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Mixed Eulerian numbers and beyond

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Pith's one-line read Matroidal mixed Eulerian numbers are shown to be exactly the universal valuative invariant, with an explicit non-recursive formula.

desk verdict Closed formula for mixed Eulerian numbers plus a clean equivalence with Derksen's G-invariant; proofs hold up, with a small scalar typo in Lemma 3.1. read the letter →

arxiv 2502.04980 v1 pith:NHAAH3YB submitted 2025-02-07 math.AG math.CO

Gaku Liu , Mateusz Micha{l}ek , Julian Weigert This is my paper · ORCID
classification math.AGmath.CO MSC 14C1714N1552A3905E1452B40
keywords matroidalmixedEuleriannumbersG-invariantvaluativeinvariantspermutohedralvarietyChowringTuttepolynomialvolumesclosedformula
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes that the collection of matroidal mixed Eulerian numbers is not a weaker shadow of matroid structure: it is precisely the G-invariant, a universal valuative invariant meaning any matroid invariant that respects base-polytope subdivisions factors through it. Two loopless matroids have the same G-invariant exactly when every corresponding matroidal mixed Eulerian number agrees. Along the way the paper derives the first explicit non-recursive closed formula for the classical mixed Eulerian numbers, together with several recursive formulas, by viewing these numbers as intersection numbers on a permutohedral variety and translating between two bases of invariant divisors. If correct, this closes a question left open in matroid theory and shows that a single family of degree data, computable by intersection products, encodes all valuative matroid invariants.

What carries the argument

The load-bearing object is the permutohedral variety $X_{\Pi_n}$ with two families of divisors: the hypersimplex classes $L_1,\ldots,L_n$, whose intersections with the matroid class $[X_M]$ define the matroidal mixed Eulerian numbers, and the exceptional divisors $S_1,\ldots,S_n$, whose intersections are easy to compute by restriction to facets. The key identity is the linear change of basis $S_i = -L_{i-1}+2L_i-L_{i+1}$ (equivalently the symmetric-power matrices $S^d A_{n,L\to S}$ and $S^d B_{n,S\to L}$), together with the restriction rule $[X_M]|_{x_F} = [X_{M|F}]\otimes [X_{M/F}]$ for every nonempty proper flat $F$, which splits an intersection into smaller permutohedral varieties and drives the inductions.

What would settle it

Run through all loopless matroids on five elements: if two of them share every matroidal mixed Eulerian number for all tuples summing to rank minus one but have different G-invariants, Theorem 1.12 is false. Alternatively, evaluate the closed formula of Corollary 1.6 for $A(1,1,1)$ and compare it with the original mixed-volume definition of mixed Eulerian numbers; disagreement would falsify the formula.

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Extended reading notes

Core claim

The central discovery is the equivalence between matroidal mixed Eulerian numbers and the G-invariant: for loopless matroids $M_1, M_2$ of rank $r+1$ on $n+1$ elements, $\mathcal{G}(M_1)=\mathcal{G}(M_2)$ if and only if $A_{M_1}(a_1,\ldots,a_n)=A_{M_2}(a_1,\ldots,a_n)$ for every nonnegative tuple summing to $r$ (Theorem 1.12). Because the G-invariant is universal among valuative invariants, this says the mixed Eulerian numbers carry all valuative information of a matroid. The paper also proves an explicit, non-recursive formula for every mixed Eulerian number (Corollary 1.6), obtained by expressing the volume polynomial of the permutohedral variety in a basis of exceptional divisors and evaluating the resulting binomial coefficients; the same formula is extended to all matroidal mixed Eulerian numbers (Theorem 1.8).

Load-bearing premise

The whole construction rests on a factorization rule imported from earlier work: the class of a matroid restricts to a facet as the tensor product of the class of the restricted matroid and the class of the contracted matroid, and if that rule failed the induction, the closed formulas, and the equivalence with the G-invariant would collapse.

Editorial extensions

If this is right

  • All classical mixed Eulerian numbers can be computed by a closed formula involving binomial coefficients and the entries of the inverse change-of-basis matrix, with no recursion.
  • The Tutte polynomial can be written as a linear combination of matroidal mixed Eulerian numbers (Proposition 3.16).
  • Any valuative matroid invariant that vanishes on matroids with loops can be recovered from the symmetrized class $[X_{M,\mathrm{sym}}]$ in the permutohedral Chow ring; the mixed Eulerian numbers are coordinates for this class.
  • The recursion in Theorem 1.9 shows all matroidal mixed Eulerian numbers are nonnegative, since the matrix coefficients and the smaller factors appearing are nonnegative.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The closed formula suggests that mixed Eulerian numbers, long believed to lack a simple form, are just the coefficients of a fixed polynomial after an integer change of basis; one could test whether the same strategy applies to other toric volume polynomials.
  • Because the G-invariant already distinguishes many matroids up to isotopy, the equivalence gives a geometric certificate: the full intersection table of $[X_M]$ with $L$-monomials is a complete valuative fingerprint, potentially useful for computational checks of matroid non-isomorphism.
  • The explicit binomial structure may point toward purely combinatorial proofs of log-concavity statements that motivated the question, bypassing the algebraic geometry used in earlier approaches.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. The paper studies matroidal mixed Eulerian numbers A_M(a), defined as intersection numbers of the matroid class [X_M] with monomials in the hypersimplex divisor classes L_i on the permutohedral variety. Its main results are: an explicit closed formula for classical mixed Eulerian numbers (Corollary 1.6, via Theorem 1.5), a closed formula for matroidal mixed Eulerian numbers in terms of flags of flats and gamma invariants (Theorem 1.8/3.5), two recursive formulas (Theorems 1.9/3.7 and 1.10/3.8), and the structural result that the collection of all matroidal mixed Eulerian numbers is equivalent to Derksen's G-invariant (Theorem 1.12/3.14). The paper also gives a formula for the Tutte polynomial in terms of these numbers (Proposition 3.16). The proofs work through the Chow ring of the permutohedral variety, using restriction to facets, the change of basis between the L_i and exceptional S_i divisors, and external results such as the factorization of matroid classes under restriction (Lemma 2.18).

Significance. The equivalence with Derksen's G-invariant is a strong and satisfying answer to the question of Berget, Spink, and Tseng, and Corollary 1.6 appears to be the first explicit non-recursive closed formula for all mixed Eulerian numbers. The approach is original in combining Schubert's classical complete-quadrics technique with modern matroid Chow-ring methods, and the resulting formulas are concrete and independently checkable. The main theorems are proved in detail, with external inputs clearly identified; I found no circularity or hidden fitting parameters. The paper is a substantial contribution to the intersection-theoretic study of matroids and should be published after the local issues below are addressed.

minor comments (5)
  1. [Section 3.A, Lemma 3.1] The displayed scalar q omits the factor a_l! = (n+1-|F_l|)!, so the equality m = q S_{a_1} ... S_{a_l} is false as written. For example, when n = 2 and F_1 = {0}, the symmetrized sum is 2 S_2, not 1 S_2. Since the proof only needs q to be nonzero, the generation claim is unaffected, but the displayed equality should be corrected.
  2. [Section 3.C, Proposition 3.13] The coefficients D_c(b) are asserted rather than defined or proved to exist. Because equation (3.6) is the bridge from S-intersection numbers to matroidal mixed Eulerian numbers in the proof of Theorem 1.12, please add an explicit definition of D_c(b) (for instance, as the coefficients obtained by applying the inverse of the linear transformation in (1.1) to the S-monomial) and a one-sentence verification of the expansion. This is a completeness issue rather than a substantive gap, since Lemma 3.1 and the invertibility of the two bases guarantee the coefficients exist.
  3. [Section 3.C, heading] The heading contains the typo "matrodal" and should read "matroidal".
  4. [Section 3.B, Theorem 3.8] The condition a_{j-1} = 0 is written inside the set over which F ranges, although it is not a condition on F. Please move this condition outside the sum, or state explicitly that the last summand is zero when a_{j-1} is nonzero.
  5. [Throughout] There are several residual typos and infelicities, including "satsify", "polyope", "explecit", and "of of"; a careful proofreading pass would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation is self-contained with external anchors.

full rationale

The paper's central results are derived from independent external theorems rather than from its own conclusions. Matroidal mixed Eulerian numbers are defined as intersection numbers on the permutohedral variety (Section 1.B), and the restriction/factorization engine Lemma 2.18 is quoted from the published independent work [7, Proposition 5.3] by Berget-Eur-Spink-Tseng, with no author overlap; Lemma 2.17 similarly imports [32, Lemma 2.6]. The explicit formulas in Theorem 3.5 and Corollary 1.6 are obtained by induction using only these external restriction rules and the elementary Lemma 2.15, so the target formulas are not assumed as inputs. For the G-invariant equivalence, the paper uses Derksen-Fink's universality theorem [18, Theorem 1.4] and Bonin-Kung's catenary-data theorem [9, Theorem 3.3] as external anchors, then expresses the catenary data as S-intersections and converts to L-intersections; this exhibits G as a linear combination of matroidal mixed Eulerian numbers rather than defining those numbers in terms of G. The self-citations that appear (e.g., [36] for mixed volumes, [6,27] for equivariant classes) are background or contextual and are not load-bearing. A small scalar typo in Lemma 3.1 (the displayed q omits a factor a_l!, so the printed equality fails in small examples) is a correctness blemish, not a circular step, since any nonzero scalar suffices for the generation claim. No fitted input is renamed as a prediction, no ansatz is smuggled via self-citation, and no known result is merely renamed. Accordingly, the derivation chain is self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters and no invented entities. The central results depend on standard toric geometry, the Kunneth formula, and four established theorems from the matroid/toric literature; these are independent anchors, so the circularity burden is minimal.

assumptions (5)
  • standard math Chow ring presentation of the permutohedral variety and perfect pairing (Proposition 2.1).
    Standard toric geometry used throughout Sections 2 and 3 for all intersection computations.
  • domain assumption The matroid class [X_M] is well-defined and satisfies the restriction formula [X_M]|_F = [X_{M|F}] tensor [X_{M/F}] (Lemma 2.18).
    Imported from [7, Proposition 5.3]; the key inductive step in Theorems 3.5, 3.7, and 3.8.
  • domain assumption Derksen's G-invariant is universal among valuative matroid invariants (Theorem 3.10, [18]).
    Provides the direction 'G equal implies all mixed Eulerian numbers equal' and the universal property used in Theorem 3.15.
  • domain assumption Bonin-Kung's theorem expresses G(M) as a nonnegative combination of catenary data gamma(b) with coefficients nu(M; b) (Theorem 3.12, [9]).
    Used in Proposition 3.13 to write G as a linear combination of matroidal mixed Eulerian numbers.
  • standard math Kunneth formula identifies CH(X_{Pi_F} x X_{Pi_{E\F}}) with the tensor product of Chow rings (used in Lemma 2.15).
    Allows splitting intersection numbers on a facet into products of two smaller permutohedral varieties.

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Pith. "Pith review of Mixed Eulerian numbers and beyond." pith.science (2026). https://pith.science/paper/NHAAH3YB

@misc{pith2026250204980,
  author       = {Pith},
  title        = {Pith review of: Mixed Eulerian numbers and beyond},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NHAAH3YB}},
  note         = {Machine review of arXiv:2502.04980}
}
abstract

We derive explicit formulas for the matroidal mixed Eulerian numbers. We resolve a question posed by Berget, Spink, and Tseng, demonstrating that the invariant defined by matroidal mixed Eulerian numbers is precisely equivalent to Derksen's $\mathcal{G}$-invariant. As an application, we provide the first explicit, non-recursive formula for mixed Eulerian numbers. Our combinatorial approach draws inspiration from the classical work of Schubert and incorporates the cutting-edge contributions of Huh.

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