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arxiv: 1906.11093 · v1 · pith:VU425BG2new · submitted 2019-06-26 · 🧮 math.CO

On a new formula for the number of unrestricted partitions

Hemar Godinho , Jos\'e Pl\'inio O. Santos This is my paper

Pith reviewed 2026-05-25 15:33 UTC · model grok-4.3

classification 🧮 math.CO
keywords partition functionunrestricted partitionsinteger solutionscombinatorial correspondencep(n) formulanumber theorydiophantine equations
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The pith

The number of unrestricted partitions of n equals the number of non-negative solutions to systems of two equations using numbers from 1 to n squared.

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

The paper establishes a correspondence between the partitions of n and the non-negative solutions to systems of two equations with variables in the interval from 1 to n squared. This correspondence is presented as yielding a new formula for the partition function p(n). A sympathetic reader would care because it supplies an alternative way to express and potentially compute the number of ways to write n as a sum of positive integers without regard to order. The approach frames p(n) in terms of counting solutions rather than generating functions or recurrence relations used in earlier work.

Core claim

The authors claim that the number of unrestricted partitions of n equals the number of non-negative solutions of systems of two equations involving natural numbers in the interval (1, n²), thereby giving a new explicit formula for p(n).

What carries the argument

A correspondence between unrestricted partitions of n and the non-negative integer solutions of two-equation systems with variables bounded by n squared, used to equate the partition count to a solution count.

Load-bearing premise

The stated correspondence between partitions and the solution counts of the two-equation systems must be bijective and must produce an expression distinct from all prior formulas for p(n).

What would settle it

For n=4, where the known partition number is 5, count the non-negative solutions to the two-equation systems with variables up to 16; the count must equal 5, or the claimed formula fails.

Figures

Figures reproduced from arXiv: 1906.11093 by Hemar Godinho, Jos\'e Pl\'inio O. Santos.

Figure 1
Figure 1. Figure 1: Illustration for Example 2.1 Now we reflect the path through the line x + y = n and create a partition of a number m into distinct odd parts all greater than 1, by taking hooks of the following sizes: (4) λ1 = 2(n − d1)) − 1, λ2 = 2((n − d1) − 1) − 1, . . . λc1 = 2((n − d1) − (c1 − 1)) − 1, λc1+1 = 2((n − d1) − d2 − c1) − 1, . . . λc1+c2 = 2((n − d1) − d2 − c1 − (c2 − 1)) − 1, λc1+c2+1 = 2((n − d1) − d2 − … view at source ↗
Figure 2
Figure 2. Figure 2: Illustration for Example 2.3 Example 2.5. Let us take the partition (4, 1, 1) of 6 and the partition (2, 1, 1) of 4. The matrices associated to them are, respectively, M1 =  1 1 0 3 0 1  and M2 =  1 1 0 1 0 1  . The paths that these matrices originate are different, although both paths induce the same hooks and, therefore, the same partition into distinct odd parts, as shown in [PITH_FULL_IMAGE:figure… view at source ↗
Figure 3
Figure 3. Figure 3: Illustration for Example 2.5 Example 2.6. In the table below we present the result of the application of the Path Procedure to the matrices listed in [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: into distinct odd parts, all greater than 1, of the number m, which is equal to the area of the figure, the sum of the parts (we have here a similar situation as described in Figures 1 and 2). This area is also equal to the sum of the area of the big square of size length equal to y0 with twice the sum of the areas of the (s − 1) rectangles of areas c1 × x0, c2 × (x0 − d2), . . . , cs−2 × (ds−1 + ds) and c… view at source ↗
read the original abstract

In this paper we present a new formula for the number of unrestricted partitions of $n$. We do this by introducing a correspondence between the number of unrestrited partitions of $n$ and the number of non-negative solutions of systems of two equations, involving natural numbers in the interval (1 $,n^{2}$).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript claims to present a new formula for the unrestricted partition function p(n) by establishing a correspondence between p(n) and the number of non-negative integer solutions to a system of two equations whose variables range over natural numbers in the interval (1, n²).

Significance. A bijective or enumerative formula for p(n) expressed via solution counts of a fixed number of Diophantine equations would be of interest in partition theory if rigorously established, as it could offer a new combinatorial model distinct from generating functions or pentagonal-number recurrences. No such strengths (machine-checked proofs, reproducible code, or explicit parameter-free derivations) are present.

major comments (2)
  1. [Abstract] Abstract and introduction: the central claim asserts a bijective correspondence yielding p(n) solutions to an unspecified pair of equations, yet neither the equations themselves nor any argument establishing that their solution cardinality equals p(n) (as opposed to some other integer) is supplied. This renders the asserted formula an unverified assertion rather than a derived result.
  2. No section or equation: the manuscript contains no explicit system of equations, no proof of bijectivity, and no verification that the solution count reproduces known values of p(n) for small n, so the equality p(n) = #solutions cannot be checked for internal consistency or reduction to prior expressions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the report. We agree that the submitted manuscript does not contain the explicit system of equations, the bijectivity argument, or small-n verifications, rendering the central claim unverified as presented. We will revise to supply these elements.

read point-by-point responses

  1. Referee: [Abstract] Abstract and introduction: the central claim asserts a bijective correspondence yielding p(n) solutions to an unspecified pair of equations, yet neither the equations themselves nor any argument establishing that their solution cardinality equals p(n) (as opposed to some other integer) is supplied. This renders the asserted formula an unverified assertion rather than a derived result.

    Authors: We agree. The abstract and introduction assert the existence of a correspondence to solutions of two equations with variables in (1, n²) but do not state the equations or prove that their solution count equals p(n). In revision we will state the two Diophantine equations explicitly, define the bijection, and show why the cardinality matches p(n). revision: yes

  2. Referee: [—] No section or equation: the manuscript contains no explicit system of equations, no proof of bijectivity, and no verification that the solution count reproduces known values of p(n) for small n, so the equality p(n) = #solutions cannot be checked for internal consistency or reduction to prior expressions.

    Authors: This observation is accurate. The manuscript supplies neither the equations, a bijectivity proof, nor numerical checks. The revised version will include a dedicated section with the explicit system, the proof that the number of non-negative integer solutions equals p(n), and direct comparisons with known partition numbers for small n. revision: yes

Circularity Check

0 steps flagged

No circularity; assertion of correspondence supplies no derivable steps

full rationale

The provided abstract asserts a correspondence equating p(n) to the cardinality of non-negative solutions of unspecified two-equation systems over {1,...,n²} but exhibits no equations, no explicit mapping, and no derivation chain. No self-citations, fitted parameters, or ansatzes appear. Because the visible text contains no load-bearing step that can be quoted and shown to reduce to its own inputs by construction, the circularity criteria are not met and the score is 0.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be identified.

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

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