---
title: Bi-Stirling-Euler-Mahonian Polynomials
url: https://www.emergentmind.com/topics/bi-stirling-euler-mahonian-polynomials
type: topic
---

# Bi-Stirling-Euler-Mahonian Polynomials

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 [2507.17667], [2509.02461].

## 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**: $\lrmin(\sigma) = \#\{i : \sigma_i < \sigma_j\ \forall j < i\}$.
- **Right-to-left minima**: $\rlmin(\sigma) = \#\{i : \sigma_i < \sigma_j\ \forall j > i\}$.
- **Longest ascent-plateaux**: For $Q_n^{(k)}$, indices $i$ with $\sigma_{i-1} < \sigma_{i} = \sigma_{i+1} = \dots = \sigma_{i+k-1}$; count denoted $\lap(\sigma)$.
- **Longest left ascent-plateaux**: As above, but with $\sigma_0=0$, so $i=1$ is allowed; count is $\lapp(\sigma)$.
- **Proper and improper ascent-plateaux** on $Q_n^{(k)}$: Proper if all values $<\sigma_i$ occur before $i$, improper otherwise; counted by $\plap(\sigma)$ and $\implap(\sigma)$.

For Mahonian refinements, the statistics include
- **Major index**: $\maj(\sigma) = \sum_{i:\sigma_i>\sigma_{i+1}} i$.
- **Mixed major index**: For parameters $\alpha,\beta\in\mathbb{N}$,
  $$
  \widetilde{\maj}(\sigma) = (1+\des(\sigma)-\lrmin(\sigma))(\alpha-1) + (n-\rlmin(\sigma))(\beta-1)+\maj(\sigma).
  $$

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

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

\[
E_n(x\,|\,\alpha,\beta,q) = \sum_{k=0}^n E_{n,k}(\alpha,\beta,q) x^k,
\]
where
\[
E_{n,k}(\alpha,\beta,q) = \sum_{\sigma\in S_{n+1},\,\des(\sigma) = k} [\alpha]^{\lrmin(\sigma)-1} [\beta]^{\rlmin(\sigma)-1} q^{\widetilde{\maj}(\sigma)},
\]
and $[m]=(1-q^m)/(1-q)$ is the standard $q$-integer [2509.02461].

This recovers several known cases:
- $q=1$ yields Carlitz–Scoville’s bi-Eulerian numbers $A(k,n-k|\alpha,\beta)$.
- $\alpha=0$ produces Butler’s Stirling–Euler–Mahonian $q$-Eulerian numbers.
- For $\alpha+\beta=r$, $q=1$, one recovers $r$-Eulerian numbers up to scaling.

Parallel to these, on Stirling permutations ($k\geq 2$), one constructs polynomials such as
\[
E_{n}^{(k)}(x,y,z) = \sum_{\sigma\in Q_n^{(k)}} x^{\implap(\sigma)}\,y^{\plap(\sigma)}\,z^{n-\lapp(\sigma)},
\]
which, upon specializations, retrieve $1/k$-Eulerian numbers and connect to symmetric decompositions via context-free grammars [2507.17667].

## 3. Symmetric and γ-Positive Decompositions

A major theme is the *partial* and *bi-* $\gamma$-positivity of these polynomials. For the $1/k$-Eulerian polynomial,
\[
A_n^{(k)}(x) = \sum_{\pi\in S_n} x^{\des(\pi)} k^{n-\des(\pi)},
\]
there exists a decomposition
\[
A_{n+1}^{(k)}(x) = \sum_{i\geq 0}(1+kx)^i\,\sum_{j\geq 0}\gamma_{n,i,j}(k)\;x^j(1+x)^{n-i-2j},
\]
with explicit, nonnegative coefficients $\gamma_{n,i,j}(k)$ defined by a two-variable recurrence.

For the bivariate ascent-plateau polynomials $P_n(x,y)$ on Stirling permutations,
\[
P_{n+1}(x,y) = x\sum_{i\geq 0}\xi_{n,i}(xy)^i(1+xy)^{n-2i} + xy\sum_{j\geq 0}\eta_{n,j}(xy)^j(1+xy)^{n-1-2j},
\]
where both summands feature nonnegative coefficients and satisfy coupled recurrences, establishing $\gamma$-positivity for $x\mapsto P_{n+1}(x,x)$ and $x\mapsto P_{n+1}(x,1)$.

The $q$-ascent-plateau polynomials $M_n(x,q)$ enjoy a *bi-$\gamma$-positive* expansion:
\[
M_n(x,q) = \sum_{i\geq 0}q f_{n-1,i}(q)x^i(1+x)^{n-1-2i} + x\sum_{j\geq 0}q g_{n-1,j}(q)x^j(1+x)^{n-2-2j},
\]
valid for $0<q\leq2$, arising from a pair of context-free grammars [2507.17667].

## 4. Grammars and Combinatorial Proof Techniques

Both [2507.17667] and [2509.02461] 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 $Q_n^{(k)}$):
\[
G: I\to I\,y,\quad x\to k x z, \quad y\to k x z, \quad z\to k x z,
\]
yields
\[
D_G^n(I) = I\sum_{\sigma\in Q_n^{(k)}} x^{\implap(\sigma)} y^{\plap(\sigma)} z^{n-\lapp(\sigma)}.
\]

Proofs for main product, convolution, and generating function formulas, as well as for (bi-)$\gamma$-positivity, are based on context-free grammars and grammatical labeling, providing uniform, combinatorial accounts of all recurrence and expansion relations [2507.17667].

## 5. Closed-Form Expressions and Recurrence Relations

The bi-Stirling-Euler-Mahonian polynomials satisfy a $q$-difference recurrence:
\[
E_n(x) = ([\beta] + q^{\beta}[n-1+\alpha]x) E_{n-1}(x) + (1-x)x q^{\beta} \delta_x E_{n-1}(x),
\]
where $\delta_x$ is the standard $q$-derivative. Explicit Worpitzky-type expansions and inversions express $E_{n,k}(\alpha,\beta,q)$ in $q$-binomial and monomial terms, and the exponential generating function
\[
\sum_{n\ge0}\frac{E_n(x)}{(x;q)_{n+\alpha+\beta}} \frac{t^n}{n!} = \sum_{j\ge0}x^j\binom{j+\alpha+\beta-1}{j}_q \exp([j+\beta]t)
\]
further encapsulates enumerative data [2509.02461].

For classical polynomials, generating functions specialize to expressions such as:
\[
\sum_{n\ge 0} E_n(x\,|\,\alpha,\beta,1)\,\frac{z^n}{n!} = \left(\frac{1-x}{1-x e^{(1-x)z}}\right)^\alpha \left(\frac{1-x}{e^{(x-1)z}-x}\right)^\beta.
\]

## 6. Specializations, Symmetry, and Generalizations

Several specializations unify and extend classical polynomial sequences:

| Parameters                  | Resulting Polynomials                                  | Features                                 |
|-----------------------------|-------------------------------------------------------|------------------------------------------|
| $q=1$                       | Carlitz–Scoville bi-Eulerian $A(k,n-k|\alpha,\beta)$  | Symmetry: $E_{n,k}(\alpha,\beta,1) = E_{n,n-k}(\beta,\alpha,1)$ |
| $\alpha=0$                  | Butler’s Stirling–Euler–Mahonian                      | Connects to $q$-Stirling-Eulerian        |
| $\beta=0$                   | “Dual” $q$-Eulerian                                   | Mirrors properties for left-to-right minima |
| $\alpha+\beta = r$, $q=1$   | $r$-Eulerian numbers                                  | $A(n+r,k;r) = r!E_{n,k}(r,1,1)$          |
| $q=2$, $M_n(x,q)$           | Type-$B$ Eulerian: $2^nA_n(x)=x^n M_n(1/x,2)$         | Interpolates signed permutations         |

This symmetry is reflected in the interchange of the roles of left-to-right and right-to-left minima and parameters $\alpha$, $\beta$, and extended by the Mahonian parameter $q$ [2509.02461].

*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 $\gamma$-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 [2507.17667], [2509.02461].

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

Source: https://www.emergentmind.com/topics/bi-stirling-euler-mahonian-polynomials