{"id":"0bd87c68-8a08-427c-98e4-5dbef4e5ccc0","arxiv_id":"2502.04980","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"First explicit formula for mixed Eulerian numbers, and proof that matroidal mixed Eulerian numbers determine Derksen's G-invariant.","lead":"This paper finds an explicit non-recursive formula for mixed Eulerian numbers, which generalize Eulerian and Catalan numbers, and shows the matroid version of these numbers is equivalent to a known universal matroid invariant. It settles an open question and gives a new tool for computing and comparing matroid invariants.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; main theorems survive scrutiny, with a minor scalar typo in Lemma 3.1 that does not affect the central claims.","rationale":"I read the paper as proving two central things: a first explicit non-recursive formula for mixed Eulerian numbers (Corollary 1.6) and the equivalence of matroidal mixed Eulerian numbers with Derksen's G-invariant (Theorem 1.12). The proof strategy is to express L-divisor monomials in the S-basis, compute S-intersections by restricting to products of smaller permutohedral varieties, and then reassemble the data. The load-bearing external input is Lemma 2.18, the factorization [X_M]|_F = [X_{M|F}] tensor [X_{M/F}], imported from [7, Proposition 5.3]. This is a known theorem, and the paper's applications are exactly the setting in which it is stated; non-flat subsets F are handled because M/F then has loops and the corresponding class vanishes. I checked the induction in Theorem 3.5 carefully. The displayed exponent c1-1 in equation (3.2) is not an error: Lemma 2.15 uses the copy of S_{b1} written as a sum of rays as the divisor being intersected, so only the remaining c1-1 copies are restricted. The subsequent binomial expansion is determined by degree conditions on the two factors, and the gamma factors in the final formula enforce the rank-increment-one flags needed for the catenary data in Proposition 3.13. I also tested small cases such as n=2, S_1^2, S_2^2, and S_1 S_2 against Theorem 1.5; the signs, binomial coefficients, and multinomial factors all agree. The only concrete issue I found is a scalar typo in Lemma 3.1: the stated q omits the final factorial a_l! = (n+1-|F_l|)!. For l=1, for example, the symmetrized ray sum is |E\\F|! times S_{a1}, not just S_{a1}. Since the proof only needs the symmetrized monomial to be a nonzero scalar multiple of a product of S_i's, the conclusion of Lemma 3.1 survives with a corrected scalar, and no later argument depends on the exact value of q. I therefore do not see a load-bearing correctness concern. The reader's weakest assumption, Lemma 2.18, is indeed the most external input, but it is a citable known theorem and its use is appropriate. The verdict ACCEPT remains appropriate, subject only to correcting the scalar typo in Lemma 3.1.","tokens_in":29057,"tokens_out":48639,"duration_ms":477229,"concrete_test":"Recompute Lemma 3.1 in the case n=3, l=2, F1={0}, F2={0,1}: count permutations mapping the chain to a fixed chain {i} subset of {i,j}; each monomial in the symmetrized sum has coefficient 2! = 2, while the printed q gives 1. Verify that inserting the missing a_l! = (n+1-|F_l|)! restores the equality, and that the corrected scalar still places the symmetrized monomial in the subalgebra generated by S_i, preserving the lemma's conclusion.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No load-bearing concern identified. The most external input is Lemma 2.18, imported from [7, Proposition 5.3], and the paper uses it in the intended setting; I found no mismatch that threatens Theorem 1.12 or Corollary 1.6. The induction in Theorem 3.5 is internally consistent: the apparent missing power in equation (3.2) is explained by Lemma 2.15, where the copy of S_{b1} written as a sum of rays is the intersected divisor, and only the remaining c1-1 copies are restricted. The gamma factors in Proposition 3.13 correctly force rank-increment-one flags via degree constraints, so the S-product intersection numbers do recover Bonin-Kung catenary data. The only concrete blemish is in Lemma 3.1: the displayed scalar q = (n+1-a1)! ... (a_{l-1}-a_l)! omits the factor a_l! = (n+1-|F_l|)!. For n=2 and F1={0}, the claimed equality fails: the symmetrized sum is 2*S_2, not 1*S_2. This does not affect the generation claim, since any nonzero scalar suffices, but the printed equality is false and should be corrected.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":29325,"tokens_out":18444,"duration_ms":177146,"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.","major_comments":[],"minor_comments":[{"comment":"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.","section":"Section 3.A, Lemma 3.1"},{"comment":"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.","section":"Section 3.C, Proposition 3.13"},{"comment":"The heading contains the typo \"matrodal\" and should read \"matroidal\".","section":"Section 3.C, heading"},{"comment":"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.","section":"Section 3.B, Theorem 3.8"},{"comment":"There are several residual typos and infelicities, including \"satsify\", \"polyope\", \"explecit\", and \"of of\"; a careful proofreading pass would be helpful.","section":"Throughout"}],"recommendation":"minor_revision","confidential_remarks":"I am comfortable with this manuscript after a minor revision. The main theorems appear correct; the only mathematical blemish I found is the scalar typo in Lemma 3.1, which does not affect the generation statement, and Proposition 3.13 is too terse about the coefficients D_c(b). The paper is a good fit for the journal and the attribution to prior work by Berget-Spink-Tseng, Katz-Kutler, Huh-Katz, Bonin-Kung, and Derksen-Fink is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Mateusz and coauthors have written a strong paper. The headline is that mixed Eulerian numbers finally have a non-recursive formula, and the matroidal version is shown to be a universal valuative invariant—precisely Derksen's G-invariant. The main theorems are proved in detail; the paper uses the Chow ring of the permutohedral variety, a restriction formula from Berget–Eur–Spink–Tseng, and the Huh–Katz/BEST machinery, all in their intended settings.\n\nWhat is genuinely new: Corollary 1.6 gives the first explicit closed formula for all mixed Eulerian numbers, side-stepping Croitoru's suggestion that a simple formula was unlikely. Theorem 1.12 answers a question of Berget–Spink–Tseng by showing matroidal mixed Eulerian numbers carry exactly the information of Derksen's G-invariant, hence all valuative matroid invariants. The Schubert-inspired change of basis from the L-divisors to the S-divisors is effective and well-motivated. The inductive proof of Theorem 3.5 is consistent; I checked the step where one copy of S_{b1} is written as a sum of rays and the restrictions are applied. The gamma factors in Proposition 3.13 do recover the Bonin–Kung catenary data.\n\nThe soft spots are minor. In Lemma 3.1, the displayed coefficient q omits the factor a_l!; for n=2 and F1={0}, the symmetrized sum is 2S_2, not 1S_2. The correction is a one-line fix. It does not affect the generation claim, since any nonzero scalar suffices, but the printed equality is false. Also, the closed formula in Corollary 1.6 involves an alternating sum over many terms; it is explicit but not efficient. The recursive formulas in Theorems 3.7 and 3.8 address that, so the paper is honest about practice. The most external input, Lemma 2.18 (restriction of [X_M] to a facet), is a known theorem from [7] and is used correctly.\n\nOverall, this is a solid contribution in algebraic combinatorics and matroid theory. Anyone working on valuative invariants, matroid Chow rings, or Eulerian numbers will want to cite it. It deserves a serious referee; I would accept with minor revisions. Bring it to reading group next week.","headline":"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.","tokens_in":29818,"tokens_out":3672,"would_cite":true,"duration_ms":33109,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14C17","14N15","52A39","05E14","52B40"],"pacs":[],"model":"deepseek-v4-flash","headline":"Matroidal mixed Eulerian numbers are shown to be exactly the universal valuative invariant, with an explicit non-recursive formula.","keywords":["matroidal mixed Eulerian numbers","G-invariant","valuative invariants","permutohedral variety","Chow ring","Tutte polynomial","mixed volumes","closed formula"],"falsifier":"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.","tokens_in":28890,"feed_emoji":"🔢","tokens_out":6911,"duration_ms":62233,"temperature":0.7,"pith_summary":"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.","feed_headline":"Mixed Eulerian numbers encode every valuative matroid invariant","feed_subtitle":"Closed formulas and an equivalence with the universal G-invariant settle the role of these intersection numbers.","key_machinery":"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.","core_discovery":"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).","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the restriction formula $[X_M]|_F=[X_{M|F}]\\otimes[X_{M/F}]$ that drives every induction and the change-of-basis computation.","marker":"[7]"},{"why":"Introduces matroidal mixed Eulerian numbers, proves log-concavity of matroid h-vectors, and poses the question answered here.","marker":"[8]"},{"why":"Gives the catenary-data decomposition of the G-invariant used to express it as a linear combination of mixed Eulerian numbers.","marker":"[9]"},{"why":"States that a simple closed formula for mixed Eulerian numbers is unlikely, providing the baseline that Corollary 1.6 overturns.","marker":"[15]"},{"why":"Defines the G-invariant whose universality is the paper's target.","marker":"[17]"},{"why":"Proves the G-invariant is universal among valuative matroid invariants, the property Theorem 1.12 exploits.","marker":"[18]"},{"why":"Identifies the boundary cases $A_M(l,0,\\dots,0,r-l)$ with coefficients of the reduced characteristic polynomial, giving the $\\gamma_M(l)$ factors.","marker":"[31]"},{"why":"Earlier work on matroidal mixed Eulerian numbers, supplying the restriction formulas for the divisors $L_i$ used in Section 2.","marker":"[32]"},{"why":"Introduces mixed Eulerian numbers as mixed volumes of hypersimplices and records the classical combinatorial sequences they generalize.","marker":"[43]"}],"fun_headline_variants":["Mixed Eulerian numbers are the universal valuative invariant","Explicit non-recursive formula for every mixed Eulerian number","Mixed Eulerian numbers equal the universal G-invariant","Universal matroid invariants reduce to mixed Eulerian numbers","Mixed Eulerian numbers: one formula to rule all valuative invariants"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Mixed Eulerian numbers are the universal valuative invariant","Explicit non-recursive formula for every mixed Eulerian number","Mixed Eulerian numbers equal the universal G-invariant","Universal matroid invariants reduce to mixed Eulerian numbers","Mixed Eulerian numbers: one formula to rule all valuative invariants"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000694,"raw_usage":{"total_tokens":3074,"prompt_tokens":815,"completion_tokens":2259,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":431,"completion_tokens_details":{"reasoning_tokens":2176}},"tokens_in":431,"tokens_out":2259,"duration_ms":16921,"temperature":1.0,"reasoning_tokens":2176,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T20:44:48.914928+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Tautological classes of matroids","cited_arxiv_id":null,"evidence_quote":"Supplies the restriction formula $[X_M]|_F=[X_{M|F}]\\otimes[X_{M/F}]$ that drives every induction and the change-of-basis computation."},{"cited_title":"Log-conc avity of matroid h-vectors and mixed Euler- ian numbers","cited_arxiv_id":null,"evidence_quote":"Introduces matroidal mixed Eulerian numbers, proves log-concavity of matroid h-vectors, and poses the question answered here."},{"cited_title":"Bonin and Joseph P.S","cited_arxiv_id":null,"evidence_quote":"Gives the catenary-data decomposition of the G-invariant used to express it as a linear combination of mixed Eulerian numbers."},{"cited_title":"Mixed volumes of hypersimplices, root systems and shifted Y oung tableaux","cited_arxiv_id":null,"evidence_quote":"States that a simple closed formula for mixed Eulerian numbers is unlikely, providing the baseline that Corollary 1.6 overturns."},{"cited_title":"Symmetric and quasi-symmetric function s associated to polymatroids","cited_arxiv_id":null,"evidence_quote":"Defines the G-invariant whose universality is the paper's target."},{"cited_title":"Valuative invariants for po lymatroids","cited_arxiv_id":null,"evidence_quote":"Proves the G-invariant is universal among valuative matroid invariants, the property Theorem 1.12 exploits."},{"cited_title":"Log-concavity of characteristi c polynomials and the Bergman fan of matroids","cited_arxiv_id":null,"evidence_quote":"Identifies the boundary cases $A_M(l,0,\\dots,0,r-l)$ with coefficients of the reduced characteristic polynomial, giving the $\\gamma_M(l)$ factors."},{"cited_title":"Matroidal Mixed Eulerian Numbers","cited_arxiv_id":"2305.19095","evidence_quote":"Earlier work on matroidal mixed Eulerian numbers, supplying the restriction formulas for the divisors $L_i$ used in Section 2."},{"cited_title":"Permutohedra, associahedra, an d beyond","cited_arxiv_id":null,"evidence_quote":"Introduces mixed Eulerian numbers as mixed volumes of hypersimplices and records the classical combinatorial sequences they generalize."}],"review_version":1}