An inequality between finite analogues of rank and crank moments

Bibekananda Maji; Garima Sood; Pramod Eyyunni

REVIEW 1 major objections 5 minor 23 references

An inequality between finite analogues of rank and crank moments

T0 review · 1 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For every size bound $N$ and every even $k$, the finite crank moment $M_{k,N}(n)$ strictly exceeds the finite rank moment $N_{k,N}(n)$ for all $n \geq 1$.

desk verdict A careful, dense proof of a natural open conjecture in partition theory; the machinery is borrowed but the result is new and the argument checks out. read the letter →

arxiv 1908.08660 v1 pith:XWODG6RR submitted 2019-08-23 math.NT

Pramod Eyyunni , Bibekananda Maji , Garima Sood This is my paper · ORCID

classification math.NT MSC 11P8011P8111P8205A17

keywords partitionsfiniteanaloguesrankmomentscranksymmetrizedsmallestpartsfunctioninequalityq-seriesidentities

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

Working with finite analogues of partition rank and crank statistics — weighted counts over vector partitions whose parts are restricted by a bound $N$ — the paper proves that the even-order moments always satisfy $M_{k,N}(n) > N_{k,N}(n)$ for all $n \geq 1$ and fixed $N$. This is the finite version of a classical inequality between crank and rank moments, and it was previously left as a conjecture in the paper that introduced these finite analogues. The proof derives an explicit generating function for the difference of the finite symmetrized moments whose coefficients are manifestly nonnegative, and then uses a positive change of basis to pass from symmetrized to ordinary moments. A byproduct is the stronger estimate $M_{2k,N}(n) - N_{2k,N}(n) \geq 2\,\mathrm{spt}(n,N)$, where $\mathrm{spt}(n,N)$ is the finite smallest-parts count.

What carries the argument

The load-bearing object is the generating function identity of Theorem 2.5 for $\mu_{2k,N}(n) - \eta_{2k,N}(n)$, the difference of the finite symmetrized crank and rank moments. It expresses that difference as a finite sum over non-increasing sequences $N \geq n_k \geq \cdots \geq n_1 \geq 1$ of terms with manifestly nonnegative coefficients, so the difference is a nonnegative integer for every $n$. The second ingredient is the family of polynomials $g_j(m)=\prod_{i=0}^{j-1}(m^2-i^2)$ together with the positive integers $S^*(k,j)$, which rewrite ordinary $2k$-th moments as positive linear combinations of symmetrized moments. These two parts turn coefficient nonnegativity into the strict moment inequality.

What would settle it

Compute the finite moments directly from the definitions (1.9) and (1.10) for small values, say $N=2$, $k=2$, and $n=1,\dots,10$, and compare with the claim; a single $n$ with $M_{2,N}(n) \leq N_{2,N}(n)$ would refute the theorem. Alternatively, expand the right-hand side of (2.9) as a $q$-series to a chosen order and check that no coefficient is negative — the proof asserts none can be.

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Extended reading notes

Core claim

The central discovery is the identity of Theorem 2.5: the generating function for the difference of the finite symmetrized crank and rank moments equals a single chain sum $$\sum_{N \geq n_k \geq \cdots \geq n_1 \geq 1} \frac{$q^{{n_1+\cdots+n_k}}$}{(1-$q^{{n_1}}$)^2 \cdots (1-$q^{{n_k}}$)^2 ($q^{{n_1+1}}$;q)_{N-n_1}},$$ which expands as a power series with nonnegative coefficients. Hence each coefficient $\mu_{2k,N}(n) - \eta_{2k,N}(n)$ is nonnegative. Using the finite analogue of the classical relations between moments and symmetrized moments — expressed through the positive integers $S^*(k,j)$ — the paper converts this coefficient nonnegativity into the strict inequality $M_{2k,N}(n) > N_{2k,N}(n)$ for every even order, and indeed shows the gap is at least $2\,\mathrm{spt}(n,N)$.

Load-bearing premise

The proof depends on a double limit in Proposition 5.1 where the expressions are divided by $(1-\rho_1)(1-\rho_2)$ and then $\rho_1, \rho_2$ are taken to $1$ inside factors that vanish at that limit; the paper invokes a limiting identity for this step without supplying a full formal justification, so if that limit is not valid the nonnegative-coefficient representation is not established.

Editorial extensions

If this is right

For each fixed $N$, the finite gap $M_{2k,N}(n) - N_{2k,N}(n)$ is at least $2\,\mathrm{spt}(n,N)$, so the difference grows at least as fast as the finite smallest-parts function in $n$.
The finite higher-order spt-function $\mathrm{spt}_k(n,N) := \mu_{2k,N}(n) - \eta_{2k,N}(n)$ is nonnegative for all $k,n,N$, and equals $\mathrm{spt}(n,N)$ when $k=1$.
Letting $N \to \infty$ in the chain-sum identity recovers the classical identity for the difference of symmetrized crank and rank moments, and hence the classical inequality between the $2k$-th crank and rank moments.
Because the coefficients $S^*(k,j)$ are positive, nonnegativity of the symmetrized differences implies nonnegativity of the ordinary moment differences for every even $k$, not just for the smallest order.

Reading between the lines

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

The same positivity mechanism may yield a stronger lower bound for $M_{2k,N}(n)-N_{2k,N}(n)$ in terms of higher-order finite spt-functions, since the proof only uses the $j=1$ term of the symmetrized expansion.
A natural test is to compute the ratio $(M_{2k,N}(n)-N_{2k,N}(n))/(2\,\mathrm{spt}(n,N))$ for small $N,k,n$; the paper's proof forces it to be at least $1$, and numerical values would show how sharp that bound is.
If the double limit in Proposition 5.1 were justified as a formal power series identity in $q$ for all $N$ and $k$, the proof would hold without any convergence assumptions, and the same style of argument could be tried on other pairings of partition statistics.
The chain sum in Theorem 2.5 resembles a count of marked chains and may admit a direct combinatorial interpretation for $\mathrm{spt}_k(n,N)$, mirroring the classical interpretation of higher-order spt-functions.

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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 conjecture was open, and this paper settles it. The main new ingredients are finite analogues of symmetrized rank and crank moments, explicit generating functions for them, and a Bailey-pair lemma (Proposition 5.1) that is used to produce a manifestly nonnegative generating function for the difference. That last step is the load-bearing part, and it holds up. I checked the flagged double limit in Proposition 5.1: identity (3.7) is correct, as a formal expansion in u = 1 - rho_1 and v = 1 - rho_2 shows, and the n_1 = 0 term vanishes linearly so no pole survives. The paper would be easier to referee if (3.7) were proved or cited rather than quoted, but this is an expository gap, not a substantive one.

The proof follows Garvan's template closely, as the authors say, but that is fair: the finite analogues are genuinely new, the generating function identities are new, and the conclusion is stronger than the original conjecture (the difference is at least 2 spt(n,N), which is positive). The reliance on the authors' earlier paper [15] for definitions and the conjecture is legitimate background, not circularity. The main theorem is proved from explicit q-series manipulations, and I did not find any place where the conjecture was assumed.

Soft spots are minor. The text is very dense; a non-specialist will struggle to follow the index juggling in Section 5, and a few more signposts would help. There is a typo in the display in the proof of Conjecture 1.2 (an extra q^n appears in the summand), and a couple of other harmless slips. The concluding remarks suggest asymptotics and congruences as future work, which is appropriate rather than a deficiency.

This paper is for partition theorists and q-series specialists. It will not reorganize the field, but it proves a natural conjecture with explicit, checkable identities. It deserves a serious referee. My recommendation: send to a specialist, ask for a proof or citation for (3.7) and a fix of the typos, and accept after minor revision.

Referee Report

1 major / 5 minor

Summary. The paper proves Conjecture 1.2 from the authors' earlier work [15]: for every fixed natural number N and every even k ≥ 2, the finite analogue of the crank moment exceeds the finite analogue of the rank moment: M_{k,N}(n) > N_{k,N}(n) for all n ≥ 1. To do this, the authors define finite analogues of symmetrized rank and crank moments η_{k,N}(n) and μ_{k,N}(n), derive their generating functions (Theorems 2.2 and 2.4), and then prove a nonnegative-coefficient generating function for the difference μ_{2k,N}(n) − η_{2k,N}(n) (Theorem 2.5). The proof of Theorem 2.5 is built on an induction using Bailey's lemma (Proposition 5.1) together with two classical Bailey pairs. The final step links ordinary finite moments to symmetrized moments through Garvan's numbers S*(k,j), yielding the stronger bound M_{2k,N}(n) − N_{2k,N}(n) ≥ 2 spt(n,N) > 0. The paper is a direct finite analogue of Garvan's proof of the classical rank-crank moment inequality.

Significance. If the proof is accepted, the paper settles a conjecture from [15] and provides a finite/restricted analogue of Garvan's theorem, extending a well-known circle of ideas in partition theory. The main technical contribution, Theorem 2.5, is a finite analogue of Garvan's difference generating function and appears to be new. The arguments are explicit q-series manipulations: Theorems 2.2 and 2.4 are derived from known partial-fraction identities, and Proposition 5.1 is a genuine application of Bailey's lemma. The positivity conclusion is clean and yields a stronger statement than the conjecture itself, namely the lower bound in terms of spt(n,N). The main weakness is that a key limiting identity, Eq. (3.7), is quoted without proof; I verified it independently and found it correct, but the manuscript should supply a derivation or reference. There are also several small typographical errors. Overall, the result is worthy of publication once the missing justification is added.

major comments (1)

[Section 3, Eq. (3.7)] The limit identity (3.7) is stated without proof or reference and is used in a load-bearing way in both the base case and the induction step of Proposition 5.1, specifically when the proof divides by (1−ρ1)(1−ρ2) and then lets ρ1,ρ2 → 1. The identity is true: writing a = 1/ρ1 = 1−u and b = 1/ρ2 = 1−v, each factor (1−q^j)(1−ab q^j)/((1−a q^j)(1−b q^j)) expands as 1 − uv q^j/(1−q^j)^2 + O(u^2 v, u v^2), which gives (3.7). However, the manuscript should provide this derivation or cite a source, and it should comment on the double limit at the points where the denominators (q/ρ1)_n(q/ρ2)_n vanish. As written, the proof of Proposition 5.1 is incomplete at a critical step, even though the gap is readily fixable.

minor comments (5)

[Section 5, proof of Conjecture 1.2] The displayed generating function in the proof of Conjecture 1.2 reads (μ_{2t,N}(n)q^n − η_{2t,N}(n))q^n, which appears to be a typo; it should be (μ_{2t,N}(n) − η_{2t,N}(n))q^n.
[Section 5, proof of Theorem 2.5] The sentence 'Divide both sides of (5.4) by (q)_N' refers to the unnumbered display in Corollary 5.4, not to equation (5.4) in Proposition 5.5. Please add a cross-reference or number the display in Corollary 5.4.
[Section 5, Corollary 5.2] In the sentence 'Substituting the above Bailey pair in Theorem 5.1', the reference should be to Proposition 5.1, not Theorem 5.1.
[Abstract and Introduction] There are several typographical slips, including 'conjectur ed' and 'c rank' in the abstract, and 'Garvan himself' in the first sentence; these should be corrected in a final revision.
[Section 5, proof of Proposition 5.1] When the proof says 'letting ρ1 → 1, ρ2 → 1 and using (3.7)', it may help the reader to note explicitly that all sums are finite (since the indices are bounded by N), so the interchange of limit and summation is immediate.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found; the conjecture is proved from explicit generating functions and standard Bailey-pair identities, not assumed or fitted.

full rationale

I walked the derivation chain from the finite moment definitions through the symmetrized generating functions to the final inequality. The finite analogues are taken as definitions from the authors' earlier paper [15], and the generating functions (3.1) and (3.4) are cited from that paper, but the conjectured inequality itself is never assumed. Theorems 2.2 and 2.4 derive the symmetrized rank and crank generating functions directly from these definitions using standard derivative manipulations. Proposition 5.1 is proved from Bailey's lemma rather than from the target inequality. Corollaries 5.2-5.4 substitute standard Bailey pairs into Proposition 5.1, and Theorem 2.5 follows by combining these identities with Theorems 2.2 and 2.4. The final step uses the nonnegative-coefficient form of the difference generating function and the positive coefficients S*(k,j), so no fitted parameter is renamed as a prediction and no uniqueness or ansatz is imported from prior work to force the conclusion. The only delicate point visible in the proof, the double-limit identity (3.7), is quoted without proof; that is an expository gap rather than a circular reduction, since the identity is a computational lemma independent of the conjecture and can be checked directly. Self-citations supply definitions and background material, not the central claim, so the proof is not circular.

Assumptions & free parameters 0 free parameters · 6 assumptions · 2 invented entities

The result is a theorem in partition theory. It rests on standard q-series machinery: Bailey's lemma, well-known Bailey pairs due to Andrews, and partial fraction expansions of generating functions. No free parameters are fitted; the only invented objects are the finite analogues of symmetrized moments, which are definitions used to organize the proof.

assumptions (6)

standard math Bailey's Lemma (Theorem 3.1)
Used to iterate Bailey pairs in the proof of Proposition 5.1, which underpins Theorem 2.5.
standard math Well-known Bailey pairs (α_n, β_n) from Andrews's book [1, pp. 27-28]
Used in Corollaries 5.2 and 5.4 to evaluate specific sums appearing in the proof of Theorem 2.5.
standard math Partial fraction identities (3.2) and (3.5) due to Andrews
Used to rewrite the rank and crank generating functions in Section 4 before differentiating.
standard math Limiting identity (3.7) for the double limit of q-products
Quoted without proof in Section 3 and used in the double limit step in Proposition 5.1.
standard math Garvan's polynomial basis relation x^{2n} = Σ_{k=1}^n S*(n,k) g_k(x) with S*(n,k) > 0
Used in Proposition 5.5 to express ordinary moments as positive combinations of symmetrized moments.
domain assumption Symmetry NS1(m,n)=NS1(-m,n) and MS2(m,n)=MS2(-m,n), established in [15, p.9]
Used to show odd symmetrized moments vanish (Propositions 2.1 and 2.3) and to simplify the expression for μ_{2k,N}(n) in Proposition 5.5.

invented entities (2)

Finite analogue of symmetrized rank moment η_{k,N}(n)
purpose: To express ordinary finite rank moments as nonnegative combinations and to obtain a generating function with nonnegative coefficients.
Defined in (2.2); an internal mathematical definition with no external falsifiable handle.
Finite analogue of symmetrized crank moment μ_{k,N}(n)
purpose: To play the same role for crank moments; the difference of this entity and η_{k,N}(n) is shown to have a nonnegative-coefficient generating function.
Defined in (2.6); an internal mathematical definition with no external falsifiable handle.

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Cite this review

Pith. "Pith review of An inequality between finite analogues of rank and crank moments." pith.science (2026). https://pith.science/paper/XWODG6RR

@misc{pith2026190808660,
  author       = {Pith},
  title        = {Pith review of: An inequality between finite analogues of rank and crank moments},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XWODG6RR}},
  note         = {Machine review of arXiv:1908.08660}
}

read the original abstract

The inequality between rank and crank moments was conjectured and later proved by Garvan himself in 2011. Recently, Dixit and the authors introduced finite analogues of rank and crank moments for vector partitions while deriving a finite analogue of Andrews' famous identity for smallest parts function. In the same paper, they also conjectured an inequality between finite analogues of rank and crank moments, analogous to Garvan's conjecture. In the present paper, we give a proof of this conjecture.

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Reference graph

Works this paper leans on

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Pith tools

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

[1] [15]

Untrodden pathways in the theory of the restricted partition function $p(n, N)$

A. Dixit, P. Eyyunni, B. Maji and G. Sood, Untrodden pathways in the theory of the restricted partitio n function p(n,N) , arXiv:1812.01424, 2018

work page Pith review arXiv 2018

[2] [1]

G. E. Andrews, q-series: their development and applicat ion in analysis, number theory, combinatorics, physics, and computer algebra, CBMS Regional Conference Se ries in Mathematics, 66, American Math- ematical Society, Providence, RI, 1986

work page 1986

[3] [2]

G. E. Andrews, Ramanujan and partial fractions , Contributions to the History of Indian Mathematics, Hindustan Book Agency, New Delhi, 2005

work page 2005

[4] [3]

Andrews, Partitions, Durfee symbols, and the Atkin-Garvan moments o f ranks , Invent

G.E. Andrews, Partitions, Durfee symbols, and the Atkin-Garvan moments o f ranks , Invent. Math. 169 (2007) 37–73

work page 2007

[5] [4]

G. E. Andrews, The number of smallest parts in the partitions on n, J. Reine Angew. Math. 624 (2008), 133–142

work page 2008

[6] [5]

G. E. Andrews, R. Askey, R. Roy, Special functions. Encyc lopedia of Mathematics and its Applications,

work page

[7] [6]

G. E. Andrews, B. C. Berndt, Ramanujan’s Lost Notebook, P art I, Springer, New York, 2005

work page 2005

[8] [7]

G. E. Andrews and F. G. Garvan, Dyson ’s crank of a partition, Bull. Amer. Math. Soc. (N.S.) 18 (1988) 167–171

work page 1988