{"id":"b13727cf-aa5e-4dfc-a218-f2eebfeec4e8","arxiv_id":"1908.07920","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A set of permutations is cyclic Schur-positive exactly when it is Schur-positive and closed under cyclic rotation of descent sets; the criterion yields new examples and proves two conjectures.","lead":"Permutations have a statistic called the descent set, and some families of permutations can be built from standard Young tableaux; this paper adds a cyclic analogue of that property. The authors show that a family has the cyclic property exactly when it is Schur-positive and invariant under cyclic rotations of descent sets, which settles two open conjectures and gives new examples.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 4.8's block-reordering step is not fully justified; a small-case verification should settle whether Lemma 4.2 and Theorem 4.1 stand.","rationale":"The reader's weakest_assumption already identifies the same point: Lemma 4.8 is the load-bearing link in the proof of Theorem 4.1. I agree that this is the most fragile part of the paper. However, I differ from the reader on the appropriate verdict: the proof of Lemma 4.8 invokes a Des-preserving reordering of more than two blocks without a complete proof that the descents at block boundaries are preserved when a starred block is moved. This is not an accusation of error; it is an identified gap in justification that a small computational check can resolve. Since the main theorem and its applications all depend on this step, and since the paper does not provide an explicit general bijection for Lemma 4.8, acceptance should be conditional on an independent verification of that lemma. If the check passes, the verdict should be ACCEPT; if it fails, the corresponding applications would need substantial revision.","tokens_in":125,"tokens_out":16110,"duration_ms":240656,"concrete_test":"For every n ≤ 8, every t ≤ 4, every weak composition γ of n−1, every k ≤ n, every decomposition α+β=γ with |α|=k, |β|=n−k, and every i ∈ [t], compute the multisets {cDes(π) : π ∈ (4.5)} and {cDes(π) : π ∈ (4.6)} from Lemma 4.8. If any multiset mismatch occurs, Lemma 4.8 is false. If no mismatch occurs, the cDes-equidistribution at least holds in all small cases, which would remove the immediate objection and support the lemma's truth.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central engine for Theorem 4.1 is Lemma 4.2, whose proof depends on Lemma 4.8. The delicate step in Lemma 4.8 is the assertion that, by 'using Lemma 4.3 repeatedly,' one can reorder blocks in the shuffles S(β_i,...,β*_t,β_1,...,β_{i-1}) and S(α_{i+1},...,α_t,α_1,...,α*_i) to obtain the reversed block orders in (4.7), with a Des-preserving bijection. Lemma 4.3 is proven only for shuffles of two increasing sequences; the extension to reordering more than two blocks is described but not proven to preserve global descents when some blocks are starred (i.e., when the position of the largest letter is fixed) and when blocks are moved across one another. If this reordering introduces or removes descents at block boundaries, the cDes multisets of the two sides of Lemma 4.8 can differ, so no cDes-preserving bijection exists. That would invalidate Lemma 4.2, and with it the inclusion-exclusion proof of Theorem 4.1, the equidistribution of cDes on CnD^{-1}_{n-1,J} versus D^{-1}_{n-1,J}Cn, and the consequences Theorem 4.9, Corollary 4.12, and the resolution of Conjecture 10.2 of [9]. The proof of Lemma 4.8 is otherwise non-bijective and does not supply an explicit map; the special cases in Sections 5 and 6 cover only |J|=1 and J=[i], not the general block-reordering claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a notion of cyclic Schur-positivity (cSp) for sets of permutations, extending classical Schur-positivity via cyclic descent sets on permutations and on standard Young tableaux of skew shapes. The central result, Theorem 3.4, characterizes cSp sets as exactly those subsets of S_n that are Schur-positive and cDes-invariant. Building on this, the authors prove that horizontal rotation closures of Schur-positive sets are cSp (Theorem 3.11) and that, for every inverse descent class, the cyclic descent set distribution is the same on vertical and horizontal rotations (Theorem 4.1). Consequences include Theorem 4.9, asserting that vertically rotated inverse descent classes are cSp; Corollary 4.10, resolving a conjecture from [2]; and Corollary 4.12, resolving Conjecture 10.2 of [9]. The paper also supplies explicit bijections for the singleton descent case (Section 5) and for arc permutations (Section 6), the latter giving a bijective proof of the cSp property for arc permutations.","tokens_in":19271,"tokens_out":10632,"duration_ms":96136,"significance":"If the main results stand, the paper makes a substantial contribution to the study of Schur-positive permutation sets. Theorem 3.4 is an elegant and useful characterization, and Theorem 4.1, together with its corollaries, resolves open conjectures and produces new families of Schur-positive sets. The explicit bijections in Sections 5 and 6 are a definite strength, as is the paper's clear organization and the largely self-contained proof of Theorem 3.4. The main concern is that the proof of the technical Lemma 4.8, on which Theorem 4.1 depends, is not fully justified in the present version. The central idea is plausible and the rest of the argument is coherent, but the missing details are load-bearing rather than cosmetic.","major_comments":[{"comment":"The assertion that 'Using Lemma 4.3 repeatedly' one can reorder the blocks in S(β_i,...,β_t*,β_1,...,β_{i-1}) to obtain the reversed order in Eq. (4.7) is not justified by Lemma 4.3 as stated. Lemma 4.3 is proved only for shuffles of two increasing sequences, while the extension to t blocks is merely described, with no proof that the composed maps preserve descents when some blocks are starred, i.e., when the maximum of a block is fixed at the end. Since Lemma 4.8 is the engine for Lemma 4.2 and hence for Theorem 4.1, this gap needs to be filled. A proof by induction on t, or an explicit verification for t=3 and all small n, would settle the point.","section":"Section 4.1, Lemma 4.8, Eq. (4.7)"},{"comment":"The two equivalences for i=1 and for i≠1 are asserted without specifying the bijections. In particular, the step 'the equivalence follows by interchanging the order of the β's just as we did above with the α's' changes the location of the starred block, and the claim that this preserves descent sets is not immediate. Because the starred block determines the fixed last letter, the cyclic descent set is sensitive to the star's position. The authors should either write down the bijections explicitly or prove the equality of the cDes multisets by a direct computation. This is a load-bearing step in the proof of Lemma 4.2.","section":"Section 4.1, Lemma 4.8, displayed equivalences after Eq. (4.7)"},{"comment":"The final step of the proof of Lemma 4.2 applies Lemma 4.3 to the set S(γ_t−1,...,(γ_1+1)*) to obtain S(γ_1,...,γ_t*). This again relies on the unproved multi-block extension of Lemma 4.3 with a starred block. The reader needs a precise statement of the extended lemma, including the behavior of the star under the reversal, and a proof that the map is cDes-preserving. Without this, the proof of Lemma 4.2, and therefore of Theorem 4.1, is incomplete.","section":"Section 4.1, proof of Lemma 4.2"}],"minor_comments":[{"comment":"The title reads 'sets of permutation'; it should be 'sets of permutations'. The typeset title also contains an odd spacing in 'PERMUT A TION'.","section":"Title and abstract"},{"comment":"The final expression 'ψσ = ¯σ c^j = 6728194356' has ten digits and cannot be a permutation in S_9; it should presumably be '672819435' or another nine-letter permutation. Please correct this typo.","section":"Section 6.2, Example 6.10"},{"comment":"In the decomposition of binary words as w = 1^{i_1}2^{j_1} 2|1 1^{i_2}2^{j_2} 2|1 ... , the notation '2|1' is easy to misread. Please clarify explicitly that the bars mark the descent positions 21 that are kept fixed.","section":"Section 4.1, proof of Lemma 4.3"},{"comment":"The label 'parallelshort' in Figure 3 appears to be a typo; it is unclear what is meant. The figure would benefit from a clearer caption explaining the two cases.","section":"Section 5, Figure 3"}],"recommendation":"major_revision","confidential_remarks":"The paper is well written and the main ideas are attractive. My recommendation is driven solely by the incomplete justification of Lemma 4.8, which is essential for Theorem 4.1. If the authors can supply a rigorous proof of the multi-block reversal with starred blocks, or verify it by an exhaustive small-case computation, I would support acceptance. I do not see grounds for rejection, as the rest of the argument is coherent and the results are significant."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read on Bloom–Elizalde–Roichman's cyclic Schur-positivity paper. The core new idea is Theorem 3.4: a set A ⊆ S_n is cyclic Schur-positive iff it is Schur-positive and cDes-invariant. That is a genuine characterization, and it makes the notion much more tractable. The applications flow naturally from it: Theorem 3.11 shows horizontal rotations of Schur-positive sets are cSp, and the equidistribution result Theorem 4.1 gives Schur-positivity for vertical rotations of inverse descent classes, settling Conjecture 10.2 of [9] and, via Corollary 4.10, a conjecture from [2]. The exposition is clear and credits prior work appropriately; the self-citations are to established theorems, not to the conjectures being solved.\n\nThe soft spot is Lemma 4.8, which is the engine behind Theorem 4.1. The proof says 'using Lemma 4.3 repeatedly' to reorder blocks, but Lemma 4.3 is proved only for shuffles of two increasing sequences. The extension to t ≥ 2 is described but not proved as a formal lemma, and the starred-block conditions, which track the position of the largest letter, are not explicitly checked in the reordering. If that reordering introduces or removes descents at block boundaries, the cDes multisets on the two sides of Lemma 4.8 could differ, and Theorem 4.1 would fail. I don't see a counterexample, and the explicit bijections in Sections 5 and 6 for the cases |J|=1 and J=[i] lend credibility to the general claim. But the gap is real enough that a referee should ask for a detailed proof of the block-reordering step, or at least a small-case verification. Everything else in the paper holds up: the characterization theorem is proved cleanly, the inclusion-exclusion argument from Lemma 4.2 to Theorem 4.1 is sound once Lemma 4.2 is granted, and the arc permutation bijection in Section 6 is explicit and checkable.\n\nWho is this for? Algebraic combinatorialists working on Schur-positivity, descent sets, and cyclic descents. It deserves a serious referee rather than a desk reject; the main theorems are new, the writing is honest about where proofs are non-bijective, and the resolved conjectures are real. The likely fix is a more careful proof of Lemma 4.8, possibly with a computer verification for small n. I would send it to peer review with a note to focus attention on that lemma.","headline":"A solid paper whose characterization theorem for cyclic Schur-positivity is clean and useful, but whose proof of the key equidistribution result depends on a block-reordering step in Lemma 4.8 that deserves a close independent check.","tokens_in":19834,"tokens_out":14324,"would_cite":true,"duration_ms":115564,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E05","05E10","05A05","05A19"],"pacs":[],"model":"deepseek-v4-flash","headline":"A set of permutations is cyclic Schur-positive exactly when it is Schur-positive and invariant under a cyclic-descent rotation.","keywords":["Schur-positive sets","cyclic descents","descent sets","standard Young tableaux","arc permutations","inverse descent classes","equidistribution","rotation closures"],"falsifier":"Test Lemma 4.8 directly on small cases: fix a composition $\\gamma$, choose a split into weak compositions $\\alpha$ and $\\beta$, and compare the multiset of descent sets on the two sets displayed in (4.5) and (4.6); any mismatch would disprove the claimed $\\mathrm{cDes}$-preserving bijection and with it Theorem 4.1.","tokens_in":18763,"feed_emoji":"🔄","tokens_out":10357,"duration_ms":168278,"temperature":0.7,"pith_summary":"The paper introduces cyclic Schur-positivity (cSp), a cyclic-descent analogue of the classical notion of Schur-positivity for sets of permutations, and proves a complete characterization: a subset of $S_n$ is cSp exactly when it is Schur-positive and $\\mathrm{cDes}$-invariant, meaning it admits a bijection that rotates every cyclic descent set. This reduces a statement about quasisymmetric functions to a checkable combinatorial symmetry. The characterization immediately makes every horizontally rotated Schur-positive set cSp, and the paper goes on to prove that vertically rotated inverse descent classes are cSp as well. The proof route yields a new equidistribution result for cyclic descent sets on the two rotation directions, and it settles two open conjectures about Schur-positivity of inverse descent classes and of arc permutations.","feed_headline":"Cyclic descents settle two conjectures on Schur-positive sets","feed_subtitle":"A new characterization shows arc permutations and rotated inverse descent classes are cyclic Schur-positive.","key_machinery":"The engine is the cyclic descent set $\\mathrm{cDes}$, defined for permutations by wrapping the descent set around the cycle, and extended to standard Young tableaux of skew shapes that are not connected ribbons. Theorem 3.4 is the main characterization. For the equidistribution result, the machinery is a sequence of descent-preserving block reorderings: shuffles of increasing sequences are encoded as binary words, reversed by the descent-preserving map $f$ from Lemma 4.3, and split through the shuffle decomposition to reduce vertical rotations to horizontal ones. Lemma 4.8 reassembles the blocks into a single shuffle while preserving descent sets, and inclusion-exclusion converts this into the theorem on inverse descent classes.","core_discovery":"The central claim is Theorem 3.4: A subset $A$ of $S_n$ is cyclic Schur-positive if and only if it is Schur-positive and $\\mathrm{cDes}$-invariant, i.e. there is a bijection $\\psi\\colon A\\to A$ with $\\mathrm{cDes}(\\psi\\pi)=1+\\mathrm{cDes}(\\pi)$ for every $\\pi$. Because horizontal rotation gives such a bijection automatically, every Schur-positive set that is invariant under horizontal rotation is cSp; conversely the theorem shows every cSp set is Schur-positive. Building on this, the paper proves that the vertical rotations $C_n D_{n-1,J}^{-1}$ of inverse descent classes are cSp, that the union of all rotations of a fixed inverse cyclic descent set is cSp, and that arc permutations are cSp, with an explicit $\\mathrm{cDes}$-preserving bijection from arc permutations to a disjoint union of SYT of shapes $(n-k-1,1^k)\\oplus(1)$.","pith_inferences":["Theorem 3.4 gives a practical certificate: to prove a $\\mathrm{cDes}$-invariant set Schur-positive, it is enough to exhibit a single rotation bijection, and conversely any $\\mathrm{cDes}$-invariant Schur-positive set decomposes explicitly into skew SYT.","The inclusion-exclusion proof of Theorem 4.1 suggests that the full equidistribution could be made bijective by composing the Section 5 and Section 6 maps; a direct map would likely extend to broader unions of grid classes.","Because arc permutations are vertical rotations of left-unimodal permutations, the same $\\mathrm{cDes}$-preserving bijection may transfer to other rotation-closed permutation classes, generating new Schur-positive sets beyond the ones listed.","The explicit SYT targets in Theorem 6.11 have a strip shape, so the same cyclic-descent generating functions may be pluggable into known cyclic sieving results for promotion."],"forward_implications":["Every Schur-positive set in $S_{n-1}$ has a horizontal rotation closure in $S_n$ that is cyclic Schur-positive (Theorem 3.11).","For every $J\\subseteq[n-2]$, the vertically rotated inverse descent class $C_n D_{n-1,J}^{-1}$ is cyclic Schur-positive, hence Schur-positive (Theorem 4.9).","For every nonempty proper $J\\subseteq[n]$, the set of permutations whose inverse has cyclic descent set $i+J$ for some $i$ is cyclic Schur-positive, resolving the conjecture from [2] (Corollary 4.10).","The distribution of $\\mathrm{Des}$ over $C_n D_{n-1,J}^{-1}$ equals that over $D_{n-1,J}^{-1} C_n$, resolving Conjecture 10.2 of [9] (Corollary 4.12).","Arc permutations form a cyclic Schur-positive set, with a bijective proof matching their $\\mathrm{cDes}$-distribution to SYT of the near-hook strip shapes (Corollary 6.3 and Theorem 6.11)."],"supporting_citations":[{"why":"establishes which skew shapes admit cyclic descent extensions, the existence fact behind the definition of cSp and the uniqueness lemma used in Theorem 3.4","marker":"[3]"},{"why":"gives the descent-set characterization of Schur-positive permutation sets on which Definition 1.4 is modeled","marker":"[4, Prop. 9.1]"},{"why":"contains the inverse descent class conjectures solved here and the earlier Schur-positivity results for these classes","marker":"[9]"},{"why":"proves that horizontal rotation closures of Schur-positive sets are Schur-positive, a key input to Theorems 3.11 and 4.9","marker":"[10, Theorem 1.1]"},{"why":"introduces fundamental quasi-symmetric functions and supplies the Schur-positivity of Knuth classes and inverse descent classes","marker":"[11]"},{"why":"provides the geometric grid class framework used in Section 6 for the arc permutation bijection","marker":"[5]"},{"why":"supplies the arc permutation classes, their pattern avoidance description, and their Schur-positivity","marker":"[8]"},{"why":"formulates the cyclic quasi-symmetric function conjecture that Corollary 4.10 resolves","marker":"[2]"},{"why":"gives the explicit combinatorial description of cyclic descent sets on general skew tableaux used in the bijections to SYT","marker":"[13]"},{"why":"supplies Gessel's identity expressing fundamental quasi-symmetric functions as skew Schur functions, used in the converse of Theorem 3.4","marker":"[16, Theorem 7.19.7]"}],"fun_headline_variants":["Cyclic Schur-positivity characterization resolves two conjectures","Arc permutations and rotated descent classes are cyclic Schur-positive","New cyclic descent bijection proves two conjectures","Rotations yield cyclic Schur-positivity for inverse descent classes","A cyclic descent bijection yields new Schur-positive examples"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of the equidistribution theorem rests on a technical claim that certain block rearrangements of permutations preserve descent sets even when distinguished entries are moved; if even one such rearrangement fails, the inclusion-exclusion argument and the resolved conjectures collapse.","fun_headline_variants_meta":{"raw":{"variants":["Cyclic Schur-positivity characterization resolves two conjectures","Arc permutations and rotated descent classes are cyclic Schur-positive","New cyclic descent bijection proves two conjectures","Rotations yield cyclic Schur-positivity for inverse descent classes","A cyclic descent bijection yields new Schur-positive examples"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0019,"raw_usage":{"total_tokens":7409,"prompt_tokens":871,"completion_tokens":6538,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":487,"completion_tokens_details":{"reasoning_tokens":6456}},"tokens_in":487,"tokens_out":6538,"duration_ms":460458,"temperature":1.0,"reasoning_tokens":6456,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:53:30.154148+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Test Lemma 4.8 directly on small cases: fix a composition $\\gamma$, choose a split into weak compositions $\\alpha$ and $\\beta$, and compare the multiset of descent sets on the two sets displayed in (4.5) and (4.6); any mismatch would disprove the claimed $\\mathrm{cDes}$-preserving bijection and with it Theorem 4.1.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"establishes which skew shapes admit cyclic descent extensions, the existence fact behind the definition of cSp and the uniqueness lemma used in Theorem 3.4"},{"cited_title":"Elizalde and Y","cited_arxiv_id":null,"evidence_quote":"contains the inverse descent class conjectures solved here and the earlier Schur-positivity results for these classes"},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"introduces fundamental quasi-symmetric functions and supplies the Schur-positivity of Knuth classes and inverse descent classes"},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the geometric grid class framework used in Section 6 for the arc permutation bijection"},{"cited_title":"Elizalde and Y","cited_arxiv_id":null,"evidence_quote":"supplies the arc permutation classes, their pattern avoidance description, and their Schur-positivity"},{"cited_title":"Cyclic quasi-symmetric functions","cited_arxiv_id":"1811.05440","evidence_quote":"formulates the cyclic quasi-symmetric function conjecture that Corollary 4.10 resolves"},{"cited_title":"Huang, Cyclic descents for general skew tableaux , J","cited_arxiv_id":null,"evidence_quote":"gives the explicit combinatorial description of cyclic descent sets on general skew tableaux used in the bijections to SYT"}],"review_version":1}