{"id":"af461d2f-8ab1-4f2f-8505-dfecad1551e3","arxiv_id":"2506.00349","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Littlewood-Richardson coefficients are counted by peelable tableaux on shuffle diagrams, and this rule proves a special case of the Lam-Postnikov-Pylyavskyy Schur log-concavity conjecture.","lead":"The authors introduce a new combinatorial rule that counts Littlewood-Richardson coefficients using peelable tableaux on shuffle diagrams, and they use it to prove a special case of a Schur log-concavity conjecture. The rule also gives a new way to compute generalized Littlewood-Richardson coefficients for Temperley-Lieb immanants and shows that Bender-Knuth involutions realize the symmetry of these coefficients.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.2 rests on Corollary 2.7, whose extension to arbitrary skew shapes is asserted without proof in Remark 2.9; this unproved removal of the NP25 shape constraints is the load-bearing gap.","rationale":"The reader's weakest assumption identifies Remark 2.9, and that is exactly the most load-bearing gap in the paper. Tracing the argument: Theorem 1.2 is proved by combining Corollary 2.7 with Proposition 2.13 and Lemma 2.19. Corollary 2.7 says that Yamanouchi shuffle tableaux count Littlewood–Richardson coefficients for arbitrary skew shapes. This is not a new theorem in the present paper; it is derived from [NP25, Theorem 5.1], which the authors themselves say was stated with additional shape constraints. Remark 2.9 asserts those constraints are unnecessary, but gives no proof. If the constraints are needed, the whole chain from shuffle tableaux to peelable tableaux can break at the very first step, before any of the paper's new bijective arguments are reached. The paper is well written and the examples are consistent, and I found no internal contradiction in the later constructions that would make the theorem false independently of this gap. The Section 4 injection is also only sketched, and several of its lemmas would need full proofs, but that section supports the application and is not the foundation of the main counting theorem. The correct scientific response is therefore to keep the conditional verdict: the results are plausible and worth developing, but the unproved removal of the NP25 shape constraints must be supplied, and a computational enumeration would be a quick way to test whether the gap is real. Since the reader already reached exactly this position, no change to the verdict is needed.","tokens_in":18153,"tokens_out":12751,"duration_ms":120978,"concrete_test":"Write a small enumerator for shuffle tableaux of shape (lambda/mu) ⊛ (nu/rho), implement the §2.1 operators E_i, and count Yamanouchi tableaux (no E_i applies) for every pair of skew shapes lambda/mu, nu/rho with total size <= 9, including shapes that violate the NP25 admissibility conditions referenced in Remark 2.9. Compare each count with c^kappa_{lambda/mu, nu/rho} computed by an independent standard Littlewood–Richardson algorithm, and also compare with the D-peelable count from Definition 1.1. A single mismatch settles that Remark 2.9 is false and Theorem 1.2 fails; agreement over a broad enumeration would strongly support the removal but would still not replace the missing proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central formula Theorem 1.2 is obtained from Corollary 2.7 (Yamanouchi shuffle tableaux count c^kappa) via the bijections phi and psi. Corollary 2.7 in turn is presented as a direct consequence of [NP25, Theorem 5.1] that Temperley–Lieb crystals are type A Kashiwara crystals. But Remark 2.9 states that [NP25] imposes additional constraints on the shapes lambda/mu and nu/rho so that the crystal operators behave well with the Temperley–Lieb type, and that these constraints are unnecessary for this section. No argument, lemma, or reference supporting this removal is supplied. If the constraints are actually necessary, then for arbitrary skew shapes either E_i and F_i may fail to be well-defined on shuffle tableaux, or the crystals may fail to decompose as type A Kashiwara crystals, and the equality 'Yamanouchi count = c^kappa' can fail even though the character identity sum_T omega(T) = s_{lambda/mu} s_{nu/rho} holds. Since the proof of Theorem 1.2 never verifies the unrestricted-shape claim independently, the main theorem currently depends on an unproved assertion. This is a proof gap, not a disagreement with consensus: the paper itself explicitly acknowledges that [NP25] has these constraints, and Remark 2.9 is the only evidence offered that they can be dropped.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a new combinatorial formula for Littlewood-Richardson coefficients: for two skew shapes λ/μ and ν/ρ, the coefficient c^κ_{λ/μ,ν/ρ} is counted by D-peelable tableaux of shape κ, where D is the shuffle diagram (λ/μ) ⊛ (ν/ρ). The proof strategy is to exhibit a bijection φ between D-compatible tableaux and Yamanouchi shuffle tableaux, invoke the result of [NP25] that Yamanouchi shuffle tableaux count Littlewood-Richardson coefficients, and then construct a second bijection ψ between D-compatible and D-peelable tableaux via standardization. The paper also derives a corollary for Temperley-Lieb immanants, proves a Bender-Knuth symmetry statement, and uses the new rule to prove a special case of the Lam-Postnikov-Pylyavskyy Schur log-concavity conjecture.","tokens_in":18459,"tokens_out":4414,"duration_ms":39636,"significance":"If correct, Theorem 1.2 gives a new, and potentially computationally efficient, way to compute Littlewood-Richardson coefficients for products of skew Schur functions, with a direct application to Temperley-Lieb immanants in Corollary 2.23. The Bender-Knuth symmetry result is elegant, and the application to Schur log-concavity is a nontrivial step toward a known conjecture. However, the paper relies at several load-bearing points on assertions that are not fully proved, most critically the unconditional validity of the Yamanouchi criterion for arbitrary skew shapes. These gaps must be repaired before the paper can be accepted.","major_comments":[{"comment":"The central equality between the number of Yamanouchi shuffle tableaux and the Littlewood-Richardson coefficient is stated in Corollary 2.7 as a consequence of [NP25, Theorem 5.1], but then Remark 2.9 asserts that the additional shape constraints imposed in [NP25] are unnecessary for this section. No proof, example, or reference is supplied for this removal. Since Theorem 1.2 and Corollary 2.23 both pass through this step for arbitrary skew shapes, this unproved assertion is load-bearing. The authors must either prove that the crystal operators remain well-defined and that the Temperley-Lieb crystals decompose as type A Kashiwara crystals without those constraints, or they must restrict the statements of Corollary 2.7, Theorem 1.2, and the subsequent applications to the shapes covered by [NP25].","section":"Section 2.1, Remark 2.9"},{"comment":"Proposition 2.13, which establishes the bijection between D-compatible tableaux and Yamanouchi shuffle tableaux, is the first pillar of the main theorem, but its proof is only a sketch. The individual lemmas (2.14-2.17) contain the main arguments, but Lemma 2.17 in particular is dense and contains unclear statements, such as 'for S to be Yamanouchi, the has to be non-(r-2,r-1)-overlapped (r-2)-squares right of h but left of i' on page 9. Given that this proposition is load-bearing for Theorem 1.2, the authors should provide a complete, self-contained proof of the bijection, with all cases in Lemma 2.17 written out clearly.","section":"Section 2.2, Proposition 2.13"},{"comment":"Lemma 2.19, which asserts that inverse standardization converts D-compatible tableaux into D-peelable tableaux and standardization converts D-peelable tableaux back into D-compatible tableaux, is stated with 'the proof ... is straightforward and is left to the readers.' This lemma is the second pillar of Theorem 1.2, as it is exactly the bridge between the D-compatible/Yamanouchi bijection and the D-peelable count. Leaving this proof to the reader is not acceptable for a central step; a detailed proof should be included, or at minimum a rigorous statement of the correspondence between the NE matching condition and the inverse-standardization recipe.","section":"Section 2.3, Lemma 2.19"},{"comment":"The proof of Theorem 1.7 rests on the injection θ constructed in Definition 4.3, but the accompanying proof sketch lists five lemmas, several of which are proved only sketchily. In particular, Lemma 4.10 is proved in a single sentence: if a required square cannot be found, 'the path of θ forms a sequence of λm increasing entries, each strictly East of the previous. Hence, by the NE matching condition of peelable tableaux, µm is at least λm.' This is a nontrivial implication that is not spelled out, and it is crucial to well-definedness. Lemma 4.12 also makes several claims (e.g., the existence of tk+1, the relative position of sk+1 and tk+1, and the recursive step) that are asserted rather than fully proved. Since the injection is the core of the Schur log-concavity application, these proofs must be completed.","section":"Section 4, Theorem 4.6 and Lemmas 4.10-4.14"},{"comment":"Lemma 4.1(iii) is garbled as printed: 'µm − µℓ−1 ≥ · · · ≥µk − µℓ−1 ≤ a − 1' is not a coherent inequality, and the quantity ℓ is not defined in the lemma (it appears elsewhere as the number of parts of λ). This lemma is used repeatedly in the proof of the injection, so the statement must be corrected and proved. As written, it is not possible for a reader to verify the argument.","section":"Section 4, Lemma 4.1"}],"minor_comments":[{"comment":"In Example 2.8, the text says 'and κ = 3' and then 'cκ = 3', but κ should be the partition (4,3,2) as in Example 1.3, not the integer 3.","section":"Section 2.1, Example 2.8"},{"comment":"In Step (4) of Definition 4.3, the list of involved squares contains a typo: 'sk+2, tk+2, sk+3, tk+2, . . .' should presumably read 'sk+2, tk+2, sk+3, tk+3, . . .'.","section":"Section 4, Definition 4.3"},{"comment":"There is a spelling error in Definition 4.3: 'repear' should be 'repeat'.","section":"Section 4, Definition 4.3"},{"comment":"The sentence 'Assume for simplicity that ℓ(λ) = ℓ(ν) (if ℓ(λ) > ℓ(µ), we can append 0's at the end of µ to get ℓ(λ) = ℓ(ν))' is confusing, as it refers to both ℓ(λ)=ℓ(ν) and ℓ(λ)>ℓ(µ) without clear relation; the theorem statement uses ℓ = ℓ(λ). Please clarify which partitions are padded in the statement and proof of Theorem 1.4.","section":"Section 1.3 and Theorem 1.4"},{"comment":"The instruction 'Sort all the rows in increasing order' appears in both definitions of φ and ψ, but it is not precise about how the sorted rows are assembled into a shuffle tableau while preserving the required row and column conditions. A formal description of the sorting procedure would improve clarity.","section":"Section 2.2 and 2.3, Definitions 2.12 and 2.20"},{"comment":"The phrase 'if there is a 2 k-square in column some c ≥ a' is ungrammatical; it should be 'in some column c ≥ a'.","section":"Section 4, Lemma 4.12"}],"recommendation":"major_revision","confidential_remarks":"The paper depends heavily on [NP25], which is authored by one of the present authors. This is not itself a problem, but the unproved removal of the NP25 shape constraints in Remark 2.9 should be resolved before publication. I would also suggest that the editor ask the authors to provide full proofs for the lemmas currently relegated to 'straightforward' or 'sketch' status, since the main theorem and the log-concavity application both rest on them."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The new thing here is a D-peelable tableau rule on shuffle diagrams, extending the Remmel–Whitney idea to products of two skew Schur functions. There is also a Bender–Knuth symmetry proof and an injection proving a special case of the Lam–Postnikov–Pylyavskyy Schur log-concavity conjecture. The paper is clearly written, with worked examples that actually help, and the bijection φ between D-compatible and Yamanouchi shuffle tableaux in Section 2 is mostly proved in detail. The BK symmetry argument is genuinely elegant, and the injection θ in Section 4 is a real construction, even if the proof is compressed.\n\nThe soft spots are in proportion to how much they carry. The biggest one is Corollary 2.7. It inherits the Yamanouchi-to-LR equality from [NP25, Theorem 5.1], and that theorem came with extra shape constraints. Remark 2.9 simply asserts those constraints are unnecessary for this section, with no argument or reference. That is load-bearing: if the crystal operators fail to be well-defined or the Temperley–Lieb crystals do not decompose as type A Kashiwara crystals for general skew shapes, the equality can fail even though the character identity holds. The proof of Theorem 1.2 never checks this independently. This is a real proof gap, not a disagreement with consensus. It may be fixable, but it has to be fixed, or the theorem must be restated under the NP25 hypotheses.\n\nSecond, Lemma 2.19, which connects D-compatible tableaux to D-peelable tableaux via inverse standardization, is left to the reader. That is the bridge to the main theorem, and it deserves a few lines. Third, Section 4's injection θ is supported by Lemma 4.10 with a one-sentence proof, and Lemma 4.1(iii) looks garbled as printed. The construction is intricate enough that a referee will need the details. The efficiency claim for peelable versus Yamanouchi tableaux is also unbenchmarked, but that is minor.\n\nWho is this for? People working on LR combinatorics, crystal bases, and Schur positivity. The formula, if it holds, is a useful addition to the toolkit, and the application to the LPP conjecture is a genuine special case. The gaps are substantial but look fillable, and the paper does not seem to be hiding anything—it explicitly flags the constraint issue in Remark 2.9. I would send it to a serious referee rather than desk reject. If the referee can close the Remark 2.9 gap, the paper is in decent shape; if not, the authors need to restrict the statement.","headline":"A new peelable-tableau rule for LR coefficients that is plausible and worth refereeing, but the unrestricted-shape claim in Corollary 2.7 and the Section 4 injection need real proofs before the results are fully established.","tokens_in":18994,"tokens_out":1613,"would_cite":true,"duration_ms":17533,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E05","05E10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper establishes that Littlewood-Richardson coefficients count peelable tableaux of a given shape on a shuffle diagram, a new rule that also speeds up Temperley-Lieb immanant computations and proves a special case of the Schur…","keywords":["Littlewood-Richardson coefficients","shuffle tableaux","peelable tableaux","Temperley-Lieb immanants","Schur log-concavity","Bender-Knuth involutions","crystal operators","Schur positivity"],"falsifier":"Enumerate $D$-peelable tableaux of shape $\\kappa$ for a pair of skew shapes outside the constrained family of the prior Temperley-Lieb crystal construction, for instance by brute-force enumeration over all partitions $\\kappa$ of size at most 5, and compare with the classical Littlewood-Richardson rule; a single mismatch would falsify Theorem 1.2 as stated.","tokens_in":17960,"feed_emoji":"🧩","tokens_out":14280,"duration_ms":120921,"temperature":0.7,"pith_summary":"The paper claims a new combinatorial formula for Littlewood-Richardson coefficients: given two skew shapes $\\lambda/\\mu$ and $\\nu/\\rho$, build the shuffle diagram $D$ by interlacing their squares, and let $a_i$ be the number of columns of $D$ containing squares in rows $i$ and $i+2$. Then the coefficient $c^{\\kappa}_{\\lambda/\\mu,\\nu/\\rho}$ equals the number of semistandard tableaux of shape $\\kappa$ that are $D$-peelable, meaning that for each $i$ there are at least $a_i$ disjoint pairs of an $i$-square and an $(i+2)$-square with the $i$-square northeast of the $(i+2)$-square. This extends the Remmel-Whitney peelable-tableau rule from products of two Schur functions to products of arbitrary skew Schur functions. It matters because the rule gives a checkable way to compute these structure constants, a faster route to the generalized Littlewood-Richardson coefficients attached to Temperley-Lieb immanants of Jacobi-Trudi matrices, and a proof of a special case of a Schur log-concavity conjecture.","feed_headline":"Peelable tableaux count Littlewood-Richardson coefficients","feed_subtitle":"A shuffle-diagram rule also speeds up Temperley-Lieb immanants and proves a Schur log-concavity case.","key_machinery":"The carrying object is the shuffle diagram $(\\lambda/\\mu) \\circledast (\\nu/\\rho)$, obtained by interlacing the squares of the two skew shapes so that rows and columns of the two factors alternate; together with its peelability condition, it packages the Littlewood-Richardson rule into a northeast-matching condition between entries differing by 2. The argument's load-bearing chain is: Yamanouchi shuffle tableaux on $D$ count $c^{\\kappa}_{\\lambda/\\mu,\\nu/\\rho}$ because Temperley-Lieb crystals are type A Kashiwara crystals; the map $\\varphi$ gives a bijection between Yamanouchi shuffle tableaux and $D$-compatible tableaux (standard tableaux obeying two relative-position conditions read from the diagram); and inverse standardization converts $D$-compatible tableaux into $D$-peelable tableaux, whose defining condition is precisely the standardized content condition. The Bender-Knuth involution sequence $BK_1 \\circ BK_3 \\circ \\cdots \\circ BK_{2\\ell-1}$ is the symmetry mechanism, and the injection $\\theta$ of Section 4 is the positivity mechanism for the Schur log-concavity special case.","core_discovery":"The central claim is Theorem 1.2: let $D$ be the shuffle diagram of shape $(\\lambda/\\mu) \\circledast (\\nu/\\rho)$, formed by placing squares of $\\lambda/\\mu$ at odd coordinates $(2i-1,2j-1)$ and squares of $\\nu/\\rho$ at even coordinates $(2i,2j)$, and let $a_i$ be the number of columns with squares on both row $i$ and row $i+2$. A semistandard Young tableau $T$ of shape $\\kappa$ is $D$-peelable if for every $i$ there are at least $a_i$ disjoint pairs of an $i$-square and an $(i+2)$-square with the $i$-square northeast of the $(i+2)$-square. The theorem asserts that the Littlewood-Richardson coefficient $c^{\\kappa}_{\\lambda/\\mu,\\nu/\\rho}$ counts exactly these $D$-peelable tableaux of shape $\\kappa$. The proof passes through a bijection between $D$-compatible standard tableaux and Yamanouchi shuffle tableaux (shuffle tableaux on which no lowering crystal operator can be applied), then identifies $D$-peelable tableaux with inverse standardizations of $D$-compatible tableaux; because Yamanouchi shuffle tableaux on a shuffle diagram are known to count Littlewood-Richardson coefficients, the count transfers.","pith_inferences":["The authors do not state it, but the peelable criterion suggests a direct enumeration algorithm: count semistandard tableaux of shape $\\kappa$ and check only the pairwise column-matching condition, which would avoid crystal operators entirely and could be benchmarked against the Yamanouchi count on random small skew shapes.","As an extension the paper does not pursue, the path of $\\theta$ in Section 4 hints that if the path argument extends to other partitions with a block of equal rows followed by a tail, the injection would prove additional cases of the Schur log-concavity conjecture beyond the vertical-strip family.","The paper treats only the swap symmetry; because the Bender-Knuth bijection is shape-preserving and local, composing odd-indexed involutions with their inverses may yield tableau-level proofs of other symmetries of Littlewood-Richardson coefficients, such as the cyclic symmetry.","The shuffle-diagram construction naturally suggests a rule for iterated products of several skew Schur functions by interlacing more than two shapes, a generalization the paper does not pursue."],"forward_implications":["Theorem 1.2 gives a new enumeration of $c^{\\kappa}_{\\lambda/\\mu,\\nu/\\rho}$ for products of arbitrary skew Schur functions, reducing the computation to counting tableaux that satisfy a simple pairwise matching condition.","Corollary 2.23 turns the coefficient of $s_{\\lambda}$ in any Temperley-Lieb immanant of a Jacobi-Trudi matrix into a count of peelable tableaux, bypassing the slower Yamanouchi-checking procedure.","Theorem 1.4 shows the symmetry $c^{\\kappa}_{\\lambda/\\mu,\\nu/\\rho}=c^{\\kappa}_{\\nu/\\rho,\\lambda/\\mu}$ is realized by a shape-preserving bijection given by odd-indexed Bender-Knuth involutions.","Theorem 1.7 proves Schur positivity of $s_{\\nu}s_{\\rho}-s_{\\lambda}s_{\\mu}$ for the family $\\lambda=(a^k,1^{n-k-1},0)$, $\\nu=\\lambda-(e_m+\\cdots+e_k)$, confirming a special case of the Schur log-concavity conjecture."],"supporting_citations":[{"why":"Supplies Temperley-Lieb crystals and the theorem that Yamanouchi shuffle tableaux on a shuffle diagram count Littlewood-Richardson coefficients, which the present paper replaces with a peelable-tableau count.","marker":"[NP25]"},{"why":"Introduced peelable tableaux for the Remmel-Whitney rule on the skew diagram $\\lambda*\\mu$, the construction generalized here to shuffle diagrams.","marker":"[RS98]"},{"why":"Gave the Remmel-Whitney rule for multiplying Schur functions, the pattern this paper adapts to products of skew Schur functions.","marker":"[R W84]"},{"why":"Defined Bender-Knuth involutions, which Theorem 1.4 uses to realize the swap symmetry of Littlewood-Richardson coefficients.","marker":"[BK72]"},{"why":"Source of the Schur log-concavity conjecture addressed by Theorem 1.7 and of the immanant-based approach to Schur positivity.","marker":"[LPP07]"},{"why":"Introduced Temperley-Lieb immanants, whose generalized Littlewood-Richardson coefficients Corollary 2.23 computes.","marker":"[RS05]"},{"why":"Provides the Hecke-algebra context and the theorem connecting determinants to dual canonical bases that motivates the Temperley-Lieb immanant coefficients.","marker":"[Hai93]"}],"fun_headline_variants":["Peelable tableaux: new formula for LR coefficients","Shuffle diagrams give new LR coefficient rule","Peelable tableaux count LR and aid log-concavity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument rests on an unproved assertion (Remark 2.9) that the prior construction of Yamanouchi shuffle tableaux works for all skew shapes, not only the restricted shapes where it was originally proved; if that assertion fails, the new formula for arbitrary skew shapes fails with it.","fun_headline_variants_meta":{"raw":{"variants":["Peelable tableaux: new formula for LR coefficients","Shuffle diagrams give new LR coefficient rule","Peelable tableaux count LR and aid log-concavity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001674,"raw_usage":{"total_tokens":6649,"prompt_tokens":962,"completion_tokens":5687,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":578,"completion_tokens_details":{"reasoning_tokens":5634}},"tokens_in":578,"tokens_out":5687,"duration_ms":48653,"temperature":1.0,"reasoning_tokens":5634,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:07:22.782888+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate $D$-peelable tableaux of shape $\\kappa$ for a pair of skew shapes outside the constrained family of the prior Temperley-Lieb crystal construction, for instance by brute-force enumeration over all partitions $\\kappa$ of size at most 5, and compare with the classical Littlewood-Richardson rule; a single mismatch would falsify Theorem 1.2 as stated.","supporting_citations":[],"review_version":1}