---
title: 'Remixed Eulerian Numbers: q-Deformations'
url: https://www.emergentmind.com/topics/remixed-eulerian-numbers
type: topic
---

# Remixed Eulerian Numbers: q-Deformations

Remixed Eulerian numbers are polynomial \(q\)-deformations \(A_c(q)\) of Postnikov’s mixed Eulerian numbers, indexed by weak compositions \(c\) of a fixed rank. They were introduced by Nadeau and Tewari as a refinement of mixed-volume coefficients for the permutahedron, and they recover classical Eulerian numbers, Gaussian binomial coefficients, and Garsia–Remmel \(q\)-hit numbers as special cases. Subsequent work supplied several equivalent definitions—algebraic, probabilistic, and geometric—together with explicit formulas for broad subfamilies and, for a distinguished remixed subfamily, a direct permutation-statistic interpretation involving left-to-right minima, right-to-left minima, descents, and a mixed major index [2208.04128] [2411.03466] [2509.02461].

## 1. From mixed Eulerian numbers to their \(q\)-deformation

For \(r\ge 1\) and
\[
W_r:=\{c=(c_1,\dots,c_r)\in \mathbb N^r: c_1+\cdots+c_r=r\},
\]
Postnikov’s mixed Eulerian numbers \(A_c\) arise in the volume expansion of the permutahedron \(\mathrm{Perm}(\lambda)\) attached to a decreasing sequence \(\lambda_1\ge \cdots \ge \lambda_{r+1}\). Writing \(\mu_i=\lambda_i-\lambda_{i+1}\), one has
\[
\mathrm{vol}(\mathrm{Perm}(\lambda))
=
\sum_{c\in W_r}
A_c\,
\frac{\mu_1^{c_1}\cdots \mu_r^{c_r}}{c_1!\cdots c_r!}.
\]
Equivalently, in hypersimplex language,
\[
A_c
=
n!\,V\bigl(\Delta(1,n)[c_1],\Delta(2,n)[c_2],\dots,\Delta(n,n)[c_n]\bigr),
\]
so the family is encoded by mixed volumes of type-\(A\) hypersimplices. In this form it contains usual Eulerian numbers, binomial coefficients, Catalan numbers, and hit numbers as specializations [2509.02461] [2411.03466].

Nadeau and Tewari defined remixed Eulerian numbers \(A_c(q)\) as a polynomial refinement of \(A_c\). The defining principle is that \(A_c(q)\) should preserve the mixed-Eulerian indexing by compositions while interpolating standard \(q\)-analogues of familiar subfamilies. This suggests a deformation that is not merely numerical but structural: the same indexing set \(W_r\) supports algebraic symmetries, probabilistic recurrences, and geometric interpretations that are already latent in the mixed-volume theory [2208.04128].

## 2. Definitions and fundamental identities

One algebraic definition uses \(q\)-divided symmetrization. For \(c\in W_r\), set
\[
y_c=x_1^{c_1}(x_1+x_2)^{c_2}\cdots(x_1+\cdots+x_r)^{c_r},
\]
and define
\[
A_c(q)=\#\mathrm{DS}_{r+1}^q(y_c).
\]
The same paper gives an equivalent formulation in the \(q\)-Klyachko algebra, where \(A_c(q)\) is \([r]_q!\) times the coefficient of \(u_{[r]}\) in the squarefree expansion of \(u_1^{c_1}\cdots u_r^{c_r}\). Here
\[
[m]=\frac{1-q^m}{1-q},\qquad
[r]!=\prod_{i=1}^r [i],\qquad
\binom{n}{k}_q=\frac{[n]!}{[k]![n-k]!}.
\]
These definitions specialize correctly at \(q=1\): \(A_c(1)=A_c\) [2208.04128].

The basic structural identities are especially rigid. The remixed Eulerian numbers satisfy
\[
A_c(q)\in \mathbb N[q],\qquad
A_{(1,\dots,1)}(q)=[r]!,
\]
and the reversal symmetry
\[
A_{(c_1,\dots,c_r)}(q)
=
q^{\binom r2}\,
A_{(c_r,\dots,c_1)}(q^{-1}).
\]
Their total sum is
\[
\sum_{c\in W_r} A_c(q)
=
[r]!\cdot \frac{1}{r+1}\binom{2r}{r},
\]
and the two-block specialization is
\[
A_{(k,0,\dots,0,r-k)}(q)
=
q^{\binom k2}\binom{r}{k}_q.
\]
As polynomials in \(q\), they are symmetric and unimodal. More precisely, if the path heights are
\[
h_i(c)=\sum_{1\le j\le i}(c_j-1),
\]
then the valuation and degree are
\[
d_c=\sum_{1\le i\le r-1}(-h_i(c))^{+},\qquad
D_c=\binom r2-\sum_{1\le i\le r-1}(h_i(c))^{+},
\]
and the reversal identity gives palindromicity about \((d_c+D_c)/2\) [2208.04128] [2509.02461].

A particularly simple product formula occurs under prefix-sum dominance:
\[
\sum_{i\le j} c_i\ge j \quad (j=1,\dots,r)
\quad\Longrightarrow\quad
A_c(q)=[1]^{c_1}[2]^{c_2}\cdots [r]^{c_r}.
\]
This is the remixed analogue of the simplest mixed-Eulerian strata and reappears later as the Łukasiewicz case [2208.04128].

## 3. Probabilistic, tree, and polyhedral models

A probabilistic definition realizes \(A_c(q)\) through an abelian particle process on \(\mathbb Z\). For a configuration \(c=(c_1,\dots,c_n)\) with \(\sum_i c_i=n\), let \(\mathrm{MSet}(c)\) contain \(c_i\) copies of \(i\). Balls are dropped one at a time at sites given by an ordering of \(\mathrm{MSet}(c)\); whenever a collision occurs, the moving ball steps left with probability \(q/(1+q)\) and right with probability \(1/(1+q)\) until it reaches an empty site. By the Diaconis–Fulton abelian property, the probability that the final support is exactly \([1;n]\) depends only on \(c\), and
\[
A_c(q)=[n]!\,P(c\to [1;n])_q.
\]
This model yields both a local “big-step” relation and a “final induction” recurrence indexed by the last dropped ball [2411.03466].

The same numbers admit weighted-tree interpretations. Given a word \(\mathbf i\in [r]^r\) of content \(c\), one considers \(\mathbf i\)-compatible decreasing binary trees \(T\) with explicitly defined interval-exit weights \(\mathrm{wt}(T,\mathbf i)\). Then
\[
A_c(q)=\sum_{T\in \mathcal P(\mathbf i)} [r]_q!\,\mathrm{wt}(T,\mathbf i),
\]
where \(\mathcal P(\mathbf i)\) is the set of \(\mathbf i\)-compatible trees. A second interpretation uses bilabeled trees:
\[
A_c(q)=\sum_{(T,\mathrm{dec},\mathrm{lr})\in B(c)} q^{\ell(u)},
\]
where \(u\) is the permutation attached to the decreasing labeling. At \(q=1\), the first model recovers Postnikov’s weighted-tree interpretation of mixed Eulerian numbers, and the second recovers Liu’s interpretation [2208.04128].

A polyhedral counterpart comes from a \(q\)-weighted cube decomposition of the permutahedron. If
\[
V^q(\lambda)=\frac{1}{[r]_q!}\,\#\mathrm{DS}_{r+1}^q\bigl((\lambda_1x_1+\cdots+\lambda_{r+1}x_{r+1})^r\bigr),
\]
then
\[
V^q(\lambda)=\sum_{u\in S_r} q^{\ell(u)}\,\mathrm{vol}(C_\lambda(u)),
\]
where the \(C_\lambda(u)\) are Bruhat-interval cubes in a subdivision of \(\mathrm{Perm}(\lambda)\). This model links remixed Eulerian numbers to Gelfand–Tsetlin faces, shifted tableaux, and Richardson-variety volumes [2208.04128].

## 4. Explicit families: connected, Łukasiewicz, and beyond

The probabilistic model makes several subfamilies explicitly computable. Write the left-to-right order of \(c\) as the nondecreasing sequence \(u_1,\dots,u_n\), and define heights
\[
H_{c,k}:=\sum_{i=1}^k (c_i-1).
\]
A configuration is Łukasiewicz if \(H_{c,i}\ge 0\) for all \(i\). In that case,
\[
A_c(q)=\prod_{a\in \mathrm{MSet}(c)} [a].
\]
For example, \(c=(3,0,0,2,0)\) is Łukasiewicz and
\[
A_{(3,0,0,2,0)}(q)=[1]^3[4]^2.
\]
This is exactly the prefix-dominant product formula viewed through the particle process [2411.03466].

Every configuration can be written uniquely as \(c=(0^k,\gamma,0^{n-m-k})\), where \(\gamma\) is the core. When \(\gamma\) has no holes, the configuration is connected, and the generating series simplifies to
\[
\frac{\sum_{i=0}^{n-m} t^i\,A_{(0^i,\gamma,0^{n-m-i})}(q)}{(t;q)_{n+1}}
=
\sum_{j\ge 0} t^j \prod_{a\in \mathrm{MSet}(\gamma)} [j+a].
\]
Coefficient extraction gives
\[
A_{(0^i,\gamma,0^{n-m-i})}(q)
=
\sum_{j=0}^{i}
(-1)^{i+j}
q^{\binom{i-j}{2}}
\binom{n+1}{i-j}_q
\prod_{a\in \mathrm{MSet}(\gamma)} [j+a].
\]
Comparing this with Garsia–Remmel’s generating function shows that connected remixed Eulerian numbers coincide with \(q\)-hit numbers for suitable Ferrers shapes [2411.03466].

The connected formula extends in several directions. If \(c\) is almost Łukasiewicz, with a unique defect index \(j\) at which \(H_{c,j}=-1\), then
\[
A_c(q)
=
\prod_{a\in \mathrm{MSet}(c)} [a]
-
\binom{n+1}{j}_q
\left(\prod_{a<j}[a]\right)
\left(\prod_{b>j}[b-j]\right).
\]
For weakly Łukasiewicz configurations, the connected generating series holds modulo \(t^{k+1}\), where \(k\) is the maximal initial string of zeros compatible with weak Łukasiewiczness, and the coefficient of \(t^k\) still has an explicit alternating \(q\)-binomial form. For cores with exactly one hole, an exact corrective series \(P_\gamma(q,t)\) is available, and coefficient extraction produces closed formulas for all coefficients in the corresponding interval of shifts [2411.03466].

These formulas establish that remixed Eulerian numbers are not confined to the connected or interval-support regime. A plausible implication is that the connected \(q\)-hit picture is only the first layer of a broader deformation theory organized by the defect pattern of the height path.

## 5. Permutation statistics and the bi-Stirling–Euler–Mahonian subfamily

A natural remixed subfamily, first isolated by Gaudin, is
\[
A(r,s\mid \alpha,\beta)_q
:=
[\alpha+\beta-1]!^{-1}\,
A_{(0^r,1^{\alpha-1},\,r+s+1,\,1^{\beta-1},0^s)}(q).
\]
Writing
\[
E_{n,k}(\alpha,\beta,q):=A(k,n-k\mid \alpha,\beta)_q,
\]
one obtains a direct permutation-statistic model. For \(n\ge k\ge 0\),
\[
E_{n,k}(\alpha,\beta,q)
=
\sum_{\substack{\sigma\in S_{n+1}\\ \mathrm{des}(\sigma)=k}}
[\alpha]^{\mathrm{ltrmin}(\sigma)-1}
[\beta]^{\mathrm{rtlmin}(\sigma)-1}
q^{\mathrm{mmaj}(\sigma)},
\]
where
\[
\mathrm{mmaj}(\sigma)
=
\bigl(1+\mathrm{des}(\sigma)-\mathrm{ltrmin}(\sigma)\bigr)(\alpha-1)
+
\bigl(n-\mathrm{rtlmin}(\sigma)\bigr)(\beta-1)
+
\mathrm{maj}(\sigma).
\]
Thus \(E_{n,k}\) simultaneously records descents, left-to-right minima, right-to-left minima, and a mixed major index [2509.02461].

Packaging all four statistics gives
\[
B_n(x,y,t;q)
:=
\sum_{\sigma\in S_{n+1}}
x^{\mathrm{ltrmin}(\sigma)-1}
y^{\mathrm{rtlmin}(\sigma)-1}
t^{\mathrm{des}(\sigma)}
q^{\mathrm{mmaj}(\sigma)},
\]
with
\[
E_{n,k}(\alpha,\beta,q)=[t^k]\,B_n([\alpha],[\beta],t;q).
\]
The coefficients satisfy the recurrence
\[
E_{n,k}(\alpha,\beta,q)
=
q^{\beta+k-1}[n-k+\alpha]\,E_{n-1,k-1}(\alpha,\beta,q)
+
[k+\beta]\,E_{n-1,k}(\alpha,\beta,q),
\]
with \(E_{0,0}=1\). The row-generating polynomial
\[
E_n(x\mid \alpha,\beta,q)=\sum_{k=0}^n E_{n,k}(\alpha,\beta,q)x^k
\]
has a Worpitzky-type expansion
\[
\frac{E_n(x\mid \alpha,\beta,q)}{(x;q)_{n+\alpha+\beta}}
=
\sum_{j\ge 0}
x^j
\binom{j+\alpha+\beta-1}{j}_q
[j+\beta]^n,
\]
together with a closed coefficient formula and an exponential generating function [2509.02461].

The specializations recover several earlier families. Setting \(q=1\) gives Carlitz–Scoville’s generalized \((\alpha,\beta)\)-Eulerian numbers; \(\alpha=0\) or \(\beta=0\) recovers Butler’s Stirling–Euler–Mahonian polynomials; \(\alpha=\beta=1\) yields the classical Eulerian numbers and Carlitz’s \(q\)-Eulerian identity. Rawlings’s \((q,r)\)-Eulerian numbers satisfy
\[
A[n+r,k;r]=[r]!\,E_{n,k}(1,r,q).
\]
In this sense, a specific remixed composition class organizes several previously separate Eulerian–Mahonian theories into a single four-statistic framework [2509.02461].

## 6. Geometric generalizations, structural properties, and open directions

The mixed-Eulerian background of the theory has been extended far beyond the permutahedron. In the matroid Chow ring, matroidal mixed Eulerian numbers are defined by
\[
A_{c_1,\dots,c_n}(M)=\deg_M(\gamma_1^{c_1}\cdots \gamma_n^{c_n}),
\]
and for perfect matroid designs one obtains analogues of the mixed-Eulerian recurrences and lopsided product formulas. In particular, for projective geometry \(\mathrm{PG}(r,q)\),
\[
A_{(c_1,\dots,c_r)_n}(\mathrm{PG}(r,q))
=
q^{\binom{r+1}{2}}\,A_c(q),
\]
so remixed Eulerian numbers acquire a direct realization as Chow-ring intersection numbers. On the mixed side, explicit formulas for matroidal mixed Eulerian numbers and their equivalence with Derksen’s \(\mathcal G\)-invariant place the \(q=1\) theory inside a broad valuative framework [2305.19095] [2502.04980].

A related geometric line studies mixed \(\Phi\)-Eulerian numbers for arbitrary root systems through toric Hessenberg varieties and Peterson Schubert calculus. Those numbers are not themselves the remixed \(A_c(q)\), but they provide the ambient Lie-theoretic geometry from which the type-\(A\) mixed Eulerian numbers arise. This suggests that remixed Eulerian numbers should be viewed as a specifically type-\(A\), \(q\)-deformed branch of a larger mixed-Eulerian landscape [2104.14083].

Several structural questions remain open. For remixed numbers beyond the connected case, natural next steps include multi-hole cores and more general corrective series. For the bi-Stirling–Euler–Mahonian subfamily, open problems include combinatorial proofs of the \(q\)-Worpitzky identity and of the exponential generating function, as well as unimodality, real-rootedness, and \(\gamma\)-positivity for suitable normalizations and specializations. Extensions to rook theory, quasisymmetric functions, and descent algebras have also been proposed. These directions indicate that the subject is no longer only a deformation of classical Eulerian distributions; it is a meeting point for mixed volumes, permutation statistics, stochastic stabilization, and geometric representation theory [2411.03466] [2509.02461].

Source: https://www.emergentmind.com/topics/remixed-eulerian-numbers