Polynomial analogues of restricted multicolor b-ary partition functions

Karl Dilcher; Larry Ericksen

REVIEW 2 major objections 4 minor 18 references

Polynomial analogues of restricted multicolor b-ary partition functions

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves that a certain multivariable polynomial encodes every Λ-restricted ρ-colored b-ary partition of n as a unique monomial, with the exponents specifying the parts.

desk verdict A useful but slightly sloppy extension of the polynomial-analogue program to multicolor restricted b-ary partitions; the central theorem holds for formal parameters or t≥2, but the t=1 case breaks the claimed bijection. read the letter →

arxiv 1908.03751 v1 pith:O3H5KJSL submitted 2019-08-10 math.NT

Karl Dilcher , Larry Ericksen This is my paper · ORCID

classification math.NT MSC 11P8111B3711B83

keywords b-arypartitionbinaryrestrictedcoloredpolynomialanaloguegeneratingfunctionrecurrencerelationmultinomialcoefficient

topics The Collatz Conjecture

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 studies partitions of an integer n into powers of a fixed base b≥2, where each power may be assigned one of ρ colors and the ℓ-th color may be used at most λ_ℓ times for each power value. The paper's main claim is that a naturally defined multivariable polynomial $\Omega^\Lambda_{b,T}(n;Z)$ has exactly one monomial for each such partition, and that the exponents of the monomial encode the partition completely. In other words, the polynomial is a lossless algebraic codebook for the full set of restricted colored b-ary partitions, not just a count. The paper proves this by a base-b recurrence, gives an explicit formula using multinomial coefficients modulo 2, and obtains a factorization theorem for the polynomials. This matters because earlier polynomial analogues handled single-color or unrestricted cases, whereas here every individual partition of a fairly general colored class is made visible as a monomial.

What carries the argument

The load-bearing object is the polynomial sequence $\Omega^\Lambda_{b,T}(n;Z)$, defined by the generating function $\sum_{n\ge0} \Omega^\Lambda_{b,T}(n;Z)q^n = \prod_{j\ge0}\prod_{\ell=1}^\rho (1+z_{\ell,1}^{t_{\ell,1}^j}q^{b^j}+\cdots+z_{\ell,\lambda_\ell}^{t_{\ell,\lambda_\ell}^j}q^{\lambda_\ell b^j})$. Each factor at level j contributes choices for parts of size $b^j$, and the exponent $t_{\ell,i}^j$ accumulates the position j. The recurrence (4.4), $\Omega^\Lambda_{b,T}(bn+j;Z)=\sum_{k=0}^{\lfloor \lambda/b\rfloor} Y_{bk+j}\Omega^\Lambda_{b,T}(n-k;Z^T)$, with $Y_\nu$ the sums of products of variables whose subscripts add to ν, is the induction engine that proves the monomial bijection.

What would settle it

Set b=2, Λ=(2), n=10. The two hyperbinary partitions 8+1+1 and 2+4+4 give the monomials $y_1^{t_1^3}y_2$ and $y_1^{t_1}y_2^{t_2^2}$; substituting $t_1=t_2=1$ makes both monomials $y_1 y_2$, showing that the theorem's bijection cannot hold for concrete parameters equal to 1.

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Extended reading notes

Core claim

The central discovery is Theorem 3.2: for every n≥1, the polynomial $\Omega^\Lambda_{b,T}(n;Z)$ expands as a sum over exactly $C^\Lambda_b(n)$ monomials, one for each Λ-restricted ρ-colored b-ary partition of n. Writing the exponent of $z_{\ell,i}$ as a polynomial $p(t_{\ell,i})=\sum_j t_{\ell,i}^{\tau_{\ell,i}(j)}$ with coefficients 0 and 1, the corresponding partition is $\sum_{\ell=1}^\rho \sum_{i=1}^{\lambda_\ell} i\,(b^{\tau_{\ell,i}(1)}+\cdots+b^{\tau_{\ell,i}(\mu_{\ell,i})})$, with the subscript on b indicating the color. Thus each monomial is a complete certificate of one partition: the variable $z_{\ell,i}$ records that i copies are used, and each exponent power $t_{\ell,i}^\tau$ records one occurrence of the part $b^\tau$ in color ℓ. The proof runs by induction on n using a recurrence that breaks $bn+j$ into a remainder part and a lifted block.

Load-bearing premise

The whole correspondence assumes the numbers t can be treated as abstract symbols: if any t equals 1, two different partitions can end up with the same monomial, so the key bijection breaks.

Editorial extensions

If this is right

Each coefficient polynomial has $C^\Lambda_b(n)$ monomials, so the polynomial refines the ordinary counting function: reading the variables at $z_{\ell,i}=1$ recovers the numerical partition count.
The recurrence (4.4) gives a base-b divide-and-conquer method to generate the full list of restricted colored b-ary partitions of n without inspecting unrelated integers.
The explicit formula (6.6) expresses the polynomial as a sum over tuples $k_{\ell,i}\in M_b$ with $\sum i\,k_{\ell,i}=n$, weighted by multinomial coefficients modulo 2, so the monomial list can be computed directly from base-b digits.
The factorization theorem (7.2) shows that within blocks of length $b^\ell$, $\Omega^\Lambda_{b,T}(nb^\ell+j;Z)$ factors as $\Omega^\Lambda_{b,T}(n;Z^{T^\ell})\Omega^\Lambda_{b,T}(j;Z)$, a self-similarity that generalizes the Stern-type identity.
When $\Lambda=(b-1,\ldots,b-1)$, Corollary 8.1 gives $C^\Lambda_b(n)=\binom{n+\rho-1}{\rho-1}$, independent of the base, because each color permits all possible multiplicities up to b−1.

Reading between the lines

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

One consequence the authors do not spell out: because monomials are in bijection with partitions, the polynomial can serve as a compact data structure, and sampling a monomial uniformly at random would sample partitions uniformly.
The factorization result and the recurrence may be combined to obtain recurrences for the counting sequence $C^\Lambda_b(n)$ along arithmetic progressions, potentially exposing congruences analogous to those known for binary and b-ary partition functions.
If the parameters t are treated as formal indeterminates, the polynomial code is more robust than a numerical evaluation; for concrete integer parameters one should demand $t\ge2$ to keep the decoding injective, since $t=1$ collapses distinct partitions.
The independence of b in Corollary 8.1 suggests the existence of an explicit bijection between such colored partitions across different bases, and the monomial code may help construct it.

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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 headline: this is a competent extension of the Dilcher–Ericksen program to ρ-colored restricted b-ary partitions. The new object Ω^Λ_{b,T} and the representation theorem are real, and the proof by induction from the recurrence is sound. But two things need attention before I'd rely on it: the monomial code in Theorem 3.2 is not injective when the parameters T equal 1, and the proof of Theorem 7.2 has a gap in the induction step.

What is genuinely new: the Λ-restricted ρ-colored setting (Definition 2.1) and the polynomial Ω (Definition 2.4) are natural generalizations, and Theorem 3.2 gives a monomial-level codebook for all such partitions. The recurrence in Theorem 4.3 is clean, and the explicit formula in Theorem 6.3 follows by combining the convolution identity with the authors' earlier theorem. These are solid, useful extensions. The reliance on their own prior work is legitimate citation, not circularity.

Where the soft spots are. The stress-test note lands. Definition 2.4 says T is a tuple of positive integer parameters, so t=1 is permitted. When t=1, exponent polynomials like t^2 and t collapse to the same integer, so two distinct partitions can map to the same monomial. The paper's own Example 3.1 contains this collision when t_{1,1}=t_{2,1}=1. The proof in Section 5 never proves injectivity of the exponent encoding, so Theorem 3.2(b) is false as stated for that parameter choice. The fix is easy: require the parameters to be formal indeterminates or integers at least 2, and state that convention explicitly. This is a gap in presentation, not a fatal flaw in the intended construction.

The second issue is smaller but real. In Theorem 7.2, the induction step picks i in [λ−b+1, b−1] and r in the ℓ-range. Not every j in the stated ℓ+1 range can be written as r b + i with i in that interval — for instance, when j mod b is below λ−b+1. The theorem may still be true, but the proof does not cover the full stated range. That needs a reworked induction.

Overall: this is a solid contribution to the partition-polynomial literature. The central construction is new and useful, and the proofs are mostly honest. It deserves a serious referee, but the referee should insist on fixing the t=1 convention and the induction gap. If those are addressed, it's a well-founded paper for a journal like J. Integer Seq.

Referee Report

2 major / 4 minor

Summary. The paper introduces Λ-restricted ρ-colored b-ary partitions, where each part b^j may be taken in color ℓ at most λ_ℓ times, and associates to them a family of multivariate polynomials Ω^Λ_{b,T}(n;Z) via the generating function (2.4). The main result, Theorem 3.2, asserts that each monomial of Ω^Λ_{b,T}(n;Z) encodes one such partition: the exponent p_{h,ℓ,i}(t), written as Σ_{τ∈S} t^τ, records the levels τ at which i copies of b^τ occur in color ℓ. The proof is by induction from the recurrence in Theorem 4.3, which is derived from the generating function by factoring out the j=0 factors. The paper also proves an explicit multinomial formula (Theorem 6.3), a convolution decomposition (Theorem 2.6), and product identities (Theorem 7.2).

Significance. The construction is natural and extends earlier work of the same authors and of Bates-Mansour and Stanley-Wilf from hyperbinary and restricted b-ary partitions to the multicolor setting. The recurrence and explicit formulas are cleanly derived, and the monomial encoding is a genuinely useful way to list individual partitions rather than only their counts. The proof of Theorem 4.3 is a straightforward and correct manipulation of generating functions, and the induction in Section 5 is structurally sound. The main reservation is the unstated convention needed for the exponent parameters; this is a localized but central flaw that must be fixed before the results are valid as stated.

major comments (2)

[Definition 2.4 and Theorem 3.2] The representation theorem is false as stated because Definition 2.4 allows t_{ℓ,i}=1. For t=1 the exponent p(t)=Σ_{τ∈S} t^τ is just |S|, so the map S ↦ Σ_{τ∈S} t^τ is not injective and different partitions collapse to the same monomial. In Example 3.1, taking T=(1,1;1,1,1) gives z_{2,1}^{t_{2,1}} z_{1,1} = z_{2,1} z_{1,1} = z_{2,1} z_{1,1}^{t_{1,1}}, while equation (3.6) assigns these two monomials to the distinct partitions 2_2+1_1 and 2_1+1_2; consequently Ω^Λ_{2,T}(3;Z) has coefficient 2 on z_{2,1}z_{1,1} and (3.4) cannot hold with distinct monomials for each of the seven partitions. The induction in Section 5 does not address this because it never proves injectivity of the exponent encoding. The theorem is repairable by requiring t_{ℓ,i}≥2 for all ℓ,i (then base-t expansion with digits 0/1 recovers S uniquely), or by stipulating that the t's are formal indeterminates and p(t) is a formal polynomial; the rest of the proof then goes through.
[Theorem 6.3 and Example 6.4] The explicit formula inherits the same parameter-convention problem. In the sum (6.6), D_{ℓ,i}=d^b_{t_{ℓ,i}}(k_{ℓ,i}); when t_{ℓ,i}=1 this is the number of 1-digits of k_{ℓ,i} in base b, so different tuples (k_{ℓ,i}) can yield the same monomial. The examples, for instance Example 6.4, explicitly read the formula as a bijective inventory of partitions, so Theorem 6.3 should carry the same t_{ℓ,i}≥2 (or formal-variable) hypothesis as Theorem 3.2. This is a second manifestation of the same missing convention rather than a separate mathematical error.

minor comments (4)

[Section 5, induction beginning] In the display after (5.1), the upper limit of the sum should be C^Λ_b(j), not C^Λ_b(n), since the induction begins with the argument j.
[Section 4, proof of Theorem 4.3] The occurrence of 'λell' in the product expansion should be λ_ℓ; this appears to be a typographical error.
[Theorem 7.1] The displayed inequality should be parenthesized as (b^ℓ−1)/(b−1) ≤ j ≤ b^ℓ−1; as printed it is ambiguous.
[Definition 2.1] The phrase 'a set of powers b^j' is slightly misleading because powers can occur with multiplicities; 'multiset' or 'sequence' would be more accurate.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the representation theorem is proved by induction from the defining generating function and its recurrence; prior self-citations are auxiliary, not load-bearing for the central claim.

full rationale

The paper's central result, Theorem 3.2, connects the polynomials Ω_{b,T}^{Λ}(n;Z) defined by the generating function (2.4) to the individual Λ-restricted ρ-colored b-ary partitions of n. This is not equivalent to the defining input by construction: the proof in Section 5 is an induction that uses the recurrence (4.4), whose proof in Theorem 4.3 is a direct manipulation of the generating function, together with the combinatorial decomposition of a partition of nb+j into a lifted partition of n−k and a low part kb+j. The exponent polynomials p_{h,ℓ,i}(t_{ℓ,i})=Σ t^{τ} are not fitted parameters; the τ are exactly the base-b exponents read from (3.6), and the monomial-to-partition map is verified explicitly in the base case and preserved by the recurrence. The paper's self-citations to [5] and [7] are used for auxiliary tools: Theorem 6.2 from [7] supplies the one-color explicit formula used in Section 6, but Theorem 3.2 is proved independently in Section 5 without invoking it, and the recurrence and factorization results are proved from the generating function in the present paper. No fitted-input-called-prediction pattern, no imported uniqueness theorem, and no ansatz smuggled by citation occur. The only substantive caveat is a correctness/interpretation condition: Definition 2.4 allows T-entries equal to 1, whereas the encoding p(t)=Σ t^{τ} is injective only when each t is an indeterminate or an integer at least 2 (for t=1, e.g., in Example 3.1 the monomials z_{2,1}^{t_{2,1}}z_{1,1} and z_{2,1}z_{1,1}^{t_{1,1}} collapse to the same monomial). That is a genuine restriction on the statement, but it is not circularity, since it concerns whether the theorem's map is well-defined/injective rather than whether the conclusion reproduces an input assumption.

Assumptions & free parameters 0 free parameters · 3 assumptions · 2 invented entities

The ledger shows the paper is not fitting anything: no free parameters. It introduces two new combinatorial/algebraic objects (the multicolor partitions and their polynomial code). The main mathematical payload rests on standard facts (unique base-b expansion, parity of multinomial coefficients) plus one imported theorem from the authors' earlier paper [7]; the only nonstandard convention is the formal-exponent reading of z^{p(t)}, which is the price of the bijection.

assumptions (3)

standard math Theorem 6.2 of [7] (explicit formula for ω^λ_{b,T}(n+1;Z)) is invoked without proof.
Used in Section 6 to prove Theorem 6.3; it is a published result in the authors' Journal of Integer Sequences paper.
standard math Unique b-ary representation of nonnegative integers, used to evaluate the product in (8.2).
Central to the proof of Corollary 8.1 in Section 8.
domain assumption Formal-exponent convention: monomials z^{p(t)} with p(t)=Σ t^{τ_j} are distinguished by the full polynomial p(t), so the multiset {τ_j} is recoverable.
Implicit in Definition 2.4 and throughout Section 3; essential for the bijection in Theorem 3.2. The paper calls t's positive integers but uses them as formal variables for exponents.

invented entities (2)

Λ-restricted ρ-colored b-ary partitions independent evidence
purpose: The combinatorial objects characterized by the polynomial code; they generalize restricted b-ary partitions and b-ary overpartitions to multiple colors with individual caps.
For ρ=1 they reduce to known restricted b-ary partitions S^λ_b(n); the generating function (2.1) can be checked on small n by direct enumeration (Example 2.2).
Multivariate polynomial sequence Ω^Λ_{b,T}(n;Z) independent evidence
purpose: Serves as a bijective monomial code for all Λ-restricted ρ-colored b-ary partitions of n, with recurrences, explicit formulas, and factorization.
When Z=(1,...,1) it recovers the numeric counts C^Λ_b(n); when ρ=1 it reduces to the earlier ω^λ_{b,T} polynomials of [5,7]; examples in Tables 1-4 match direct enumerations.

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Pith. "Pith review of Polynomial analogues of restricted multicolor b-ary partition functions." pith.science (2026). https://pith.science/paper/O3H5KJSL

@misc{pith2026190803751,
  author       = {Pith},
  title        = {Pith review of: Polynomial analogues of restricted multicolor b-ary partition functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O3H5KJSL}},
  note         = {Machine review of arXiv:1908.03751}
}

abstract

Given an integer base $b\geq 2$, a number $\rho\geq 1$ of colors, and a finite sequence $\Lambda=(\lambda_1,\ldots,\lambda_\rho)$ of positive integers, we introduce the concept of a $\Lambda$-restricted $\rho$-colored $b$-ary partition of an integer $n\geq 1$. We also define a sequence of polynomials in $\lambda_1+\cdots+\lambda_\rho$ variables, and prove that the $n$th polynomial characterizes all $\Lambda$-restricted $\rho$-colored $b$-ary partitions of $n$. In the process we define a recurrence relation for the polynomials in question, obtain explicit formulas and identify a factorization theorem.

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Reference graph

Works this paper leans on

18 extracted references · 17 canonical work pages

[1]

G. E. Andrews, Congruence properties of the m-ary partition function, J. Number Theory 3 (1971), 104–110

work page 1971
[2]

Bates and T

B. Bates and T. Mansour, The q-Calkin-Wilf tree, J. Combin. Theory Ser. A 118 (2011), 1143–1151

work page 2011
[3]

R. F. Churchhouse, Congruence properties of the binary p artition function, Proc. Cambridge Philos. Soc. 66 (1969), 371–376

work page 1969
[4]

Corteel and J

S. Corteel and J. Lovejoy, Overpartitions, Trans. Amer. Math. Soc. 356 (2004), no. 4, 1623– 1635

work page 2004
[5]

Dilcher and L

K. Dilcher and L. Ericksen, Generalized Stern polynomia ls and hyperbinary representations, Bull. Pol. Acad. Sci. Math. 65 (2017), 11–28

work page 2017
[6]

Dilcher and L

K. Dilcher and L. Ericksen, Polynomials characterizing hyper b-ary representations, J. Integer Seq. 21 (2018), Article 18.4.3, 11 pp

work page 2018
[7]

Dilcher and L

K. Dilcher and L. Ericksen, Polynomial analogues of rest ricted b-ary partition functions, J. Integer Seq. 22 (2019), Article 19.3.2

work page 2019
[8]

Dilcher and L

K. Dilcher and L. Ericksen, Properties of multivariate b -ary Stern polynomials, Ann. Comb. , to appear

work page

Show all 18 references

[9]

J. M. Dumont, N. Sidorov, and A. Thomas, Number of represe ntations related to a linear recurrent basis, Acta Arith. 88 (1999), no. 1, 371–396

1999
[10]

Euler, De partitione numerorum, Novi Commentarii Academiae Scientiarum Petropoli- tanae 3 (1753), 125–169

L. Euler, De partitione numerorum, Novi Commentarii Academiae Scientiarum Petropoli- tanae 3 (1753), 125–169. In Opera Omnia , Series 1, Volume 2, pp. 254–294. Also available from The Euler Archive , http://eulerarchive.maa.org, Paper E191
[11]

H. W. Gould, Combinatorial Identities , revised edition, Gould Publications, Morgantown, W.Va., 1972. POLYNOMIAL ANALOGUES 19

1972
[12]

W. J. Keith, Restricted k-color partitions, Ramanujan J. 40 (2016), 71–92

2016
[13]

Mahler, On a special functional equation, J

K. Mahler, On a special functional equation, J. London Math. Soc. 15 (1940), 115–123

1940
[14]

Reznick, Some binary partition functions, in Analytic Number Theory: Proceedings of a Conference in Honor of Paul T

B. Reznick, Some binary partition functions, in Analytic Number Theory: Proceedings of a Conference in Honor of Paul T. Bateman , (B. C. Berndt et al., Eds.), Birkh¨ auser, Boston, 1990, 451–477

1990
[15]

R¨ odseth, Some arithmetical properties of m-ary partitions, Proc

¨O. R¨ odseth, Some arithmetical properties of m-ary partitions, Proc. Cambridge Philos. Soc. 68 (1970), 447–453

1970
[16]

Rødseth and J

Ø. Rødseth and J. A. Sellers, On m-ary overpartitions, Ann. Comb. 9 (2005), no. 3, 345–353

2005
[17]

R. P. Stanley and H. S. Wilf, Reﬁning the Stern diatomic s equence, Preprint, 2010, http://www-math.mit.edu/~rstan/papers/stern.pdf

2010
[18]

Ulas and B

M. Ulas and B. ˙Zmija, On p-adic valuations of colored p-ary partitions, Monatsh. Math. 188 (2019), no. 2, 351–368. Department of Mathematics and Statistics, Dalhousie Univer sity, Halifax, Nova Sco- tia, B3H 4R2, Canada E-mail address : dilcher@mathstat.dal.ca P.O. Box 172, M...

2019

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

G. E. Andrews, Congruence properties of the m-ary partition function, J. Number Theory 3 (1971), 104–110

work page 1971

[2] [2]

Bates and T

B. Bates and T. Mansour, The q-Calkin-Wilf tree, J. Combin. Theory Ser. A 118 (2011), 1143–1151

work page 2011

[3] [3]

R. F. Churchhouse, Congruence properties of the binary p artition function, Proc. Cambridge Philos. Soc. 66 (1969), 371–376

work page 1969

[4] [4]

Corteel and J

S. Corteel and J. Lovejoy, Overpartitions, Trans. Amer. Math. Soc. 356 (2004), no. 4, 1623– 1635

work page 2004

[5] [5]

Dilcher and L

K. Dilcher and L. Ericksen, Generalized Stern polynomia ls and hyperbinary representations, Bull. Pol. Acad. Sci. Math. 65 (2017), 11–28

work page 2017

[6] [6]

Dilcher and L

K. Dilcher and L. Ericksen, Polynomials characterizing hyper b-ary representations, J. Integer Seq. 21 (2018), Article 18.4.3, 11 pp

work page 2018

[7] [7]

Dilcher and L

K. Dilcher and L. Ericksen, Polynomial analogues of rest ricted b-ary partition functions, J. Integer Seq. 22 (2019), Article 19.3.2

work page 2019

[8] [8]

Dilcher and L

K. Dilcher and L. Ericksen, Properties of multivariate b -ary Stern polynomials, Ann. Comb. , to appear

work page