Dimensions of compositions modulo a prime

Jia Huang

REVIEW 4 major objections 5 minor 17 references

Dimensions of compositions modulo a prime

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2506.16953 v1 pith:UZUY532P submitted 2025-06-20 math.CO

Jia Huang This is my paper · ORCID

classification math.CO MSC 05A1505A1705E1020C0811B65

keywords compositionribbonnumberdescentclass0-HeckealgebramultinomialcoefficientcongruenceLucas-typetheoremCoxetergroupprojectiveindecomposablemodule

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

It 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.

The 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.

The 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.

Referee Report

4 major / 5 minor

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.

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 (4)

[Section 3, Example 3.4] 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.
[Sections 3 and 4, Corollaries 3.5 and 4.7] 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 5, Theorem 5.3] 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 5, Corollary 5.5(iii)] 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.

minor comments (5)

[Section 5, first paragraph] 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 5, Theorem 5.3] There is a typo in the sentence about |P|: 'of b0 >0' should be 'if b0 >0'.
[Section 5, Corollary 5.5] 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 3, Corollary 3.9 proof] In the case T={pd} the text has 'β =∈{(n), (pd,pd+pe)}'; this should be 'β ∈ {(n), (pd,pd+pe)}'.
[Tables 3.1, 4.1, 5.1] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central counts are derived from Dickson's external theorem and inclusion-exclusion, with no fitted inputs or load-bearing self-citations.

full rationale

The paper's main result (Theorem 3.1) derives c_{p,i}(n) from the ribbon-number formula r_alpha = sum_{beta ≼ alpha} (-1)^{ell(alpha)-ell(beta)} binomial(n, beta) and Dickson's theorem on multinomial coefficients modulo p. The proof reduces the enumeration to a fixed binary string on the digit set P and counts completions, which is a direct combinatorial derivation rather than a restatement of the target. Corollaries specialize the same theorem to n = m p^d, sums of distinct powers of p, and 2p^d + p^e, each time recomputing the relevant finite sums from the same external theorem. The self-citation [9] appears only as contextual support for the existence of a tableau realization of P_alpha and is not load-bearing for any congruence count. The type B and type D results similarly apply inclusion-exclusion and Dickson's theorem to the explicitly stated ribbon-number formulas of Propositions 4.1 and 5.1; no parameter is fitted to data and no 'prediction' is statistically forced. Complaints about Example 3.4, if correct, concern the accuracy of a special-case computation, not circularity: a false claim can be wrong without being circular. The computation of c_p(n) for general n is reduced to counting subsets of P, which is a genuinely simpler finite enumeration and not equivalent to the original definition by construction. Therefore no circular step is present.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new entities and fits no free parameters. All tools are standard external results: Dickson's theorem, Fermat's little theorem, inclusion-exclusion, and the Coxeter group descent formalism. The counting formulas are derived rather than fitted, so the axiom ledger is minimal.

assumptions (5)

standard math Dickson's theorem on congruence of multinomial coefficients modulo p
Quoted as Theorem 2.1 without proof; it is the main tool for reducing multinomial coefficients mod p and is cited to Dickson [6].
domain assumption Norton's classification of 0-Hecke algebra modules
Invoked in Section 2 to identify ribbon numbers with dimensions of projective indecomposable modules of H_n(0) and its Coxeter generalizations.
standard math Inclusion-exclusion formula (2.1) for r_α
Derived in Section 2 from counting permutations with descent sets contained in D(β); the derivation is standard and self-contained.
standard math Fermat's little theorem for reducing powers of 2 modulo p
Used in Theorems 4.3 and 5.3 to replace 2^{n-1} and 2^{n-β1} with products of digitwise powers modulo p.
domain assumption Longest element symmetry r_I = r_{S\I} for finite Coxeter groups
Proved in Section 2 via w0; used in Corollaries 3.2, 4.4, and 5.4 to deduce symmetry of the p-vectors.

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Cite this review

Pith. "Pith review of Dimensions of compositions modulo a prime." pith.science (2026). https://pith.science/paper/UZUY532P

@misc{pith2026250616953,
  author       = {Pith},
  title        = {Pith review of: Dimensions of compositions modulo a prime},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UZUY532P}},
  note         = {Machine review of arXiv:2506.16953}
}

abstract

The (ordinary) representation theory of the symmetric group is fascinating and has rich connections to combinatorics, including the Frobenius correspondence to the self-dual graded Hopf algebra of symmetric functions. The $0$-Hecke algebra (of type $A$) is a deformation of the group algebra of the symmetric group, and its representation theory has an analogous correspondence to the dual graded Hopf algebras of quasisymmetric functions and noncommutative symmetric functions. Macdonald used the hook length formula for the number of standard Young tableaux of a fixed shape to determine how many irreducible representations of the symmetric group have dimensions indivisible by a prime $p$. In this paper, we study the dimensions of the projective indecomposable modules of the $0$-Hecke algebra modulo $p$; such a module is indexed by a composition and its dimension is given by a ribbon number, i.e., the cardinality of a descent class. Applying a result of Dickson on the congruence of multinomial coefficients, we count how many ribbon numbers belong to each congruence class modulo $p$. We also extend the result to other finite Coxeter groups.

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Reference graph

Works this paper leans on

17 extracted references · 16 canonical work pages

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Amrutha and T

P. Amrutha and T. Geetha, On the Degrees of Representations of Groups not divisible by 2 k, unpublished preprint, 2022

work page 2022

[2] [2]

G. E. Andrews, The theory of compositions. II. Simon Newcomb’s problem, Utilitas Math. 7 (1975), 33–54

work page 1975

[3] [3]

Ayyer, A

A. Ayyer, A. Prasad and S. Spallone, Odd partitions in Young’s lattice, S´ em. Lothar. Combin. 75 ([2015–2019]), Art. B75g, 13 pp

work page 2015

[4] [4]

Bj¨ orner and F

A. Bj¨ orner and F. Brenti, Combinatorics of Coxeter groups , Graduate Texts in Mathematics, 231, Springer, New York, 2005

work page 2005

[5] [5]

K. S. Davis and W. A. Webb, Lucas’ theorem for prime powers, European J. Combin. 11 (1990), no. 3, 229–233

work page 1990

[6] [6]

L. E. Dickson, The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group, Ann. of Math. 11 (1896/97), no. 1-6, 65–120

work page

[7] [7]

Giannelli, A

E. Giannelli, A. Kleshchev, G. Navarro, and P. H. Tiep, Restriction of odd degree characters and natural correspondences, Int. Math. Res. Not. IMRN 2017, no. 20, 6089–6118

work page 2017

[8] [8]

Ganguly, A

J. Ganguly, A. Prasad and S. Spallone, On the divisibility of character values of the symmetric group, Electron. J. Combin. 27 (2020), no. 2, Paper No. 2.1, 6 pp

work page 2020