Bi-Stirling-Euler-Mahonian Polynomials

• Bi-Stirling-Euler-Mahonian polynomials are comprehensive generating functions combining classical and Stirling permutation statistics, capturing descents, minima, and Mahonian parameters.
• They unify classical sequences like Carlitz–Scoville bi-Eulerian numbers and Butler’s Stirling–Euler–Mahonian numbers through parameter specializations and q-analogs.
• The study leverages context-free grammars and combinatorial proofs to reveal symmetric decompositions and γ-positivity, essential for understanding unimodality and structural properties.

The bi-Stirling-Euler-Mahonian polynomials constitute a comprehensive family of enumerative generating functions intertwining classical and Stirling permutation statistics, encapsulating descent, minimum, and Mahonian-type parameters. They form a unifying framework generalizing the bi-Eulerian polynomials of Carlitz–Scoville, Butler’s Stirling–Euler–Mahonian numbers, and recent remixed Eulerian constructions, and admit deep connections with symmetric decompositions, γ-positivity, and context-free grammars on the associated permutation and Stirling permutation sets (Ma et al., 23 Jul 2025, Xu et al., 2 Sep 2025).

1. Key Statistics and Definitions

Let $S_n$ denote the symmetric group on $n$ symbols, and $Q_n^{(k)}$ the set of $k$-Stirling permutations of order $n$—that is, sequences in which each of $1,\ldots,n$ appears $k$ times, subject to Gessel–Stanley’s plateau condition.

For $\sigma\in S_n$ or $\sigma\in Q_n^{(k)}$, relevant statistics include:

• Descent: $\des(\sigma) = \#\{i : \sigma_i > \sigma_{i+1}\}$.
• Left-to-right minima: $n$0.
• Right-to-left minima: $n$1.
• Longest ascent-plateaux: For $n$2, indices $n$3 with $n$4; count denoted $n$5.
• Longest left ascent-plateaux: As above, but with $n$6, so $n$7 is allowed; count is $n$8.
• Proper and improper ascent-plateaux on $n$9: Proper if all values $Q_n^{(k)}$0 occur before $Q_n^{(k)}$1, improper otherwise; counted by $Q_n^{(k)}$2 and $Q_n^{(k)}$3.

For Mahonian refinements, the statistics include

• Major index: $Q_n^{(k)}$4.
• Mixed major index: For parameters $Q_n^{(k)}$5,

$Q_n^{(k)}$6

2. Bi-Stirling-Euler-Mahonian and Related Generating Polynomials

The principal objects of study are four-variable generating polynomials encoding generalized permutation statistics:

$Q_n^{(k)}$7

where

$Q_n^{(k)}$8

and $Q_n^{(k)}$9 is the standard $k$0-integer (Xu et al., 2 Sep 2025).

This recovers several known cases:

• $k$1 yields Carlitz–Scoville’s bi-Eulerian numbers $k$2.
• $k$3 produces Butler’s Stirling–Euler–Mahonian $k$4-Eulerian numbers.
• For $k$5, $k$6, one recovers $k$7-Eulerian numbers up to scaling.

Parallel to these, on Stirling permutations ($k$8), one constructs polynomials such as

$k$9

which, upon specializations, retrieve $n$0-Eulerian numbers and connect to symmetric decompositions via context-free grammars (Ma et al., 23 Jul 2025).

3. Symmetric and γ-Positive Decompositions

A major theme is the partial and bi- $n$1-positivity of these polynomials. For the $n$2-Eulerian polynomial,

$n$3

there exists a decomposition

$n$4

with explicit, nonnegative coefficients $n$5 defined by a two-variable recurrence.

For the bivariate ascent-plateau polynomials $n$6 on Stirling permutations,

$n$7

where both summands feature nonnegative coefficients and satisfy coupled recurrences, establishing $n$8-positivity for $n$9 and $1,\ldots,n$0.

The $1,\ldots,n$1-ascent-plateau polynomials $1,\ldots,n$2 enjoy a bi-$1,\ldots,n$3-positive expansion: $1,\ldots,n$4 valid for $1,\ldots,n$5, arising from a pair of context-free grammars (Ma et al., 23 Jul 2025).

4. Grammars and Combinatorial Proof Techniques

Both (Ma et al., 23 Jul 2025) and (Xu et al., 2 Sep 2025) provide grammar-based proofs for polynomial expansions and recurrences. For example, grammars are exploited to encode insertion of letters in permutations or plateaux in Stirling permutations, tracking the corresponding statistics through labeled derivations.

A generic example (for joint distributions on $1,\ldots,n$6): $1,\ldots,n$7 yields

$1,\ldots,n$8

Proofs for main product, convolution, and generating function formulas, as well as for (bi-)$1,\ldots,n$9-positivity, are based on context-free grammars and grammatical labeling, providing uniform, combinatorial accounts of all recurrence and expansion relations (Ma et al., 23 Jul 2025).

5. Closed-Form Expressions and Recurrence Relations

The bi-Stirling-Euler-Mahonian polynomials satisfy a $k$0-difference recurrence: $k$1 where $k$2 is the standard $k$3-derivative. Explicit Worpitzky-type expansions and inversions express $k$4 in $k$5-binomial and monomial terms, and the exponential generating function

$k$6

further encapsulates enumerative data (Xu et al., 2 Sep 2025).

For classical polynomials, generating functions specialize to expressions such as: $k$7

6. Specializations, Symmetry, and Generalizations

Several specializations unify and extend classical polynomial sequences:

Parameters	Resulting Polynomials	Features
$k$8	Carlitz–Scoville bi-Eulerian $k$9	Symmetry: $\sigma\in S_n$0
$\sigma\in S_n$1	Butler’s Stirling–Euler–Mahonian	Connects to $\sigma\in S_n$2-Stirling-Eulerian
$\sigma\in S_n$3	“Dual” $\sigma\in S_n$4-Eulerian	Mirrors properties for left-to-right minima
$\sigma\in S_n$5, $\sigma\in S_n$6	$\sigma\in S_n$7-Eulerian numbers	$\sigma\in S_n$8
$\sigma\in S_n$9, $\sigma\in Q_n^{(k)}$0	Type-$\sigma\in Q_n^{(k)}$1 Eulerian: $\sigma\in Q_n^{(k)}$2	Interpolates signed permutations

This symmetry is reflected in the interchange of the roles of left-to-right and right-to-left minima and parameters $\sigma\in Q_n^{(k)}$3, $\sigma\in Q_n^{(k)}$4, and extended by the Mahonian parameter $\sigma\in Q_n^{(k)}$5 (Xu et al., 2 Sep 2025).

This suggests that the bi-Stirling-Euler-Mahonian polynomials offer a uniform approach to diverse classical enumerative results through explicit joint statistics, variable substitutions, and generating formulations.

7. Context and Significance

The construction and systematic study of bi-Stirling-Euler-Mahonian polynomials illuminate the interplay between permutation and Stirling permutation statistics, providing:

• Uniform combinatorial interpretations for multivariate and Mahonian generalizations of Eulerian numbers.
• Explicit $\sigma\in Q_n^{(k)}$6-positive and symmetric decompositions, central to unimodality and real-rootedness investigations.
• Transparent combinatorial proofs leveraging context-free grammars, insertion bijections, and weight preservations.
• A generalized setting accommodating Mahonian, Eulerian, and Stirling-statistic generating functions under unifying parameterizations (Ma et al., 23 Jul 2025, Xu et al., 2 Sep 2025).

These polynomials underpin a contemporary direction in enumerative combinatorics, linking algebraic, combinatorial, and symmetric properties across classical and Stirling-type statistics.