Remixed Eulerian Numbers: q-Deformations

• Remixed Eulerian numbers are polynomial q-deformations of mixed Eulerian numbers indexed by weak compositions, refining mixed-volume coefficients for the permutahedron.
• They possess multiple equivalent definitions—algebraic via q-divided symmetrization, probabilistic models, and geometric cube decompositions—that recover classical Eulerian numbers and other combinatorial invariants.
• Their study links permutation statistics, tree models, and matroid Chow rings, offering unified insights into classical Eulerian distributions and q-hit numbers.

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 (Nadeau et al., 2022, Gaudin, 2024, Xu et al., 2 Sep 2025).

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 $A_c(q)$0, one has

$A_c(q)$1

Equivalently, in hypersimplex language,

$A_c(q)$2

so the family is encoded by mixed volumes of type-$A_c(q)$3 hypersimplices. In this form it contains usual Eulerian numbers, binomial coefficients, Catalan numbers, and hit numbers as specializations (Xu et al., 2 Sep 2025, Gaudin, 2024).

Nadeau and Tewari defined remixed Eulerian numbers $A_c(q)$4 as a polynomial refinement of $A_c(q)$5. The defining principle is that $A_c(q)$6 should preserve the mixed-Eulerian indexing by compositions while interpolating standard $A_c(q)$7-analogues of familiar subfamilies. This suggests a deformation that is not merely numerical but structural: the same indexing set $A_c(q)$8 supports algebraic symmetries, probabilistic recurrences, and geometric interpretations that are already latent in the mixed-volume theory (Nadeau et al., 2022).

2. Definitions and fundamental identities

One algebraic definition uses $A_c(q)$9-divided symmetrization. For $c$0, set

$c$1

and define

$c$2

The same paper gives an equivalent formulation in the $c$3-Klyachko algebra, where $c$4 is $c$5 times the coefficient of $c$6 in the squarefree expansion of $c$7. Here

$c$8

These definitions specialize correctly at $c$9: $q$0 (Nadeau et al., 2022).

The basic structural identities are especially rigid. The remixed Eulerian numbers satisfy

$q$1

and the reversal symmetry

$q$2

Their total sum is

$q$3

and the two-block specialization is

$q$4

As polynomials in $q$5, they are symmetric and unimodal. More precisely, if the path heights are

$q$6

then the valuation and degree are

$q$7

and the reversal identity gives palindromicity about $q$8 (Nadeau et al., 2022, Xu et al., 2 Sep 2025).

A particularly simple product formula occurs under prefix-sum dominance: $q$9 This is the remixed analogue of the simplest mixed-Eulerian strata and reappears later as the Łukasiewicz case (Nadeau et al., 2022).

3. Probabilistic, tree, and polyhedral models

A probabilistic definition realizes $q$0 through an abelian particle process on $q$1. For a configuration $q$2 with $q$3, let $q$4 contain $q$5 copies of $q$6. Balls are dropped one at a time at sites given by an ordering of $q$7; whenever a collision occurs, the moving ball steps left with probability $q$8 and right with probability $q$9 until it reaches an empty site. By the Diaconis–Fulton abelian property, the probability that the final support is exactly $r\ge 1$0 depends only on $r\ge 1$1, and

$r\ge 1$2

This model yields both a local “big-step” relation and a “final induction” recurrence indexed by the last dropped ball (Gaudin, 2024).

The same numbers admit weighted-tree interpretations. Given a word $r\ge 1$3 of content $r\ge 1$4, one considers $r\ge 1$5-compatible decreasing binary trees $r\ge 1$6 with explicitly defined interval-exit weights $r\ge 1$7. Then

$r\ge 1$8

where $r\ge 1$9 is the set of $$W_r:=\{c=(c_1,\dots,c_r)\in \mathbb N^r: c_1+\cdots+c_r=r\},$$0-compatible trees. A second interpretation uses bilabeled trees: $$W_r:=\{c=(c_1,\dots,c_r)\in \mathbb N^r: c_1+\cdots+c_r=r\},$$1 where $$W_r:=\{c=(c_1,\dots,c_r)\in \mathbb N^r: c_1+\cdots+c_r=r\},$$2 is the permutation attached to the decreasing labeling. At $$W_r:=\{c=(c_1,\dots,c_r)\in \mathbb N^r: c_1+\cdots+c_r=r\},$$3, the first model recovers Postnikov’s weighted-tree interpretation of mixed Eulerian numbers, and the second recovers Liu’s interpretation (Nadeau et al., 2022).

A polyhedral counterpart comes from a $$W_r:=\{c=(c_1,\dots,c_r)\in \mathbb N^r: c_1+\cdots+c_r=r\},$$4-weighted cube decomposition of the permutahedron. If

$$W_r:=\{c=(c_1,\dots,c_r)\in \mathbb N^r: c_1+\cdots+c_r=r\},$$5

then

$$W_r:=\{c=(c_1,\dots,c_r)\in \mathbb N^r: c_1+\cdots+c_r=r\},$$6

where the $$W_r:=\{c=(c_1,\dots,c_r)\in \mathbb N^r: c_1+\cdots+c_r=r\},$$7 are Bruhat-interval cubes in a subdivision of $$W_r:=\{c=(c_1,\dots,c_r)\in \mathbb N^r: c_1+\cdots+c_r=r\},$$8. This model links remixed Eulerian numbers to Gelfand–Tsetlin faces, shifted tableaux, and Richardson-variety volumes (Nadeau et al., 2022).

4. Explicit families: connected, Łukasiewicz, and beyond

The probabilistic model makes several subfamilies explicitly computable. Write the left-to-right order of $$W_r:=\{c=(c_1,\dots,c_r)\in \mathbb N^r: c_1+\cdots+c_r=r\},$$9 as the nondecreasing sequence $A_c$0, and define heights

$A_c$1

A configuration is Łukasiewicz if $A_c$2 for all $A_c$3. In that case,

$A_c$4

For example, $A_c$5 is Łukasiewicz and

$A_c$6

This is exactly the prefix-dominant product formula viewed through the particle process (Gaudin, 2024).

Every configuration can be written uniquely as $A_c$7, where $A_c$8 is the core. When $A_c$9 has no holes, the configuration is connected, and the generating series simplifies to

$\mathrm{Perm}(\lambda)$0

Coefficient extraction gives

$\mathrm{Perm}(\lambda)$1

Comparing this with Garsia–Remmel’s generating function shows that connected remixed Eulerian numbers coincide with $\mathrm{Perm}(\lambda)$2-hit numbers for suitable Ferrers shapes (Gaudin, 2024).

The connected formula extends in several directions. If $\mathrm{Perm}(\lambda)$3 is almost Łukasiewicz, with a unique defect index $\mathrm{Perm}(\lambda)$4 at which $\mathrm{Perm}(\lambda)$5, then

$\mathrm{Perm}(\lambda)$6

For weakly Łukasiewicz configurations, the connected generating series holds modulo $\mathrm{Perm}(\lambda)$7, where $\mathrm{Perm}(\lambda)$8 is the maximal initial string of zeros compatible with weak Łukasiewiczness, and the coefficient of $\mathrm{Perm}(\lambda)$9 still has an explicit alternating $\lambda_1\ge \cdots \ge \lambda_{r+1}$0-binomial form. For cores with exactly one hole, an exact corrective series $\lambda_1\ge \cdots \ge \lambda_{r+1}$1 is available, and coefficient extraction produces closed formulas for all coefficients in the corresponding interval of shifts (Gaudin, 2024).

These formulas establish that remixed Eulerian numbers are not confined to the connected or interval-support regime. A plausible implication is that the connected $\lambda_1\ge \cdots \ge \lambda_{r+1}$2-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

$\lambda_1\ge \cdots \ge \lambda_{r+1}$3

Writing

$\lambda_1\ge \cdots \ge \lambda_{r+1}$4

one obtains a direct permutation-statistic model. For $\lambda_1\ge \cdots \ge \lambda_{r+1}$5,

$\lambda_1\ge \cdots \ge \lambda_{r+1}$6

where

$\lambda_1\ge \cdots \ge \lambda_{r+1}$7

Thus $\lambda_1\ge \cdots \ge \lambda_{r+1}$8 simultaneously records descents, left-to-right minima, right-to-left minima, and a mixed major index (Xu et al., 2 Sep 2025).

Packaging all four statistics gives

$\lambda_1\ge \cdots \ge \lambda_{r+1}$9

with

$A_c(q)$00

The coefficients satisfy the recurrence

$A_c(q)$01

with $A_c(q)$02. The row-generating polynomial

$A_c(q)$03

has a Worpitzky-type expansion

$A_c(q)$04

together with a closed coefficient formula and an exponential generating function (Xu et al., 2 Sep 2025).

The specializations recover several earlier families. Setting $A_c(q)$05 gives Carlitz–Scoville’s generalized $A_c(q)$06-Eulerian numbers; $A_c(q)$07 or $A_c(q)$08 recovers Butler’s Stirling–Euler–Mahonian polynomials; $A_c(q)$09 yields the classical Eulerian numbers and Carlitz’s $A_c(q)$10-Eulerian identity. Rawlings’s $A_c(q)$11-Eulerian numbers satisfy

$A_c(q)$12

In this sense, a specific remixed composition class organizes several previously separate Eulerian–Mahonian theories into a single four-statistic framework (Xu et al., 2 Sep 2025).

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(q)$13

and for perfect matroid designs one obtains analogues of the mixed-Eulerian recurrences and lopsided product formulas. In particular, for projective geometry $A_c(q)$14,

$A_c(q)$15

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 $A_c(q)$16-invariant place the $A_c(q)$17 theory inside a broad valuative framework (Katz et al., 2023, Liu et al., 7 Feb 2025).

A related geometric line studies mixed $A_c(q)$18-Eulerian numbers for arbitrary root systems through toric Hessenberg varieties and Peterson Schubert calculus. Those numbers are not themselves the remixed $A_c(q)$19, but they provide the ambient Lie-theoretic geometry from which the type-$A_c(q)$20 mixed Eulerian numbers arise. This suggests that remixed Eulerian numbers should be viewed as a specifically type-$A_c(q)$21, $A_c(q)$22-deformed branch of a larger mixed-Eulerian landscape (Horiguchi, 2021).

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 $A_c(q)$23-Worpitzky identity and of the exponential generating function, as well as unimodality, real-rootedness, and $A_c(q)$24-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 (Gaudin, 2024, Xu et al., 2 Sep 2025).