{"id":"b87cca17-6d7e-4790-b6d1-143d785905ed","arxiv_id":"2412.10556","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A graph's chromatic quasisymmetric function is symmetric only under strong constraints; the paper proves connected DAGs with multiple sources/sinks are nonsymmetric and identifies a new symmetric family.","lead":"This paper studies when the chromatic quasisymmetric function of a graph is actually symmetric, a property that underlies e-positivity conjectures. It proves a striking product theorem in quasisymmetric functions, settles which trees have symmetric functions, and introduces a new symmetric family called mixed mountain graphs.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the Lemma 4.10 inversion is valid at the multiset level, so the reader's proposed gap does not land.","rationale":"The reader's conditional verdict rests mainly on an alleged gap in Lemma 4.10: that the recolored vertex of color k+1 must be the source of every changed chain. I re-derived the injection map and found that the position of k+1 in the chain is irrelevant. After sorting, the image chain is just the sorted multiset with 1 replacing k+1; replacing the unique 1 by k+1 and sorting recovers the original multiset for any position of k+1. Since a maximum-ascent coloring is strictly increasing along each chain, the sorted assignment is exactly the original chain coloring. Thus injectivity holds. The non-surjectivity argument also holds because phi never creates a new k+1: it only changes k+1 into 1 and permutes existing colors within a chain, so the constructed coloring with v colored k+1 cannot be an image. I found no other load-bearing issue in Theorem 1.4 or the mountain graph constructions; the only minor weakness is the one-sentence proof of Lemma 4.5, which is easily repairable and does not affect the main theorem. Since the reader's specific objection does not land, I would not adjust the reader's verdict on that basis, although the conditional verdict remains defensible for other reasons such as the lack of reproducible computational code.","tokens_in":23285,"tokens_out":45751,"duration_ms":405372,"concrete_test":"Implement Lemma 4.10 for all connected DAGs up to n=8: enumerate the maximum-ascent coloring classes K, fix a Dilworth chain decomposition R, build phi, and verify that the stated inversion recovers every preimage and that the image set is strictly smaller. The reader's concern predicts a failure when k+1 is not the source of a changed chain; the expected outcome of this test is that all inversions succeed and the inequality of coefficients at q^|E| holds as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I do not find a load-bearing flaw in the central claim. The reader's concern about Lemma 4.10 appears to rest on a misreading of the inversion. If a changed chain originally has colors c1 < ... < cj = k+1 < ... < cm, then phi replaces k+1 by 1 and sorts, giving (1, c1, ..., c_{j-1}, c_{j+1}, ..., cm). Replacing the image's source color 1 with k+1 and sorting returns (c1, ..., c_{j-1}, k+1, c_{j+1}, ..., cm), exactly the original chain, regardless of the position j of the k+1. The map on multisets 'replace k+1 by 1' is invertible by 'replace 1 by k+1', and sorting is compatible because every chain is already strictly increasing in a maximum-ascent coloring. Non-surjectivity also stands: phi never creates a new k+1, since it only replaces k+1 by 1 and otherwise permutes the existing colors within a chain, so a preimage of the constructed coloring would have to give v color k+1, contradicting Lemma 4.8. The only weakness I noticed is that Lemma 4.5's proof that S(G) is nonempty is terse ('every sink paired with a unique source... disconnected'); the connectivity argument can be completed by considering the first merge point of two source-chains, so this is a presentational gap, not a threat to the theorem.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies when the chromatic quasisymmetric function X_G(x;q) of a labeled graph is symmetric. It proves a product theorem (Theorem 1.4): if a product of two quasisymmetric functions in countably infinitely many variables is symmetric, then both factors are symmetric, which reduces the symmetry question for CQFs to connected graphs. It then proves (Theorem 1.6) that any connected directed acyclic graph with more than one source or sink has a nonsymmetric CQF, yielding as a corollary that a tree has a symmetric CQF if and only if it is a directed path. Finally, it introduces a family of mixed mountain graphs and proves their CQFs are symmetric (Theorem 1.8) by constructing ascent-preserving involutions and automorphisms that swap adjacent color multiplicities. The authors also report Sage verification on all connected labeled graphs up to eight vertices.","tokens_in":23513,"tokens_out":26283,"duration_ms":850119,"significance":"If the results stand, Theorem 1.6 settles the tree question posed in [4] and, together with Theorem 1.4, reduces the global symmetry question to connected DAGs with exactly one source and one sink. Theorem 1.4 is a clean standalone statement with useful consequences beyond the graph application. The proofs are mostly explicit and combinatorial, with no fitted parameters, no circular dependencies, and a constructive treatment of the mixed mountain graph family. The main limitation is that Theorem 1.8 establishes symmetry but not e-positivity, so the paper does not directly resolve Question 1.3; the computational evidence is suggestive but not load-bearing for the theorems.","major_comments":[],"minor_comments":[{"comment":"The claim that QSym is a UFD is justified by the text 'there are a finite number of generators of each degree, so any given polynomial lies in a finitely generated ring that has unique factorization'; finite generation alone does not imply unique factorization. Please cite the algebraic independence of the Hazewinkel generators, or Grinberg–Reiner Corollary 6.5.33, explicitly, since this step is load-bearing for Theorem 1.4.","section":"§3, Theorem 1.4 proof"},{"comment":"The induction hypothesis is stated for a product of two factors, but the proof concludes that every irreducible factor a_i is symmetric from u = a_1...a_r. This requires iterating the two-factor statement degree by degree; the iteration is straightforward but should be said explicitly.","section":"§3, Theorem 1.4, Case 2"},{"comment":"The proof that S(G) is nonempty is compressed: it says that if every sink were paired with a unique source then G would be disconnected, without explaining why two source-to-sink paths in a connected graph must merge before reaching distinct sinks. Spelling out the first merge point would make the argument complete.","section":"§4.2, Lemma 4.5"},{"comment":"The injectivity argument is very terse. It would help to state explicitly that unchanged chains still contain color k+1, changed chains have lost it, and the inverse simply recolors the source of a changed chain with k+1 and re-sorts; this makes the inversion valid regardless of where k+1 sat in the original chain.","section":"§4.2, Lemma 4.10"},{"comment":"The number of swap applications should be m(p-m), the number of adjacent pairs needed to reverse the order of the m k-cliques and p-m bottomless k+1-cliques; the printed expression 'a(p-a)' appears to use the color parameter a and should be corrected.","section":"§5.3, Proposition 5.26"},{"comment":"The terminology 'bottomless k+1-mountain' used in the introduction and Theorem 1.8 is not obviously the same as the graph B_{p,k} defined in §5.2, where a k-clique has the edge between its lower vertices removed. Please reconcile the definitions so the number of vertices in a bottomless mountain is unambiguous.","section":"§5.2–5.3"},{"comment":"There are several small typos, including 'Supopse' in Remark 3.9 and 'CHROMA TIC' in the running header, which should be corrected in a final revision.","section":"Miscellaneous"}],"recommendation":"minor_revision","confidential_remarks":"I examined the stress-test concern about Lemma 4.10 and it does not land: the map φ is injective at the level of chain color multisets because replacing k+1 by 1 and sorting is inverted by replacing the source 1 with k+1 and sorting again, independent of the original position of k+1. I found no load-bearing mathematical error. The main needs are exposition and notation cleanup, so minor revision is appropriate. The paper fits the journal's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I read the paper with the reader's Lemma 4.10 worry in mind. I think it is a misreading. The inversion does not require knowing where the k+1 sat before sorting; the map acts on each chain by replacing the unique k+1 with 1 and then sorting. Since every chain in a maximum-ascent coloring is strictly increasing, the output chain is the sorted version of that multiset. Replacing the unique 1 with k+1 and sorting recovers the original chain, whatever its length or the position of k+1. The non-surjectivity argument also stands: phi never introduces a new k+1, so a preimage of a coloring where a vertex of S(G) is colored k+1 would need that vertex to have color k+1 already, which Lemma 4.8 forbids. So the main DAG theorem is supported.\n\nWhat is genuinely new: Theorem 1.4 is a striking factorization result in infinite variables, and it cleanly reduces the symmetry question to connected graphs. Theorem 1.6 and its tree corollary answer an open question and extend the earlier path/star results by a real margin. The mixed mountain family is a nice addition, especially because it is not unit interval in general, so it broadens the known symmetric territory. The proofs are long but mostly careful; the mountain section has a lot of moving parts and would benefit from more figures or a few more checks, but I do not see a gap.\n\nSoft spots in proportion: Lemma 4.5's proof that S(G) is nonempty is terse enough to need a sentence or two extra; the connectivity argument is not hard but as written it can trip a reader. The Sage computation is mentioned without code or data; that is a reproducibility nit, not a mathematical flaw. The mountain graph swap map relies on a case analysis that takes some effort to verify, but it appears correct.\n\nWho should read it: anyone working on chromatic symmetric functions or on quasisymmetric factorization more broadly. It is a solid contribution that deserves a serious referee. I would send it out; if I were the referee, I would accept after minor revision, mostly for exposition and for supplying the computational data.","headline":"The product theorem and the DAG nonsymmetry result are real and appear sound; the internal-review concern about Lemma 4.10 does not hold up on reading.","tokens_in":24102,"tokens_out":5180,"would_cite":true,"duration_ms":44751,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E05","05C15","05C20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that the chromatic quasisymmetric function of a connected acyclic directed graph is symmetric only when the graph has exactly one source and one sink, that directed paths are the only trees with symmetric CQF, and that…","keywords":["chromatic quasisymmetric function","quasisymmetric functions","symmetric functions","directed acyclic graphs","graph colorings","chain decompositions","mountain graphs","e-positivity"],"falsifier":"Enumerate all labeled connected directed acyclic graphs with exactly two sources and two sinks up to eight or nine vertices and, for each, compute the coefficients of $M_{(1^k,a,1^{n-k-a})}$ and $M_{(a,1^{n-a})}$ at $q^{|E|}$ in $X_G(x;q)$; the theorem predicts the two coefficients always differ, so a graph where they agree would refute the classification.","tokens_in":23027,"feed_emoji":"🏔️","tokens_out":10253,"duration_ms":89532,"temperature":0.7,"pith_summary":"The paper asks which graphs have a chromatic quasisymmetric function (the $q$-weighted generating function of proper colorings) that is actually symmetric rather than merely quasisymmetric. It proves three things: a product of two quasisymmetric functions in infinitely many variables is symmetric only if both factors are, so the symmetry question splits into connected components; every connected directed acyclic graph with more than one source or more than one sink has a nonsymmetric CQF, which makes directed paths the only trees with symmetric CQF; and a new family called mixed mountain graphs always has symmetric CQF. The first result matters because it removes disjoint unions from consideration, the second settles the tree question, and the third provides a nontrivial family with which to test whether every symmetric CQF is e-positive.","feed_headline":"Only one-source, one-sink acyclic directed graphs can be symmetric","feed_subtitle":"A new proof classifies acyclic graphs and adds the mixed-mountain family of examples.","key_machinery":"The main objects are the chromatic quasisymmetric function $X_G(x;q)$, which records proper colorings by the monomial $x_{\\kappa(1)}\\cdots x_{\\kappa(n)}$ times $q^{\\#\\text{ascents}}$, and the monomial quasisymmetric functions $M_\\alpha$ in which its non-symmetry is detected. For the negative results, the load-bearing mechanism is a comparison of the coefficients of two $M_\\alpha$'s at the maximal $q$-power $q^{|E|}$: a chain decomposition of the graph (minimum number of disjoint chains equal to the largest antichain) is used to build a map between the two weight classes whose failure to be surjective is visible in these coefficients. For the positive results, the machinery is a family of ascent-preserving bijections—an involution on colorings whose $(a,a+1)$-colored subgraph avoids the bottom edge, and 'cycle', 'reflect', and 'swap' maps on colorings that include it—which together swap the counts of any two adjacent colors.","core_discovery":"The central claim is a structural dichotomy. On the nonsymmetric side, for any connected directed acyclic graph whose sources outnumber one or whose sinks outnumber one, the coefficient of $q^{|E|}$ in the expansion of $X_G(x;q)$ in the monomial quasisymmetric basis distinguishes two weight classes: the maximum-ascent colorings of weight $(1^k,a,1^{n-k-a})$ force the weight-$(a,1^{n-a})$ class to miss $q^{|E|}$, so the function cannot be symmetric. On the symmetric side, the paper shows that every mixed mountain graph—a cycle of cliques and bottomless cliques strung together and oriented left-to-right—admits ascent-preserving bijections that swap the counts of any two adjacent colors, which is exactly what symmetry of a CQF requires. Together with the product theorem, these results reduce the symmetry problem to connected graphs and give the first tree classification.","pith_inferences":["If the dichotomy in the nonsymmetry theorem holds, the remaining open classification is the connected single-source single-sink directed acyclic graphs that contain a directed Hamiltonian path; mixed mountain graphs are one family inside that class, and unit interval orders appear to be another up to eight vertices.","The paper's open question—whether every symmetric CQF is e-positive—can be tested on mixed mountain graphs: if any mixed mountain graph fails e-positivity the answer is no, and if all pass that supports a positive answer.","The product theorem genuinely depends on having infinitely many variables: in finitely many variables, $x_1^2x_2\\cdot x_1x_2^2 = x_1^3x_2^3$ is symmetric while the factors are not, so any finite-variable analogue would need a different mechanism."],"forward_implications":["Symmetry of a disjoint union forces each connected component's CQF to be symmetric, so the classification needs only connected graphs.","A connected directed acyclic graph with two or more sources or two or more sinks has a nonsymmetric CQF; a symmetric CQF forces exactly one source and one sink and a directed path through every vertex.","Among oriented trees, the directed path is the unique graph with a symmetric CQF.","Among directed acyclic cycles, exactly the naturally oriented cycles are symmetric.","The mixed mountain graphs are a new infinite family of symmetric CQFs that are not generally natural unit interval graphs."],"supporting_citations":[{"why":"It supplies the symmetry result for natural unit interval graphs used for directed paths and as the baseline for the mountain-graph involutions.","marker":"[30]"},{"why":"It supplies the Lyndon-word generators of QSym used in the proof that the product of two quasisymmetric functions is symmetric only if both are.","marker":"[22]"},{"why":"It gives the factorization theorem in the full power-series ring that extends the product theorem to arbitrary power series.","marker":"[25]"},{"why":"It states the chain-decomposition theorem used to construct the map $\\phi_{G,R}$ in the nonsymmetry proof.","marker":"[11]"},{"why":"It proves symmetry for naturally oriented cycles, the comparison case for the cycle corollary.","marker":"[13]"},{"why":"It establishes the earlier partial path and star nonsymmetry results whose tree question this paper answers.","marker":"[4]"},{"why":"It is the source of the observation that unequal numbers of sources and sinks force nonsymmetry.","marker":"[27]"}],"fun_headline_variants":["Symmetric CQF demands at most one source and one sink","Among trees, only directed paths admit symmetric CQF","Mixed mountain graphs: a new symmetric CQF family","Product theorem simplifies CQF symmetry to connected case"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof that the map between the two weight classes is injective but not surjective rests on the assumption that the recolored vertex of color $k+1$ is the source of every chain that changes; the argument only shows each changed chain contains a color-$k+1$ vertex, and if a chain also contains one of the singleton colors $2,\\dots,k$, its source has the smaller color and the inversion step breaks.","fun_headline_variants_meta":{"raw":{"variants":["Symmetric CQF demands at most one source and one sink","Among trees, only directed paths admit symmetric CQF","Mixed mountain graphs: a new symmetric CQF family","Product theorem simplifies CQF symmetry to connected case"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000799,"raw_usage":{"total_tokens":3475,"prompt_tokens":863,"completion_tokens":2612,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":479,"completion_tokens_details":{"reasoning_tokens":2548}},"tokens_in":479,"tokens_out":2612,"duration_ms":21404,"temperature":1.0,"reasoning_tokens":2548,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T15:52:32.006391+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate all labeled connected directed acyclic graphs with exactly two sources and two sinks up to eight or nine vertices and, for each, compute the coefficients of $M_{(1^k,a,1^{n-k-a})}$ and $M_{(a,1^{n-a})}$ at $q^{|E|}$ in $X_G(x;q)$; the theorem predicts the two coefficients always differ, so a graph where they agree would refute the classification.","supporting_citations":[{"cited_title":"Chromatic quasisymmetric functions","cited_arxiv_id":null,"evidence_quote":"It supplies the symmetry result for natural unit interval graphs used for directed paths and as the baseline for the mountain-graph involutions."},{"cited_title":"Explicit polynomial generators for the ring of quasisymmetric functions over the integers","cited_arxiv_id":null,"evidence_quote":"It supplies the Lyndon-word generators of QSym used in the proof that the product of two quasisymmetric functions is symmetric only if both are."},{"cited_title":"P-partition products and fundamental quasi-symmetric function positivity","cited_arxiv_id":null,"evidence_quote":"It gives the factorization theorem in the full power-series ring that extends the product theorem to arbitrary power series."},{"cited_title":"On enumerators of Smirnov words by descents and cyclic descents","cited_arxiv_id":null,"evidence_quote":"It proves symmetry for naturally oriented cycles, the comparison case for the cycle corollary."},{"cited_title":"Chromatic Quasisymmetric Functions of the Path Graph","cited_arxiv_id":null,"evidence_quote":"It establishes the earlier partial path and star nonsymmetry results whose tree question this paper answers."},{"cited_title":"personal communication","cited_arxiv_id":null,"evidence_quote":"It is the source of the observation that unequal numbers of sources and sinks force nonsymmetry."}],"review_version":1}