Overpartitions and Kaur, Rana, and Eyyunni's mex sequences

Brian Hopkins; James A. Sellers

REVIEW 3 major objections 3 minor 15 references

Overpartitions and Kaur, Rana, and Eyyunni's mex sequences

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that partitions whose mex sequence has length at least $r$ are counted by a single family of overpartitions, and derives the two requested bijective proofs from that correspondence.

desk verdict The bijection is real and worth having, but the written proof of Theorem 2.3 has two correctable errors. read the letter →

arxiv 2505.22588 v1 pith:7ZP4XFXR submitted 2025-05-28 math.CO math.NT

Brian Hopkins , James A. Sellers This is my paper

classification math.COmath.NT MSC 05A1711P81

keywords partitionsoverpartitionsmexsequencesminimalexcludantbijectiveproofGlaishermapgeneratingfunctionsno-gaps

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

The paper's central claim is that for every $r \ge 1$ and $n \ge 0$, the partitions of $n$ whose mex sequence has length at least $r$ are equinumerous with the overpartitions of $n$ in which every non-overlined part is greater than $r$ and has the same parity as $r+1$. The authors prove this with a generating function identity and, more substantively, with an explicit two-way bijection. They then compose this bijection with Glaisher's map to give combinatorial proofs of the two identities in Kaur, Rana, and Eyyunni's Corollary 11, one for odd $r$ and one for even $r$, which those authors had requested. A sympathetic reader should care because the result replaces two analytically proved identities with one uniform combinatorial object, the restricted overpartitions, and demonstrates that a single bijection can explain both the even-part and two-colored-odd-part interpretations of mex-sequence partitions.

What carries the argument

The central object is the mex sequence itself—the longest run of consecutive missing part sizes beginning at the minimal excludant—together with the overpartition family $\overline{P}_r(n)$ defined by the parity condition on non-overlined parts. The argument is carried by an explicit decomposition of a mex-sequence partition $\kappa$ as $(\kappa_1-\sigma_1,\dots,\kappa_i-\sigma_i) \oplus (\sigma_1,\dots,\sigma_j)$, where the second summand is a no-gaps partition whose conjugate provides the distinct overlined parts, and the first summand has all parts at least $r$. In the reverse direction the same decomposition is inverted by the componentwise sum $\oplus$, whose effect on mex sequences is asserted rather than proved in the intermediate case. Glaisher's classical bijection between distinct parts and odd parts then bridges $\overline{P}_r(n)$ to the two restricted partition sets of Corollary 1.7.

What would settle it

Enumerate all overpartitions $(\lambda,\mu) \in \overline{P}_r(n)$ for small $n$ and $r$ with $1 \le m < \lambda_1$ (for instance $r=4$, $\lambda=(6,4,2,1)$, $\mu=(5,5)$), compute $\lambda' \oplus \mu$ term by term, and check whether its mex sequence has length at least $r$; the first counterexample would refute Theorem 2.3's inverse map.

Watch

Extended reading notes

Core claim

Theorem 2.3 states that $\overline{p}_r(n) = p^{\mathrm{mex}}_r(n)$ for all $r,n$. The proof exhibits a bijection from $P^{\mathrm{mex}}_r(n)$ to $\overline{P}_r(n)$: a partition with infinite mex sequence conjugates to a partition into distinct parts and is sent to an all-overlined overpartition; otherwise a backward recursion splits $\kappa$ into a partition whose parts are all at least $r$ and a no-gaps partition, the latter conjugating to the overlined parts and the former supplying the non-overlined parts. The inverse map sends an overpartition $(\lambda,\mu)$ to $\lambda' \oplus \mu$, the componentwise sum of the conjugate of the overlined parts and the non-overlined parts. The paper also proves two Glaisher-based bijections showing that $\overline{P}_r(n)$ is equinumerous with partitions avoiding even parts below $r$ when $r$ is odd, and with two-colored odd parts above $r$ when $r$ is even, thereby establishing Corollary 1.7 combinatorially.

Load-bearing premise

The inverse half of the bijection assumes, with a one-sentence justification, that the componentwise sum $\lambda' \oplus \mu$ has a run of at least $r$ consecutive missing integers starting at its mex whenever the number of non-overlined parts is between 1 and the largest overlined part; if that step fails, the inverse map lands outside the target set.

Editorial extensions

If this is right

For every $r$ and $n$, any enumeration of $\overline{P}_r(n)$ immediately yields $p^{\mathrm{mex}}_r(n)$, so the two statistics are the same object in disguise.
Corollary 1.7 of Kaur, Rana, and Eyyunni now has bijective proofs: the odd-$r$ case via Glaisher's map on odd parts, the even-$r$ case via a two-coloring of odd parts.
Partitions with infinite mex sequences correspond exactly to overpartitions with no non-overlined parts, so the no-gaps partition of size $n$ is naturally identified with a distinct-part partition of $n$.
The generating function derivation shows that Euler's identity, not Heine's transformation, is enough to connect mex sequences to overpartitions.

Reading between the lines

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

The decomposition used in the forward map is essentially a greedy reduction of a partition against a no-gaps template; the same recursion may produce bijections for other 'chain' statistics such as second mex or $r$-chain excludants, though the paper does not state this.
The inverse map's unresolved step—that $\lambda' \oplus \mu$ always begins with a run of at least $r$ missing integers when $1 \le m < \lambda_1$—could be tested computationally, and a formal componentwise definition of $\oplus$ would make the bijection fully rigorous.
Because Glaisher's map is the only ingredient separating the odd- and even-$r$ cases, any partition statistic invariant under Glaisher's map will inherit a mex-sequence interpretation from the overpartition family; this suggests a modular analogue where parity is replaced by congruence modulo $k$.
The result implies that the two families $P_e^{>r}(n)$ and $P_{o,2}^{>r}(n)$ are Glaisher-equivalent to the same overpartition set, so a single generating function underlies both, a fact that could simplify future proofs of identities involving these sets.

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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

Dear colleague,

The punchline: the main bijection is genuine and the paper is a real contribution, but the proof of Theorem 2.3 as printed contains two concrete mistakes that need correcting before publication. Do not desk-reject; send it to a referee with a request for revision.

What's new: the explicit bijection between partitions with mex sequence length at least r and the overpartition family P_r (non-overlined parts >r and parity r+1), and the two parity-specific bijections to KRE's sets. That gives the requested combinatorial proofs of Corollary 11. The uniform overpartition model is a nice unifying perspective, and the construction via componentwise sum and the sigma recursion is clever.

The analytic identity was already known from KRE; the novelty is the combinatorial maps. They are explicit and checkable, and the paper includes tables that help.

Soft spots, in proportion. First, the sigma recursion in the forward direction has the parity condition reversed. As printed, the non-overlined parts come out with parity r, not r+1; the examples use the opposite rule. That's a typo-level error but it is in a proof, so it matters. Second, in the inverse direction, the assertion that for 1≤m<λ1, λ'_{m+1} is either λ_m or λ_m−1 is false. For λ=(5,3,2,1), λ'=(4,3,2,1,1) and m=1, λ'_2=3 while λ_1=5. The conclusion that κ has mex sequence of length at least r is still true, but the given argument does not establish it. A correct argument uses the fact that λ' has no gaps, so the tail contains every integer down to 1, and the first m parts after adding μ are each at least λ'_{m+1}+r+1, so the interval [λ'_{m+1}+1, λ'_{m+1}+r] is missing. The proof should be rewritten along those lines.

I also would like to see a formal definition of the ⊕ operation rather than just an example, and a bit more detail on why the maps are mutual inverses; "straightforward to confirm" is doing work in a bijection paper.

The theorem itself appears sound, and the generating function proof is clean. The paper is well-written and cites the relevant literature, including KRE's request.

Who should read it: partition theorists and anyone working on mex-type statistics. It is a solid, short contribution that deserves a serious referee, though not a high-impact one.

Recommendation: accept with minor revision after the two proof errors are fixed.

Referee Report

3 major / 3 minor

Summary. The paper introduces a family of overpartitions P_r(n) in which non-overlined parts are greater than r and have the same parity as r+1, and claims in Theorem 2.3 that the number of such overpartitions of n equals the number of ordinary partitions of n whose mex sequence has length at least r. The proof is given both by generating functions and by a purported explicit bijection. The paper then uses this bijection to give combinatorial proofs of Corollary 1.7 of Kaur, Rana, and Eyyunni, splitting into odd and even r via Propositions 2.5 and 2.6. The paper is clearly written, with many examples and tables.

Significance. If the main bijection is correct, the paper provides a unified overpartition interpretation of mex sequences and answers the request in KRE for bijective proofs of Corollary 1.7. The construction is genuinely combinatorial and does not depend on the analytic theorem except as an external benchmark. The tables and worked examples are a strength. However, the proof of Theorem 2.3 as written contains two explicit incorrect statements, both localized and repairable, and the closing injectivity claim is not substantiated.

major comments (3)

[Section 2, proof of Theorem 2.3] The displayed recursion for σℓ has the parity condition reversed. The paper states that σℓ = σℓ+1 when κℓ − σℓ+1 ≡ r mod 2 and σℓ = σℓ+1 + 1 otherwise. To ensure that the non-overlined part κℓ − σℓ has parity r+1, as required by Definition 2.2, the first case must instead be κℓ − σℓ+1 ≡ r+1 mod 2. As printed, the forward map produces non-overlined parts of parity r. For example, with r=2 and κ=(8,7,3,2,1,1), at ℓ=2 we have κ2 − σ3 = 7−3 = 4 ≡ 0 ≡ r, so the printed rule forces σ2=3 and a non-overlined part 4, whereas Definition 2.2 and the paper's own example require σ2=4 and a non-overlined part 3.
[Section 2, proof of Theorem 2.3] In the inverse direction, the assertion that for 1 ≤ m < λ1, κ_{m+1} = λ'_{m+1} is either λ_m or λ_m − 1 is false. For λ = (5,3,2,1), one has λ' = (4,3,2,1,1), and for m=1, λ'_2 = 3 while λ_1 = 5. The desired conclusion that κ has a mex sequence of length at least r is nevertheless true, but it requires a different argument: because λ' has no gaps, the tail λ'_{m+1},...,λ'_{λ1} contains every integer from 1 to λ'_{m+1}, while the first m parts of κ exceed λ'_{m+1}+r, so the interval [λ'_{m+1}+1, λ'_{m+1}+r] is missing from κ. The proof should be revised to use this argument.
[Section 2, proof of Theorem 2.3] The proof closes with “It is straightforward to confirm that each map is injective, so that together the maps establish a bijection,” but no such confirmation is provided. Because a generating-function equality has already been established in the same proof, injectivity of both maps would indeed suffice for a bijection, but the manuscript should either prove that the two maps are mutual inverses or verify that both maps preserve the defining conditions and are injective. In particular, it is not shown that applying the forward map to κ = λ' ⊕ μ recovers (λ, μ) in the general case; the examples do not cover the full case distinction.

minor comments (3)

[Throughout] The overlines in Tables 3–6 are difficult to see in the typeset version, which makes several examples hard to verify; please ensure that overlined parts are clearly rendered.
[Section 1] The sentence “With this is hand” should read “With this in hand”.
[Section 2, Example 2.4] The examples are helpful, but adding one explicit two-line demonstration that the two maps are inverses on a single example would improve readability and compensate for the missing general verification.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main bijection is constructed directly and self-contained.

full rationale

The paper's central result, Theorem 2.3, asserts a numerical equality and provides two proofs: a generating-function proof that combines Kaur–Rana–Eyyunni's Theorem 1.5 with Euler's identity, and a combinatorial proof that builds explicit maps in both directions. The generating-function argument legitimately uses the external analytic result as an input; the paper's contribution is the identification of the overpartition family P_r(n) and a direct bijection. The combinatorial proof does not assume the counting identity p_r(n)=p_mex_r(n); it constructs the correspondence by conjugation, Glaisher's map, and part-wise addition. The maps are described explicitly and invertibility is argued. Definitions 1.4 and 2.2 are independent: P_mex_r(n) is defined by mex-sequence length and P_r(n) by parity/size restrictions on non-overlined parts; the equality is not built into either definition. The self-citations [7]–[10] appear only in a survey sentence about mex generalizations and are not load-bearing for Theorem 2.3 or Corollary 1.7. No fitted parameter is renamed as a prediction, and no uniqueness theorem from the authors' prior work is invoked to force the choice of overpartitions. The skeptic's report of two local proof errors (the displayed σℓ parity condition and the claim κ_{m+1}=λ_m or λ_m−1) concerns correctness of a supporting sentence, not circularity; even if the printed proof needs repair, the reduction involved is not a self-referential one. The paper is self-contained against the external benchmark and warrants a circularity score of 0.

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

The paper contains no fitted parameters. The assumptions are classical bijections and q-series identities, plus the prior analytic theorem being given a combinatorial proof. The new overpartition family P_r is introduced by definition rather than postulated as an unexplained entity.

assumptions (5)

standard math Glaisher's theorem: bijection between partitions into distinct parts and partitions into odd parts
Used in Propositions 2.1, 2.5, and 2.6 to convert between odd parts and overlined distinct parts.
standard math Sylvester-Franklin conjugation: partitions into distinct parts biject to no-gap partitions (Proposition 1.8)
Used in Theorem 2.3 to identify (sigma_1,...,sigma_j)' as a distinct-part partition when sigma has no gaps.
standard math Standard q-Pochhammer identities, including (-q;q)_infinity = 1/(q;q^2)_infinity
Used in the generating-function proof of Theorem 2.3 to connect the P_r generating function to KRE's expression.
domain assumption KRE's analytic Theorem 1.5 giving the generating function for p_mex_r(n)
Used as the external statement whose combinatorial proof is provided; not needed for the bijection construction itself.
standard math The componentwise sum operation, denoted plus inside a circle, applied to two partitions padded with zeros yields a partition
Assumed throughout Theorem 2.3 and its examples; follows from nonincreasing parts but is not stated formally in the paper.

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Cite this review

Pith. "Pith review of Overpartitions and Kaur, Rana, and Eyyunni's mex sequences." pith.science (2026). https://pith.science/paper/7ZP4XFXR

@misc{pith2026250522588,
  author       = {Pith},
  title        = {Pith review of: Overpartitions and Kaur, Rana, and Eyyunni's mex sequences},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7ZP4XFXR}},
  note         = {Machine review of arXiv:2505.22588}
}

read the original abstract

Kaur, Rana, and Eyyunni recently defined the mex sequence of a partition and established, by analytic methods, connections to two disparate types of partition-related objects. We make a bijection between partitions with certain mex sequences and a uniform family of overpartitions which allows us to provide combinatorial proofs of their results, as they requested.

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Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[1]

G. E. Andrews and K. Eriksson.Integer Partitions, Cambridge Univ. Press, Cambridge, 2004

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[2]

G. E. Andrews and D. Newman. Partitions and the minimal excludant.Ann. Comb.23 (2019) 249–254

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S. C. Bhoria, P. Eyyunni, and R. Li. On the combinatorics ofr-chain minimal and maximal excludants.Discrete Math.347 (2024) 114187

work page 2024
[4]

Corteel and J

S. Corteel and J. Lovejoy. Overpartitions.Trans. Amer. Math. Soc.356 (2004) 1623–1635

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J. W. L. Glaisher. A theorem in partitions.Messenger Math.12 (1883) 158–170

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P. J. Grabner and A. Knopfmacher. Analysis of some new partition statistics.Ramanujan J. 12 (2006) 439–454

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Hopkins and J

B. Hopkins and J. A. Sellers. Turning the partition crank.Amer. Math. Monthly127 (2020) 654–657

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Hopkins and J

B. Hopkins and J. A. Sellers. On Blecher and Knopfmacher’s fixed points for integer partitions. Discrete Math.347 (2024) 113938

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

G. E. Andrews and K. Eriksson.Integer Partitions, Cambridge Univ. Press, Cambridge, 2004

work page 2004

[2] [2]

G. E. Andrews and D. Newman. Partitions and the minimal excludant.Ann. Comb.23 (2019) 249–254

work page 2019

[3] [3]

S. C. Bhoria, P. Eyyunni, and R. Li. On the combinatorics ofr-chain minimal and maximal excludants.Discrete Math.347 (2024) 114187

work page 2024

[4] [4]

Corteel and J

S. Corteel and J. Lovejoy. Overpartitions.Trans. Amer. Math. Soc.356 (2004) 1623–1635

work page 2004

[5] [5]

J. W. L. Glaisher. A theorem in partitions.Messenger Math.12 (1883) 158–170

work page

[6] [6]

P. J. Grabner and A. Knopfmacher. Analysis of some new partition statistics.Ramanujan J. 12 (2006) 439–454

work page 2006

[7] [7]

Hopkins and J

B. Hopkins and J. A. Sellers. Turning the partition crank.Amer. Math. Monthly127 (2020) 654–657

work page 2020

[8] [8]

Hopkins and J

B. Hopkins and J. A. Sellers. On Blecher and Knopfmacher’s fixed points for integer partitions. Discrete Math.347 (2024) 113938

work page 2024