{"id":"057c27f8-7c2f-4e06-b52f-ba076480eea8","arxiv_id":"2501.12055","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The bi-gamma coefficients of the 1/k-Eulerian polynomials are counted by increasing pruned even k-ary forests with a specified number of old leaves.","lead":"This paper gives a new way to count the coefficients that appear when the 1/k-Eulerian polynomials are expanded in a special basis. For combinatorists, the interest is a forest model whose leaves explain why these polynomials have alternatingly increasing coefficients.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 5.11, the omitted bijection between hat Y_n(k) and hat F_n(k), is the load-bearing gap: without it, the gamma expansion for x b_n^{(k)}(x) in Theorem 2.4 is unsupported.","rationale":"The paper aims to give the first combinatorial interpretation of the bi-gamma coefficients of the 1/k-Eulerian polynomials for general k. The framework is coherent: the bijections in §3 check out, the GFS-action in §4 is a plausible extension with Lemma 4.3 providing the correct orbit weight, and the derivations in §5.3 are algebraically consistent given the stated bijections. However, the printed proof is not self-contained. Proposition 5.11 is explicitly omitted, and Proposition 5.10 relies on two inverse identities that are asserted as routine. These are precisely the steps that transfer the weight-preserving bijection from pairs to forests on the hat (complement) side. Without them, the b-side gamma expansion is not established. The reader's verdict of CONDITIONAL is appropriate: the authors should provide the missing proof of Proposition 5.11 (and ideally a fully written verification of the Ψ-inverse identities) before the result is treated as proven. No evidence of a false statement was found; the concern is proof completeness, not correctness. Thus no change to the verdict is needed.","tokens_in":20396,"tokens_out":19328,"duration_ms":187410,"concrete_test":"Write a program that, for n ≤ 7 and k = 2, 3, generates all forests in hat F_n(k) and all pairs in hat Y_n(k), implements Γ and Γ′ exactly as defined in §5.2, and checks (i) Γ′(Γ(F,S)) = (F,S) for every (F,S) ∈ hat Y_n(k), and (ii) lleaf(Γ(F,S)) − si(Γ(F,S)) = lleaf(F) − si(F) + |S|. A counterexample would falsify Proposition 5.11 and thus the proof of (2.2); if all cases pass, it gives strong evidence that the omitted proof is routine, though the missing proof would still need to be supplied for full rigor.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Theorem 2.4, giving combinatorial interpretations of the bi-gamma coefficients of A_n^{(k)}(x). The b-side expansion (2.2) is derived in §5.3 from Proposition 5.13, which is Γ∘Θ between hat X_n(k) and hat F_n(k). Proposition 5.13 is obtained by composing Proposition 5.6 (hat version of Θ, sketched) with Proposition 5.11, which states that Γ and Γ′ induce a bijection between hat Y_n(k) and hat F_n(k). Proposition 5.11 is not proved; the text says it 'can be verified by the same reasoning as in the proof of Proposition 5.10 and the proof is omitted here.' If this bijection fails, the equality ∑_{F∈hat F_n(k)} x^{lleaf(F)−si(F)} = ∑_{F′∈hat F*_n(k)} x^{oleaf(F′)}(1+x)^{n−2oleaf(F′)} does not follow, and the coefficient interpretation for xb_n^{(k)}(x) collapses. The proof of the analogous Proposition 5.10 itself contains two 'routine' inverse identities Ψ_y(Ψ_x(F′)) = F′ that are not written out; these are nontrivial because Ψ_x has four cases and the ordering of removable leaves matters for β to pick the correct leaf. In the hat case, the rightmost tree has first k−1 children not all leaves, so case (ii) of Ψ_x (moving a singleton into T_m) behaves differently and the claimed smallest-removable-leaf property must be re-verified. No numerical counterexample is apparent, and the supporting sections are coherent, but the printed argument is incomplete at exactly the point that carries the b-side gamma expansion.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the 1/k-Eulerian polynomials A_n^{(k)}(x) of Savage and Viswanathan. Its main result, Theorem 2.4, asserts that in the symmetric decomposition A_n^{(k)}(x)=a_n^{(k)}(x)+x b_n^{(k)}(x), the polynomials a_n^{(k)}(x) and x b_n^{(k)}(x) have γ-expansions whose coefficients count increasing pruned even k-ary forests with a prescribed number of old leaves and no young leaves or removable old leaves. The proof proceeds by bijecting k-Stirling permutations to such forests (Section 3), introducing a generalized Foata–Strehl action on the trees (Section 4), and then constructing two transformations Θ and Γ on forests that are meant to prove the γ-expansions (Section 5).","tokens_in":20774,"tokens_out":9627,"duration_ms":98863,"significance":"If Theorem 2.4 is correct, it gives the first combinatorial interpretation of the bi-γ-coefficients of A_n^{(k)}(x) for all k, resolving an open problem noted in the paper. The forest model is natural, the γ-coefficients are defined by explicit forest statistics rather than fitted algebraically, and Theorem 4.1 is a useful standalone γ-positivity result. The paper also credits prior work accurately. The main caveat is proof completeness: the b-side expansion depends on Proposition 5.11, whose proof is omitted.","major_comments":[{"comment":"This proposition states that Γ and Γ′ induce a bijection between ̂Y_n(k) and ̂F_n(k), but the proof is omitted: the text says it 'can be verified by the same reasoning as in the proof of Proposition 5.10'. This is load-bearing: Proposition 5.13 composes this bijection with Proposition 5.6 to prove (5.4), the b-side of Theorem 2.4. In the hat case the rightmost tree does not have its first k−1 children as leaves, so the behavior of Ψ_x in case (ii) differs and the order of removable leaves under β must be re-verified. A full proof is required.","section":"Section 5.2, Proposition 5.11"},{"comment":"The inverse identities Ψ_y(Ψ_x(F′))=F′ are asserted as 'routine to check' twice, once in each direction of the inverse proof. These identities are not local formalities: Ψ_x has four cases and β chooses the smallest removable leaf, so the proof must establish that the singleton produced by α and the removable leaf chosen by β have the correct relative order. The same identities are needed in the omitted Proposition 5.11. Please write out the case analysis or provide a precise invariant that makes the cancellation immediate.","section":"Section 5.2, Proposition 5.10"},{"comment":"The proof that Θ′ maps ̂Y_n(k) into ̂X_n(k) is skeletal. The assertion 'Since F does not contain any removable young leaves, we have Φ_S2(F)∈̂F_n(k)' and the final 'similar arguments' for rleaf(F′)=0 hide exactly the verification that the hat condition is preserved. Because this proposition is one of the two ingredients of Proposition 5.13, the b-side expansion is not fully supported without a detailed argument.","section":"Section 5.1, Proposition 5.6"},{"comment":"This observation is stated as 'can be checked routinely', but it is used in Proposition 5.10 to conclude that y is the greatest element of S′∪{y}, which is what allows α to choose the correct singleton on the next step. Please supply a proof; the observation is not a mere remark.","section":"Section 5.2, Observation 5.9"}],"minor_comments":[{"comment":"Bijectivity of ξ is justified by 'It is apparent that the construction of ξ is reversible'; please state the inverse reconstruction explicitly, since the map is used to establish the first equalities in (2.1) and (2.2).","section":"Section 3, Proposition 3.1"},{"comment":"Only the equality Θ′(Θ(F,S))=(F,S) is shown; the reverse equality is delegated to 'similar arguments'. Please include the reverse direction or explain the symmetry.","section":"Section 5.1, Proposition 5.4"},{"comment":"Commutativity of Φ_x and Φ_y is asserted after the listed properties; since the orbit method in Theorem 4.1 depends on it, a short proof or a more explicit justification would improve readability.","section":"Lemma 4.2"},{"comment":"The extension of Lemma 4.3 from trees to forests deserves a sentence, because singleton components require separate treatment: a singleton contributes one to n but zero to oint and oleaf, which is exactly why si(F) appears in the forest identity.","section":"Section 5.3, equations (5.5) and (5.9)"},{"comment":"The submitted text contains many encoding artifacts (e.g., '1/slash.left k', '/summation.disp', '/parenleft.alt3'), which make the paper difficult to read; please ensure the published version uses a clean TeX rendering.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"I would not reject: the approach is promising, the small cases check out, and the main claim is plausibly correct. My recommendation hinges on the authors supplying a complete proof of Proposition 5.11 and the inverse identities in Section 5.2. If those cannot be supplied, the b-side theorem should be presented as conditional. The paper's scope and citation practice are appropriate for math.CO."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: the paper gives the first uniform forest model for the bi-gamma coefficients of the 1/k-Eulerian polynomials, settling an explicitly open problem from Ma-Ma-Yeh-Yeh. It is likely right. But the printed proof is incomplete at one load-bearing point: Proposition 5.11, the hat-forest analogue of the Gamma/Gamma-prime bijection, is stated with 'the proof is omitted here.' That bijection carries the x b_n^{(k)}(x) expansion in Theorem 2.4. Until it is supplied, the central claim for the b-side is conditional.\n\nWhat is actually new: Theorem 2.4 interprets the gamma coefficients gamma_{n,k,i} and hat-gamma_{n,k,i} as counts of increasing pruned even k-ary forests with i old leaves and no young or removable leaves. The construction of the xi and zeta bijections in Section 3 is clean, and the generalized Foata-Strehl action in Section 4 is a genuine contribution with independent interest: it yields gamma-positivity of the longest ascent-plateau polynomials c_n^{(k)}(x). The forest machinery is a good idea and the worked examples check out.\n\nWhere it is soft: besides the omitted Proposition 5.11, the proof of Proposition 5.10 contains two 'routine' inverse identities Psi_y(Psi_x(F')) = F'. Those are not trivial: Psi_x has four cases, and the ordering of removable leaves matters. In the hat case, the behavior of case (ii) changes when the rightmost tree's first k-1 children are not all leaves. The reader and stress-test are right to flag this. I want to stress that there is no apparent numerical counterexample; the surrounding structure is coherent and the proposition is quite likely true. But the paper as printed does not prove the second half of its main theorem.\n\nWho this is for: people working on gamma-positivity, Stirling permutations, and bi-gamma expansions. A serious editor should send this to a referee, not desk-reject. My recommendation: accept peer review, and require the missing proof of Proposition 5.11 plus a written verification of the inverse identities. If the authors deliver that, the paper is a real contribution.","headline":"A promising forest model that is likely correct, but the printed proof omits Proposition 5.11—the bijection carrying the b-side of Theorem 2.4—so the main theorem is conditional as written.","tokens_in":21320,"tokens_out":2179,"would_cite":true,"duration_ms":23528,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05A05","05A19","05C05","05E18"],"pacs":[],"model":"deepseek-v4-flash","headline":"A family of ordered labeled forests counts the bi-gamma coefficients of every $1/k$-Eulerian polynomial, giving the first purely combinatorial interpretation for all $k$.","keywords":["Stirling permutations","gamma-positivity","bi-gamma-positivity","1/k-Eulerian polynomials","increasing pruned even k-ary forests","symmetric decomposition","involution action on trees","ordered labeled forests"],"falsifier":"Enumerate, for a small case such as $n=4$ and $k=3$, all forests in $\\mathcal{F}_n(k)$ and $\\widehat{\\mathcal{F}}_n(k)$, compute the weighted sums $\\sum_F x^{\\mathrm{lleaf}(F)-\\mathrm{si}(F)}$ on each family, and compare them with the symmetric decomposition of $A_4^{(3)}(x)$ obtained from the known recurrence or from the ascent-plateau formula; any mismatch refutes Theorem 2.4. To target the omitted step directly, list $\\widehat{Y}_n(k)$ and $\\widehat{\\mathcal{F}}_n(k)$ for the same small case and verify that the maps $\\Gamma$ and $\\Gamma'$ are mutual inverses on those lists.","tokens_in":20206,"feed_emoji":"🌳","tokens_out":10655,"duration_ms":98004,"temperature":0.7,"pith_summary":"The paper claims to give the first combinatorial interpretation, valid for every positive integer $k$, of the bi-$\\gamma$-coefficients of the $1/k$-Eulerian polynomials $A_n^{(k)}(x)$, which at $k=1$ reduce to the classical Eulerian polynomials. Bi-$\\gamma$-positivity of these polynomials was known by algebraic methods, but the coefficients themselves had no combinatorial meaning. The new content is a model: the two summands in the symmetric decomposition of $A_n^{(k)}(x)$ are written as weighted sums over increasing pruned even $k$-ary forests, and the $\\gamma$-expansion is obtained by counting forests with a single distinguished statistic, the number of old leaves. A reader would care because combinatorial interpretations of $\\gamma$-coefficients turn abstract positivity statements into concrete enumerative meaning and typically open the door to refined structural properties such as unimodality and real-rootedness.","feed_headline":"Labeled forests count 1/k-Eulerian bi-gamma coefficients","feed_subtitle":"For every k, both gamma expansions come from forests with no young or removable leaves.","key_machinery":"The central objects are increasing pruned even $k$-ary forests: ordered forests whose trees have even-level nodes of degree $k$, pruned so no even-level node has only leaf children, with labels increasing along root-to-leaf paths and left-to-right among children. In such a forest a labeled leaf is old if it has the largest label among all grandchildren of its grandparent, and young otherwise; a singleton is a one-node tree. The paper's machinery consists of three pieces: a bijection from $k$-Stirling permutations to these forests matching the longest-ascent-plateau statistic to the number of labeled leaves minus singletons; a generalized involution action on trees that swaps an old internal node with a young leaf and leaves the rest of the tree unchanged, so each orbit has exactly one representative with no young leaves; and two forest transformations that re-root singletons and remove or insert removable leaves, which the authors use to convert the weighted forest sum into the claimed $\\gamma$-expansion.","core_discovery":"The central claim is Theorem 2.4. It states that if $A_n^{(k)}(x)=a_n^{(k)}(x)+xb_n^{(k)}(x)$ is the symmetric decomposition, then $a_n^{(k)}(x)=\\sum_{F\\in\\mathcal{F}_n(k)}x^{\\mathrm{lleaf}(F)-\\mathrm{si}(F)}$ expands as $\\sum_i\\gamma_{n,k,i}x^i(1+x)^{n-1-2i}$, and $xb_n^{(k)}(x)=\\sum_{F\\in\\widehat{\\mathcal{F}}_n(k)}x^{\\mathrm{lleaf}(F)-\\mathrm{si}(F)}$ expands as $\\sum_i\\widehat{\\gamma}_{n,k,i}x^i(1+x)^{n-2i}$. Here $\\gamma_{n,k,i}$ and $\\widehat{\\gamma}_{n,k,i}$ count forests in the respective families with exactly $i$ old leaves and with no young leaves and no removable old leaves. The proof constructs a bijection between $k$-Stirling permutations and these forests that sends the ascent-plateau statistic to a leaf statistic, introduces an involution action on the trees that collapses each orbit to a unique forest without young leaves, and then applies two transformations that move singletons and removable leaves in a weight-preserving way. The result is the first purely combinatorial description of the bi-$\\gamma$-coefficients for all $k$.","pith_inferences":["A natural extension the authors do not pursue is a $q$-refinement tracking the labels of old leaves or internal nodes, which would give a bivariate refinement of the bi-$\\gamma$ expansions and might connect to known $q$-Eulerian polynomials.","Because the omitted proof of Proposition 5.11 is the only step supporting the $\\widehat{\\gamma}$-side, a computer check for small $n$ and $k$ would be a cheap way to validate the second half of the theorem before relying on it.","The same pruning and leaf statistics might transfer to multiset or colored Stirling permutations, where analogous bi-$\\gamma$ questions are open.","The generalized involution action could be profitably compared with descent-set cyclic sieving phenomena, suggesting the forest model may carry finer homological or representation-theoretic meaning."],"forward_implications":["The bi-$\\gamma$-coefficients of every $1/k$-Eulerian polynomial are nonnegative integers with an explicit enumerative meaning, rather than merely an existence statement from algebra.","Specializing $k=1$ gives a forest model for the bi-$\\gamma$-expansion of the classical Eulerian polynomials.","The generalized involution action gives a new $\\gamma$-positivity proof for the longest ascent-plateau polynomials over $k$-Stirling permutations starting with 1, independently of the bi-$\\gamma$ statement.","Because bi-$\\gamma$-positivity implies the alternating-increasing property, the forest model gives a direct combinatorial witness to that structural property of these polynomials.","The two forest transformations provide a template that may apply to other polynomials whose symmetric decompositions are known."],"supporting_citations":[{"why":"Introduces the 1/k-Eulerian polynomials and their k-inversion sequence interpretation.","marker":"[26]"},{"why":"Proves the ascent-plateau interpretation of the 1/k-Eulerian polynomials over k-Stirling permutations.","marker":"[24]"},{"why":"Provides the symmetric decomposition and the known bi-gamma-positivity proof that Theorem 2.4 reinterprets.","marker":"[23]"},{"why":"Supplies the weakly increasing tree framework and the involution action on trees that Section 4 adapts.","marker":"[16]"},{"why":"Gives the classical permutation action that the tree action generalizes.","marker":"[11]"},{"why":"Introduces Stirling permutations, the objects the initial bijection starts from.","marker":"[13]"},{"why":"Establishes the symmetric decomposition and bi-gamma-positivity framework used throughout.","marker":"[5]"},{"why":"Articulates the program of purely combinatorial bi-gamma-positivity that this paper's interpretation addresses.","marker":"[1]"}],"fun_headline_variants":["Forests decode bi-γ coefficients of 1/k-Eulerian polynomials","Pruned forests give first combinatorial bi-γ proof for all k","Bi-γ coefficients from leaf-free k-ary forests","Combinatorial bi-γ positivity via ordered labeled forests","Forest rules for 1/k-Eulerian bi-γ coefficients"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the unproved bijection between the auxiliary pairs $\\widehat{Y}_n(k)$ and the forests $\\widehat{\\mathcal{F}}_n(k)$ (Proposition 5.11) holds exactly as stated; the paper says it follows by the same reasoning as an earlier proved case and omits the details. If that bijection fails, the claimed combinatorial interpretation of the coefficients $\\widehat{\\gamma}_{n,k,i}$ for the $xb_n^{(k)}(x)$ summand is unsupported.","fun_headline_variants_meta":{"raw":{"variants":["Forests decode bi-γ coefficients of 1/k-Eulerian polynomials","Pruned forests give first combinatorial bi-γ proof for all k","Bi-γ coefficients from leaf-free k-ary forests","Combinatorial bi-γ positivity via ordered labeled forests","Forest rules for 1/k-Eulerian bi-γ coefficients"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000701,"raw_usage":{"total_tokens":3233,"prompt_tokens":1085,"completion_tokens":2148,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":701,"completion_tokens_details":{"reasoning_tokens":2059}},"tokens_in":701,"tokens_out":2148,"duration_ms":16363,"temperature":1.0,"reasoning_tokens":2059,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T17:32:50.231304+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate, for a small case such as $n=4$ and $k=3$, all forests in $\\mathcal{F}_n(k)$ and $\\widehat{\\mathcal{F}}_n(k)$, compute the weighted sums $\\sum_F x^{\\mathrm{lleaf}(F)-\\mathrm{si}(F)}$ on each family, and compare them with the symmetric decomposition of $A_4^{(3)}(x)$ obtained from the known recurrence or from the ascent-plateau formula; any mismatch refutes Theorem 2.4. To target the omitted step directly, list $\\widehat{Y}_n(k)$ and $\\widehat{\\mathcal{F}}_n(k)$ for the same small case and verify that the maps $\\Gamma$ and $\\Gamma'$ are mutual inverses on those lists.","supporting_citations":[{"cited_title":"Athanasiadis, Gamma-positivity in combinatorics and geometr y, S´ em","cited_arxiv_id":null,"evidence_quote":"Articulates the program of purely combinatorial bi-gamma-positivity that this paper's interpretation addresses."},{"cited_title":"Savage and G","cited_arxiv_id":null,"evidence_quote":"Introduces the 1/k-Eulerian polynomials and their k-inversion sequence interpretation."},{"cited_title":"Ma and T","cited_arxiv_id":null,"evidence_quote":"Proves the ascent-plateau interpretation of the 1/k-Eulerian polynomials over k-Stirling permutations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the symmetric decomposition and the known bi-gamma-positivity proof that Theorem 2.4 reinterprets."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the weakly increasing tree framework and the involution action on trees that Section 4 adapts."},{"cited_title":"Foata and V","cited_arxiv_id":null,"evidence_quote":"Gives the classical permutation action that the tree action generalizes."},{"cited_title":"Gessel and R","cited_arxiv_id":null,"evidence_quote":"Introduces Stirling permutations, the objects the initial bijection starts from."},{"cited_title":"Br¨ and´ en and L","cited_arxiv_id":null,"evidence_quote":"Establishes the symmetric decomposition and bi-gamma-positivity framework used throughout."}],"review_version":1}