{"id":"dd993617-10e1-4d4a-8cb9-e653c4fa3109","arxiv_id":"2506.16953","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper gives formulas for the number of compositions of n whose ribbon number is congruent to i modulo p, including explicit cases n=mp^d and sums of distinct powers of p, with extensions to Coxeter groups of types B and D.","lead":"This paper counts, for each prime p, how many compositions of n have ribbon number congruent to i modulo p, where ribbon numbers are the sizes of descent classes in the symmetric group and also the dimensions of projective indecomposable modules of the 0-Hecke algebra. It derives formulas via Dickson's theorem on multinomial congruences and extends the count to other finite Coxeter groups, where all ribbon numbers are shown to be odd in types B and D.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Example 3.4 is internally inconsistent with Corollary 3.3: for n=3p^d and p>5, c_{p,±3}(n)=0, not 2^{n-3}, so the paper's advertised special-case counts contain a false statement.","rationale":"After checking Theorem 3.1's parity and exponent bookkeeping on small cases (n=2,p=2,3; n=5,p=2; n=8,p=3), I believe the central type A theorem is correct. The reader's weakest assumption about Theorem 5.3 is also real: the second bullet's 'as in the last case' is ambiguous and, read literally, drops the 1/2 factor. That supports a CONDITIONAL verdict. But the reader's verdict missed a concrete false statement inside Section 3's own examples: the n=3p^d, p>3 bullet contradicts Corollary 3.3 for every p>5. This is not a matter of interpretation; it is a wrong count in the paper's advertised special cases. The fix is mechanical (replace p>3 by p=5 and add the p>5 cases), so no theorem is invalidated, but the manuscript needs correction before acceptance. I therefore keep the reader's CONDITIONAL verdict.","tokens_in":28321,"tokens_out":32794,"duration_ms":314937,"concrete_test":"Take n=21, p=7 (n=3*7). Using Corollary 3.3 with m=3 and the four ribbon values {1,1,2,2}, compute c_{7,i}(21): c_{7,±1}=2^{18}, c_{7,±2}=2^{18}, and all other residues are 0. Example 3.4 instead predicts c_{7,±3}=2^{18}. A brute-force computation over the 2^20 compositions of 21, or the paper's Sage code, will confirm which is correct. This single case settles whether the p>3 claim is an isolated typo or a systematic misstatement.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Example 3.4 (Section 3) states that if n=3p^d and p>3, then c_{p,±1}(n)=c_{p,±3}(n)=2^{n-3} and c_{p,i}(n)=0 for all other i. This contradicts the paper's own Corollary 3.3, which is proved from Theorem 3.1 and is correct. For m=3, the ribbon numbers of compositions of 3 are r_{(3)}=r_{(1,1,1)}=1 and r_{(1,2)}=r_{(2,1)}=2. Corollary 3.3 gives c_{p,±3}(n)=2^{n-4} times the number of compositions gamma of 3 with r_gamma congruent to ±3 modulo p. For p=5, residue 2 is -3, so there are exactly two such gamma and the stated 2^{n-3} is right. For every p>5, neither 1 nor 2 is congruent to ±3, so the count is 0, not 2^{n-3}; moreover the zero-for-all-other-i clause is false because residues ±2 occur. Thus an advertised special case of the central theorem is wrong, not merely a typo in the proof.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the number of ribbon-number congruence classes for compositions of n modulo a prime p. For type A, Theorem 3.1 reduces the count c_p(n) to counting subsets T of a digit-derived set P, using Dickson's multinomial congruence theorem and inclusion-exclusion; several special cases are then worked out. The method is extended to types B and D, and computational data for small n and for exceptional Coxeter groups are provided. The core reduction in Theorem 3.1 is valid, and the Sage tables agree with the derivation, but several displayed special-case statements and one definition in Section 5 are incorrect or ambiguous as written.","tokens_in":28514,"tokens_out":20891,"duration_ms":186650,"significance":"If the displayed errors are repaired, the paper makes a solid contribution: it gives a uniform, parameter-free reduction of the type A ribbon-number congruence counts to a finite subset-counting problem, applies it cleanly to sums of distinct prime powers, and extends the method to types B and D with machine-checked tables and a public Sage code link. The use of Dickson's theorem is appropriate, and the proof of the central reduction is self-contained modulo standard external results. The main weakness is not the method but the reliability of the stated corollaries and examples, several of which contain false formulas that would mislead a reader relying on the advertised special cases.","major_comments":[{"comment":"The bullet for n=3p^d states that for p>3 one has c_{p,±1}(n)=c_{p,±3}(n)=2^{n-3} and c_{p,i}(n)=0 for all other i. This is false for p>5. The paper's own Corollary 3.3 gives c_{p,±1}(n)=c_{p,±2}(n)=2^{n-3} and c_{p,±3}(n)=0 when p>5, because the only ribbon numbers of compositions of 3 are 1 and 2. The case p=5 works as stated because 2≡-3 (mod 5), but the p>5 clause must be separated and corrected. As printed, this advertised special case contradicts Corollary 3.3 and is internally inconsistent.","section":"Section 3, Example 3.4"},{"comment":"The exponents in the displayed formulas contain a systematic typo. In Corollary 3.5, 2^{n−2k+1} and 2^{n−2k} should be 2^{n−2^k+1} and 2^{n−2^k}. The same correction applies in Corollary 4.7, where 2^{n−2d+1} and 2^{n−2d} should be 2^{n−2^k+1} and 2^{n−2^k}. The examples already use the corrected powers: for k=3 they use 2^{n−7} and 2^{n−8}, so the literal displayed formulas are wrong for every k≠2.","section":"Sections 3 and 4, Corollaries 3.5 and 4.7"},{"comment":"The second bullet, for β1=1, says to define β′=(0,1+β2,β3,…) and then take ν_p(β′) 'as in the last case'. Read literally with the bullet order, this points to the β1>1 rule and omits the factor 1/2, which would double every type D congruence contribution. The correct coefficient is obtained by applying the β1=0 rule to β′, namely ν_p(β′)=1/2 ∏_{j=0}^d 2^{n_j} ∏_{j=0}^d binom(n_j; β′_{1j},…,β′_{ℓj}). Since this definition is used in every type D computation in Section 5, it must be stated unambiguously.","section":"Section 5, Theorem 5.3"},{"comment":"The displayed p=7 vector for n=3p^d is not a valid length-7 vector; it is a copy of the p=5 vector. Using the rD(T) values computed in the proof, the corrected vector is (2^{n−3}, 2^{n−4}, 2^{n−4}, 5·2^{n−4}, 5·2^{n−4}, 2^{n−4}, 2^{n−4}). As printed, the entries do not sum to 2^n, so this is a mathematical error, not merely a typographical shorthand.","section":"Section 5, Corollary 5.5(iii)"}],"minor_comments":[{"comment":"The formula for the type D descent set is missing a closing brace: it should read D(w)={i∈{0,1,…,n−1} : w(i)>w(i+1)}. Also, 'generated by π0,π1,…,π2' should be 'generated by π0,π1,…,π_{n−1}'.","section":"Section 5, first paragraph"},{"comment":"There is a typo in the sentence about |P|: 'of b0 >0' should be 'if b0 >0'.","section":"Section 5, Theorem 5.3"},{"comment":"Several occurrences of cD_{n,i} should be cD_{p,i} (for example in parts (i) and (ii)). The same symbol is used correctly elsewhere in the corollary, so this is clearly a typo.","section":"Section 5, Corollary 5.5"},{"comment":"In the case T={pd} the text has 'β =∈{(n), (pd,pd+pe)}'; this should be 'β ∈ {(n), (pd,pd+pe)}'.","section":"Section 3, Corollary 3.9 proof"},{"comment":"The common-factor notation in the tables, such as '2(0,1,0)', is easy to misread as a literal list; writing '2·(0,1,0)' or using brackets would improve clarity, especially because some rows also contain a standalone power of 2 before the vector.","section":"Tables 3.1, 4.1, 5.1"}],"recommendation":"major_revision","confidential_remarks":"The core method and the main reduction of Theorem 3.1 appear sound, and the Sage-based verification is a real asset. However, the number of incorrect displayed formulas is high: Example 3.4 contains a false special case, Corollaries 3.5 and 4.7 have systematically wrong exponents, Theorem 5.3 has an ambiguous and literally wrong coefficient definition, and Corollary 5.5(iii) contains an invalid p=7 vector. These are all fixable, but they must be corrected and checked numerically before the paper can be relied upon. I would ask the author to re-run every special-case display against the supplied Sage code during revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core result is genuinely new and the method is sound. The paper gives the first count of ribbon numbers (descent-class sizes, equivalently dimensions of 0-Hecke projective modules) modulo p, the natural analogue of Macdonald's theorem for symmetric-group character degrees. The type A theorem is built on Dickson's congruence for multinomial coefficients plus inclusion-exclusion, and the derivation is clean and self-contained. The type B and D sections earn their place: all ribbon numbers in those types are odd, a strong and simple contrast with type A. The Sage tables match the theorems and the code is linked, so the computational claims are reproducible. This is a solid, citable piece of enumerative combinatorics, not a breakthrough, but a real gap correctly filled.\n\nIt is not publishable as is. Example 3.4 is false. For n = 3p^d, p > 3, the example claims c_{p,±1}(n) = c_{p,±3}(n) = 2^{n−3} and all other entries zero. That is right only for p = 5, where −3 ≡ 2. For every p ≥ 7, no composition of 3 has ribbon number congruent to ±3, so c_{p,±3}(n) = 0, and the zero-elsewhere clause fails because residues ±2 each get 2^{n−3}. The correct statement for p > 5 is support on ±1 and ±2. Since this is an advertised special case of the main theorem, it needs to be the first fix.\n\nThe other problems are presentation-level. Theorem 5.3 says to handle β1 = 1 'as in the last case'; the paper's own corollaries only work if the β1 = 0 rule is applied to β′ = (0, 1+β2, ...), and the literal reading doubles the contribution. The cross-reference is wrong and must be rewritten. Corollary 3.5's exponents read n−2k+1 and n−2k; Examples 3.7–3.8 confirm the intended values are n−2^k+1 and n−2^k, so either correct the display or confirm it is a typesetting artifact. Minor typos ('c1p(n)', 'of b0 > 0') are not worth listing.\n\nThe exceptional-type section is honest data without a theorem, and the author states plainly that the method is tedious outside the special families; both are limitations, not flaws. Nothing circular, no fitted parameters. The reader's conditional verdict is fair, and the stress-test note is correct: this is a genuine error, not a typo. Send it to peer review; a referee should check the examples and the type D formula carefully. With those fixes, I would expect acceptance. The paper is for people working on descent classes, 0-Hecke representation theory, or p-valuation questions in enumerative combinatorics.","headline":"Sound, genuinely new Macdonald-type enumeration for ribbon numbers mod p, but Example 3.4 is false for p ≥ 7 and Theorem 5.3's cross-reference needs fixing; conditional accept.","tokens_in":29135,"tokens_out":11611,"would_cite":true,"duration_ms":100624,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05A15","05A17","05E10","20C08","11B65"],"pacs":[],"model":"deepseek-v4-flash","headline":"The number of compositions of n whose ribbon number is i modulo a prime p is governed, in all but one symmetric case, by subsets of a digit-built set P derived from the base-p digits of n.","keywords":["composition","ribbon number","descent class","0-Hecke algebra","multinomial coefficient congruence","Lucas-type theorem","Coxeter group","projective indecomposable module"],"falsifier":"Take p=3 and n=8. Directly enumerate all 128 compositions, compute each ribbon number r_\\$\\alpha$ by counting permutations of {1,\\ldots,8} with the corresponding descent set, reduce modulo 3, and compare the three class sizes with (42,34,52), the vector predicted by Theorem 3.1 and Table 3.1; any mismatch falsifies the formula.","tokens_in":28018,"feed_emoji":"🔢","tokens_out":7717,"duration_ms":74519,"temperature":0.7,"pith_summary":"This paper proves an exact formula for the number of compositions of n, out of all $2^{{n-1}}$, whose ribbon number—the size of a descent class and the dimension of a projective indecomposable module of the type-A 0-Hecke algebra—is congruent to i modulo a prime p. The formula expresses each count through subsets of a set P built from the base-p digits of n, so the computation no longer scans all compositions. It also carries the same reduction through to the type B and type D Coxeter groups, and records the resulting residue counts for the exceptional Coxeter groups. The motivation is Macdonald's classical count of symmetric-group representations with p-coprime dimension, now transferred to the 0-Hecke side.","feed_headline":"Base-p digits control composition dimensions mod p","feed_subtitle":"The count reduces from all 2^{n-1} compositions to subsets of a digit-built set P.","key_machinery":"The machinery has two parts. First, the ribbon-number identity r_\\$\\alpha$ = \\sum_{\\$\\beta$ \\preceq \\$\\alpha$} (-1)^{\\ell(\\$\\alpha$)-\\ell(\\$\\beta$)} \\binom{n}{\\$\\beta$}, which is inclusion-exclusion over descent sets. Second, Dickson's theorem: \\binom{n}{\\$\\beta$} mod p equals \\prod_j \\binom{n_j}{\\beta_{1j},...,\\beta_{\\ell j}}, and is zero unless the parts of \\$\\beta$ add digit-wise to the digits n_j. The digit set P collects all possible digit-wise sums that can appear, and the signed residue r(T) is what the alternating sum becomes once the binary string of \\$\\alpha$ is fixed on P.","core_discovery":"The central discovery is Theorem 3.1: write n in base p as n = \\sum n_j p^j and set \\Pi = \\prod_j (n_j+1). Then c_{p,i}(n) equals $2^{{n+1-\\Pi}}$ times the number of subsets T of P = \\{ \\sum b_j p^j : 0 \\leq b_j \\leq n_j \\} \\setminus \\{0,n\\} for which a signed residue r(T) is congruent to i, when p=2, or i=0, or all lower base-p digits are p-1; otherwise it equals $2^{{n-\\Pi}}$ times the number of T with r(T) \\equiv \\pm i. The residue r(T) is an alternating sum over compositions \\$\\beta$ refined by T of products of multinomial coefficients \\prod_j \\binom{n_j}{\\beta_{1j},...,\\beta_{\\ell j}}, so Dickson's congruence theorem turns the whole count into digit combinatorics.","pith_inferences":["The same digit-set skeleton likely survives for prime-power moduli, where Lucas-type congruences for multinomial coefficients modulo p^a would replace Dickson's theorem; this is a natural next computation.","For n built from few distinct powers of p, the weights r(T) reduce to chain counts in a Boolean poset, so the residue vectors can be read as evaluations of order-complex Euler characteristics; this may yield formulas for k-term sums without case-by-case Hasse diagrams.","Because the zero residue class is treated by a different multiplier in the theorem, the relative frequency of r_\\alpha \\equiv 0 mod p versus nonzero classes may deviate from uniformity in a way controlled by \\prod_j (n_j+1); comparing c_{p,0}(n)/2^{n-1} across digit patterns would test this.","If the type D ambiguity flagged below is resolved in the intended way, the same machinery should extend to all finite Coxeter groups whose ribbon numbers admit a multinomial-coefficient formula paralleling Proposition 5.1."],"forward_implications":["For n=p^d, the theorem gives c_2(n)=(0,2^{n-1}) and, for odd p, c_p(n)=(0,2^{n-2},0,\\ldots,0,2^{n-2}), so only the residues 0 and \\pm 1 occur.","For n equal to a sum of two distinct powers of p, the counts are c_{p,0}(n)=2^{n-2}, c_{p,\\pm1}(n)=2^{n-3}, and zero otherwise when p is odd, while c_2(n)=(2^{n-2},2^{n-2}).","The same digit-set reduction, applied to type B and type D Coxeter groups, shows that every type B and type D ribbon number is odd.","The formulas imply the palindromic symmetry c_{p,i}(n)=c_{p,-i}(n) except when the lower base-p digits are all p-1, and they give a guaranteed power of 2 dividing each count.","For the exceptional Coxeter groups, the descent-class sizes are listed, so the residue counts modulo p are immediately read off from the tables."],"supporting_citations":[{"why":"Supplies Dickson's congruence theorem for multinomial coefficients modulo p, the engine that makes the reduction to base-p digits work.","marker":"[6]"},{"why":"Macdonald's hook-length and p-core count of symmetric-group representations with p-coprime dimension is the motivational template and the source of the digit-based counting strategy.","marker":"[12]"},{"why":"Norton's description of 0-Hecke algebra representations identifies the dimensions of projective indecomposable modules with descent-class sizes, which is what the ribbon numbers count.","marker":"[14]"},{"why":"Provides the Coxeter group background, descent-set formulas, and parabolic quotient arguments used in the inclusion-exclusion formula and in the type B and type D extensions.","marker":"[4]"}],"fun_headline_variants":["Base-p digits decide mod-p ribbon numbers","Counting ribbon numbers mod p via digit subsets","Mod-p composition dimensions: a subset count","Dickson congruence yields ribbon number census"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The type D result stands or falls on an unstated rule: when \\beta_1=1, the coefficient must be computed by the \\beta_1=0 case after shifting \\$\\beta$ to (0,1+\\beta_2,\\ldots), including its factor of 1/2, rather than by literally repeating the phrase 'as in the last case', which would double the contribution.","fun_headline_variants_meta":{"raw":{"variants":["Base-p digits decide mod-p ribbon numbers","Counting ribbon numbers mod p via digit subsets","Mod-p composition dimensions: a subset count","Dickson congruence yields ribbon number census"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000215,"raw_usage":{"total_tokens":1424,"prompt_tokens":937,"completion_tokens":487,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":553,"completion_tokens_details":{"reasoning_tokens":433}},"tokens_in":553,"tokens_out":487,"duration_ms":5642,"temperature":1.0,"reasoning_tokens":433,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:18:38.558605+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take p=3 and n=8. Directly enumerate all 128 compositions, compute each ribbon number r_\\$\\alpha$ by counting permutations of {1,\\ldots,8} with the corresponding descent set, reduce modulo 3, and compare the three class sizes with (42,34,52), the vector predicted by Theorem 3.1 and Table 3.1; any mismatch falsifies the formula.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Dickson's congruence theorem for multinomial coefficients modulo p, the engine that makes the reduction to base-p digits work."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Macdonald's hook-length and p-core count of symmetric-group representations with p-coprime dimension is the motivational template and the source of the digit-based counting strategy."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Norton's description of 0-Hecke algebra representations identifies the dimensions of projective indecomposable modules with descent-class sizes, which is what the ribbon numbers count."},{"cited_title":"Bj¨ orner and F","cited_arxiv_id":null,"evidence_quote":"Provides the Coxeter group background, descent-set formulas, and parabolic quotient arguments used in the inclusion-exclusion formula and in the type B and type D extensions."}],"review_version":1}