

$S_2(n,k\mid x)$12

appear as a Stirling-transform counterpart of ordered Bell numbers and related enumerators (Boyadzhiev, 2020).

Polynomial extensions also enter arithmetic questions. The Stirling functions

$S_2(n,k\mid x)$13

satisfy $S_2(n,k\mid x)$14 on the zero set of a polynomial $S_2(n,k\mid x)$15. For $S_2(n,k\mid x)$16, the real roots of $S_2(n,k\mid x)$17 are simple, all lie in $S_2(n,k\mid x)$18, and are exactly $S_2(n,k\mid x)$19 and, when $S_2(n,k\mid x)$20 is even, $S_2(n,k\mid x)$21. These functions are then used to study divisibility of $S_2(n,k\mid x)$22 and to prove a generalization of Wilson’s theorem (Williams, 2016).

Recent work also uses Stirling polynomials to compute special values of multiple zeta functions and multiple zeta star functions at non-positive integers. With

$S_2(n,k\mid x)$23

the reverse values of multiple zeta functions are expressed as sums over products of the Stirling polynomials $S_2(n,k\mid x)$24, factorial ratios, and lower-depth reverse values at zero. The same framework relates these values to Bernoulli numbers and to generalized Gregory coefficients (Ishii et al., 18 Aug 2025).

Outside classical special-function theory, Stirling numbers govern coefficients of braid matroid Kazhdan–Lusztig polynomials: for braid matroids, restricted Whitney numbers of the first kind become Stirling numbers of the first kind, yielding a non-recursive formula for Kazhdan–Lusztig coefficients and new identities between the two kinds of Stirling numbers (Karn et al., 2018). In Coxeter-theoretic direction, type $S_2(n,k\mid x)$25 $S_2(n,k\mid x)$26-Stirling numbers are expressed by complete homogeneous and elementary symmetric polynomials,

$S_2(n,k\mid x)$27

and are connected to conjectural Hilbert series for super coinvariant algebras (Sagan et al., 2022).

Taken together, these developments show that Stirling polynomials are less a single object than a recurrent algebraic pattern: they mediate between permutation and partition statistics, basis changes among polynomial sequences, stability and real-rootedness phenomena, $S_2(n,k\mid x)$28-partition enumerators, and analytic expressions for zeta-type and Bernoulli-type special values.