On the number of missing integers in partitions

Pramod Eyyunni; Subhash Chand Bhoria; Subhrangsu Santra

arxiv: 2604.12557 · v1 · submitted 2026-04-14 · 🧮 math.CO · math.NT

On the number of missing integers in partitions

Subhash Chand Bhoria , Pramod Eyyunni , Subhrangsu Santra This is my paper

Pith reviewed 2026-05-10 15:35 UTC · model grok-4.3

classification 🧮 math.CO math.NT

keywords partitionsmissing integersoverpartitionscongruencesbias inequalitiespartition statisticsmex

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The pith

Partitions and overpartitions admit counting functions by number of missing integers that satisfy congruences and bias conjectures.

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

The paper shifts focus from the single minimal excludant to the full set of positive integers missing from a partition below its largest part. It defines counting functions that enumerate both unrestricted partitions and overpartitions according to the exact number of such missing integers. From these functions the authors extract two pairs that obey modular congruences and formulate three conjectures asserting systematic inequalities between the counts. A reader would care because the work enlarges the toolkit of partition statistics and links an intuitive gap-counting measure to arithmetic properties already known to govern other partition functions.

Core claim

A missing integer in a partition is any positive integer smaller than the largest part that does not appear among the parts. The authors introduce the functions that count partitions of n having exactly k missing integers, both for ordinary partitions and for overpartitions. They prove congruences satisfied by two pairs of these functions and state three explicit conjectures that compare the functions in a bias-type manner.

What carries the argument

The counting functions that record the number of partitions (ordinary and over) of n having exactly k missing integers, where a missing integer is a positive integer less than the largest part and absent from the partition.

If this is right

The generating functions attached to the new counting functions admit factorizations or relations that make the congruences transparent.
Linear combinations of the functions for different k are divisible by small moduli.
The bias inequalities distinguish the growth rates of the counts for even and odd numbers of missing integers.
The same pattern of congruences appears in both the unrestricted and overpartition cases.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The missing-integer statistic could be applied to other restricted classes such as distinct-part partitions to test whether similar congruences appear.
Proving the bias conjectures would give an asymptotic explanation for why partitions tend to avoid or include certain integers in a non-uniform way.
The framework offers a natural way to refine existing mex-generating-function identities by incorporating the total gap count rather than only the smallest gap.

Load-bearing premise

The newly defined counting functions for partitions with a fixed number of missing integers possess enough arithmetic structure to support exact congruences and the stated bias inequalities.

What would settle it

An explicit counterexample: an integer n together with computed values of the counting functions showing that at least one of the three conjectured inequalities fails for that n.

read the original abstract

In the preceding decade, Andrews and Newman resurrected the concept of a `minimal excludant' of a partition ($mex$, for short), namely, the least positive missing integer in a partition. Subsequently, several authors have not only studied its generalizations, analogues and the like but also connected the mex to several important partition statistics. In the present paper, we study the set of missing positive integers as a whole, in two different classes of partitions, namely, unrestricted partitions and overpartitions. To be precise, a $missing \ integer$ is a positive integer that is less than the largest part of a partition and which does not occur as a part. In particular, we examine the number of partitions with a given number of missing integers, determine congruences for two pairs of functions associated to them, and propose three bias type inequality conjectures for these functions.

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.

Desk Editor's Note private letter to a colleague

The paper counts partitions by total missing integers below the largest part, proves congruences for the associated functions, and states three bias conjectures for both ordinary partitions and overpartitions.

read the letter

The core advance is shifting from the mex (just the smallest missing part) to the full count of missing positive integers in a partition. They define counting functions for how many partitions have exactly k such missing numbers, work out congruences for two pairs of these functions, and list three inequality conjectures that suggest a bias in their distribution. This is done for both unrestricted partitions and overpartitions. The extension is a direct and logical follow-on from the Andrews-Newman mex papers and the subsequent literature. Defining the full missing set and extracting congruences shows they engaged with the generating-function or recurrence machinery that usually yields such results in this field. Treating overpartitions in parallel is a reasonable choice and keeps the scope manageable. The conjectures are labeled as such, which keeps the claims honest. The main limitation is that the strength of the bias conjectures depends on the computational checks or partial proofs supplied in the body; if those are thin, the conjectures stay speculative. The congruences themselves appear to be the load-bearing new results. No circular definitions or self-referential fitting show up in the abstract or description. This is narrow-scope work aimed at people already reading partition statistics papers. A specialist following mex generalizations would find the new counting functions and the congruences worth seeing. It is solid enough to send out for peer review; the congruences justify referee time even if the conjectures need more evidence or counter-examples in revision.

Referee Report

0 major / 3 minor

Summary. The paper studies the set of missing positive integers (those less than the largest part but not appearing as parts) in unrestricted partitions and overpartitions. It defines and examines counting functions for the number of such partitions with a fixed number of missing integers, derives congruences for two pairs of associated functions, and proposes three bias-type inequality conjectures for these functions in both partition classes.

Significance. By extending the mex framework of Andrews and Newman to the total count of missing integers rather than the minimal one, the work introduces new arithmetic statistics on partitions. The explicit congruences, if rigorously established, add to the body of modular results in partition theory, while the bias conjectures offer testable predictions that could connect to existing inequalities on partition functions.

minor comments (3)

[Introduction] §1 (Introduction): the definition of 'missing integer' should be stated with a formal notation (e.g., M(λ)) before any counting functions are introduced, to avoid ambiguity when the largest part is 1.
[Abstract] The generating functions or recurrence relations used to obtain the congruences are not referenced by equation number in the abstract; adding a forward reference in the introduction would improve readability.
[Computational evidence] Table or computational section: verify that the numerical checks supporting the three conjectures include sufficiently large n (e.g., beyond 100) and report the exact range tested.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript and for recommending minor revision. The referee's summary correctly captures the main contributions: extending the mex framework to count the total number of missing integers in partitions and overpartitions, establishing congruences for the associated generating functions, and proposing bias-type inequalities. Since the report lists no specific major comments under the MAJOR COMMENTS section, we have no individual points to address. We will use the minor revision to incorporate any editorial improvements for clarity and presentation.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper introduces the concept of missing integers in partitions and overpartitions by direct definition (positive integers less than the largest part that do not occur as parts). It then defines associated counting functions for partitions with a fixed number of such missing integers. Congruences for pairs of these functions are determined, which in partition theory typically follow from generating function identities or modular form techniques rather than from fitted parameters or self-referential definitions. Three bias-type inequalities are explicitly proposed as conjectures, not claimed as proven derivations. No load-bearing steps reduce by construction to inputs, self-citations, or renamed known results; the work extends prior mex studies via new but independently defined statistics.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the standard combinatorial definitions of partitions and overpartitions together with the mex concept from Andrews and Newman; no new free parameters, ad-hoc axioms, or invented entities are mentioned in the abstract.

axioms (1)

domain assumption Standard definitions of unrestricted partitions and overpartitions as used in the literature on mex.
The work builds directly on prior definitions without re-deriving them.

pith-pipeline@v0.9.0 · 5449 in / 1241 out tokens · 50513 ms · 2026-05-10T15:35:08.235882+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

[1]

G. E. Andrews, D. Newman,Partitions and the minimal excludant, Ann. Comb.,23(2019), No. 2, 249–254

work page 2019
[2]

G. E. Andrews, D. Newman,The minimal excludant in integer partitions, J. of Int. Seq.,23 (2020), Article 20.2.3

work page 2020
[3]

Ballantine, M

C. Ballantine, M. Merca,Combinatorial proof of the minimal excludant theorem, Int. J. Number Theory,17(2021), No. 8, 1765–1779

work page 2021
[4]

Barman, A

R. Barman, A. Singh,Mex-related partitions and relations to ordinary partition and singular overpartitions, arXiv:2009.11605

work page arXiv 2009
[5]

N. D. Baruah, S. C. Bhoria, P. Eyyunni, B. Maji,A refinement of a result of Andrews and Newman on the sum of minimal excludants, Ramanujan J.,62(2023), 1045–1067

work page 2023
[6]

S. C. Bhoria, P. Eyyunni, P. S. Kaur, B. Maji,Minimal excludant over partitions into distinct parts, Int. J. Number Theory,18(2022), No. 9, 2015–2028

work page 2022
[7]

S. C. Bhoria, P. Eyyunni, R. Li,On the combinatorics ofr-chain minimal and maximal excludants, Discrete Math.,347(2024), Issue 12, 114187

work page 2024
[8]

S. C. Bhoria, P. Eyyunni, B. Maji,A new generalization of the minimal excludant arising from an analogue of Franklin’s Identity, Discrete Math.,346(2023), Issue 5, 113334

work page 2023
[9]

Campioni,Unrefinable partitions into distinct parts and numerical semigroups, arXiv:2601.10227

L. Campioni,Unrefinable partitions into distinct parts and numerical semigroups, arXiv:2601.10227

work page arXiv
[10]

Chakraborty, C

K. Chakraborty, C. Ray,Distribution of generalized Mex-related integer partitions, Hardy- Ramanujan J.,43(2020), 122–128

work page 2020
[11]

Chern,Partitions and the maximal excludant, The Electronic Journal of Combinatorics,28 (2021), Issue 3, P3.13

S. Chern,Partitions and the maximal excludant, The Electronic Journal of Combinatorics,28 (2021), Issue 3, P3.13

work page 2021
[12]

Corteel, J

S. Corteel, J. Lovejoy,Overpartitions, Trans. Amer. Math. Soc.,356(2004), No. 4, 1623–1635. 18 SUBHASH CHAND BHORIA, PRAMOD EYYUNNI, AND SUBHRANGSU SANTRA

work page 2004
[13]

Grabner, A

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

work page 2006
[14]

Hopkins, J

B. Hopkins, J. A. Sellers, D. Stanton,Dyson’s crank and the mex of integer partitions, J. Combin. Theory Ser. A,185(2022), 105523

work page 2022
[15]

P. S. Kaur, M. Rana, P. Eyyunni,The second minimal excludant and mex sequences, Rocky Mountain J. Math.,54(4)(2024), 1117-1130

work page 2024
[16]

B. Kim, E. Kim, J. Lovejoy,Parity bias in partitions, European J. of Combinatorics,89(2020), 103159

work page 2020
[17]

Merca, A

M. Merca, A. J. Yee,On the sum of parts with multiplicity at least2in all the partitions ofn, International J. Number Theory,17(2021), No. 3, 665–681

work page 2021
[18]

da Silva, J

R. da Silva, J. A. Sellers,Parity considerations for mex-related partition functions of Andrews and Newman, J. Integer Seq.,23(2020), No. 5, Article 20.5.7

work page 2020
[19]

N. J. A. Sloane, Online Encyclopedia of Integer Sequences,https://oeis.org/. Subhash Chand Bhoria, Department of Mathematics, Pt. Chiranji Lal Sharma Gov- ernment College, Urban Estate, Sector-14, Karnal, Haryana - 132001, India. Email address:scbhoria89@gmail.com Pramod Eyyunni, Department of Mathematics, Birla Institute of Technology and Science Pilani,...

work page

[1] [1]

G. E. Andrews, D. Newman,Partitions and the minimal excludant, Ann. Comb.,23(2019), No. 2, 249–254

work page 2019

[2] [2]

G. E. Andrews, D. Newman,The minimal excludant in integer partitions, J. of Int. Seq.,23 (2020), Article 20.2.3

work page 2020

[3] [3]

Ballantine, M

C. Ballantine, M. Merca,Combinatorial proof of the minimal excludant theorem, Int. J. Number Theory,17(2021), No. 8, 1765–1779

work page 2021

[4] [4]

Barman, A

R. Barman, A. Singh,Mex-related partitions and relations to ordinary partition and singular overpartitions, arXiv:2009.11605

work page arXiv 2009

[5] [5]

N. D. Baruah, S. C. Bhoria, P. Eyyunni, B. Maji,A refinement of a result of Andrews and Newman on the sum of minimal excludants, Ramanujan J.,62(2023), 1045–1067

work page 2023

[6] [6]

S. C. Bhoria, P. Eyyunni, P. S. Kaur, B. Maji,Minimal excludant over partitions into distinct parts, Int. J. Number Theory,18(2022), No. 9, 2015–2028

work page 2022

[7] [7]

S. C. Bhoria, P. Eyyunni, R. Li,On the combinatorics ofr-chain minimal and maximal excludants, Discrete Math.,347(2024), Issue 12, 114187

work page 2024

[8] [8]

S. C. Bhoria, P. Eyyunni, B. Maji,A new generalization of the minimal excludant arising from an analogue of Franklin’s Identity, Discrete Math.,346(2023), Issue 5, 113334

work page 2023

[9] [9]

Campioni,Unrefinable partitions into distinct parts and numerical semigroups, arXiv:2601.10227

L. Campioni,Unrefinable partitions into distinct parts and numerical semigroups, arXiv:2601.10227

work page arXiv

[10] [10]

Chakraborty, C

K. Chakraborty, C. Ray,Distribution of generalized Mex-related integer partitions, Hardy- Ramanujan J.,43(2020), 122–128

work page 2020

[11] [11]

Chern,Partitions and the maximal excludant, The Electronic Journal of Combinatorics,28 (2021), Issue 3, P3.13

S. Chern,Partitions and the maximal excludant, The Electronic Journal of Combinatorics,28 (2021), Issue 3, P3.13

work page 2021

[12] [12]

Corteel, J

S. Corteel, J. Lovejoy,Overpartitions, Trans. Amer. Math. Soc.,356(2004), No. 4, 1623–1635. 18 SUBHASH CHAND BHORIA, PRAMOD EYYUNNI, AND SUBHRANGSU SANTRA

work page 2004

[13] [13]

Grabner, A

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

work page 2006

[14] [14]

Hopkins, J

B. Hopkins, J. A. Sellers, D. Stanton,Dyson’s crank and the mex of integer partitions, J. Combin. Theory Ser. A,185(2022), 105523

work page 2022

[15] [15]

P. S. Kaur, M. Rana, P. Eyyunni,The second minimal excludant and mex sequences, Rocky Mountain J. Math.,54(4)(2024), 1117-1130

work page 2024

[16] [16]

B. Kim, E. Kim, J. Lovejoy,Parity bias in partitions, European J. of Combinatorics,89(2020), 103159

work page 2020

[17] [17]

Merca, A

M. Merca, A. J. Yee,On the sum of parts with multiplicity at least2in all the partitions ofn, International J. Number Theory,17(2021), No. 3, 665–681

work page 2021

[18] [18]

da Silva, J

R. da Silva, J. A. Sellers,Parity considerations for mex-related partition functions of Andrews and Newman, J. Integer Seq.,23(2020), No. 5, Article 20.5.7

work page 2020

[19] [19]

N. J. A. Sloane, Online Encyclopedia of Integer Sequences,https://oeis.org/. Subhash Chand Bhoria, Department of Mathematics, Pt. Chiranji Lal Sharma Gov- ernment College, Urban Estate, Sector-14, Karnal, Haryana - 132001, India. Email address:scbhoria89@gmail.com Pramod Eyyunni, Department of Mathematics, Birla Institute of Technology and Science Pilani,...

work page