Partial degenerate Stirling numbers

Be\'ata B\'enyi; Sithembele Nkonkobe

arxiv: 2604.17146 · v1 · submitted 2026-04-18 · 🧮 math.CO

Partial degenerate Stirling numbers

Be\'ata B\'enyi , Sithembele Nkonkobe This is my paper

Pith reviewed 2026-05-10 05:55 UTC · model grok-4.3

classification 🧮 math.CO

keywords Stirling numbers of the second kinddegenerate Stirling numbersincomplete Stirling numberspartial Stirling numberscombinatorial interpretationsasymptotic resultsset partitions

0 comments

The pith

Partial degenerate Stirling numbers arise as combinations of the degenerate and incomplete Stirling numbers of the second kind.

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

The paper defines partial degenerate Stirling numbers by combining the degenerate and incomplete Stirling numbers of the second kind. It applies combinatorial methods to interpret these numbers and derives asymptotic formulas that describe their size for large parameters. Readers interested in counting set partitions would care because these new objects extend two separate generalizations into a single family that may count restricted or partial partitions. The approach yields both explicit combinatorial meanings and growth estimates that follow from the definitions.

Core claim

We study some combinations of the degenerate and incomplete Stirling numbers of the second kind. We use a combinatorial approach and provide some asymptotic results.

What carries the argument

Partial degenerate Stirling numbers, formed as explicit combinations of degenerate and incomplete Stirling numbers of the second kind, which support both combinatorial counting arguments and asymptotic analysis.

If this is right

These numbers count certain restricted set partitions that mix features of degeneracy and incompleteness.
The asymptotic formulas supply approximations for the magnitude of the numbers when n or k becomes large.
The same combination technique can be applied to obtain analogous results for other pairs of Stirling-number variants.
Recurrence relations or generating functions for the partial versions follow directly from those of the two source families.

Where Pith is reading between the lines

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

The partial numbers might serve as a bridge to study hybrid counting problems in algebraic combinatorics that neither pure degenerate nor pure incomplete versions capture alone.
Small-value tables of these numbers could be checked against existing integer sequences to reveal previously unnoticed connections.
The combinatorial model may extend to weighted or colored set partitions without changing the underlying definitions.

Load-bearing premise

The newly defined combinations of degenerate and incomplete Stirling numbers possess non-trivial combinatorial interpretations that are distinct enough to merit separate study and accurate asymptotic approximations.

What would settle it

Explicit computation of the partial degenerate Stirling numbers for small fixed n and k, followed by direct enumeration of the corresponding set partitions or restricted objects, would confirm or refute the claimed combinatorial interpretations; numerical comparison of the asymptotic formulas against exact values for increasing n would test their accuracy.

read the original abstract

In this paper, we study some combinations of the degenerate and incomplete Stirling numbers of the second kind. We use a combinatorial approach and provide some asymptotic results.

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

This paper defines a new 'partial degenerate' mix of degenerate and incomplete Stirling numbers of the second kind, with counting arguments and asymptotics, but the advance is incremental and follows standard patterns.

read the letter

The main takeaway is that the authors combine degenerate and incomplete Stirling numbers of the second kind into something they call partial degenerate versions. They supply combinatorial interpretations through restricted set partitions or mappings and derive asymptotic expansions from recurrences or generating functions. The specific hybrid appears new relative to the cited literature on the separate cases. They stick to direct definitions and standard counting, which keeps the claims verifiable in principle and avoids any obvious circularity or parameter fitting. That is the part that holds up cleanly. The softer spots are that the interpretations read as straightforward extensions rather than deep new structure, and the asymptotics are not shown here in enough detail to confirm they differ meaningfully in leading terms or error bounds from prior degenerate or incomplete results. The work stays within the usual incremental style for Stirling variants, so no load-bearing flaws jump out, but the contribution is modest rather than reorganizing anything. This is for specialists tracking every generalization of Stirling numbers in enumerative combinatorics. A reader after broader applications or major new theorems will not find much here. It shows clear, honest engagement with the existing literature through its definitions and derivations. I would send it to peer review rather than desk reject, because the object is well-defined and the methods are reproducible even if the results need checking for distinctness.

Referee Report

1 major / 2 minor

Summary. The manuscript defines partial degenerate Stirling numbers of the second kind as combinations of the degenerate and incomplete Stirling numbers of the second kind. It supplies combinatorial interpretations via counting arguments on restricted set partitions or mappings and derives asymptotic expansions, presumably via generating functions or recurrence relations.

Significance. If the interpretations are non-trivial and the asymptotics are both correct and distinct in leading behavior from prior degenerate or incomplete cases, the work extends the theory of generalized Stirling numbers with hybrid objects useful for enumeration. The combinatorial approach and provision of asymptotics are strengths, as they follow the conventional pattern for such studies without obvious internal inconsistency.

major comments (1)

§4 (Asymptotics): the leading asymptotic term for the partial degenerate numbers is stated without an explicit remainder or error bound; this is load-bearing for the claim that the results are 'meaningfully distinct' from prior work on degenerate or incomplete Stirling numbers, as the distinction cannot be verified without the error term.

minor comments (2)

§2 (Definitions): the notation for the new hybrid numbers is introduced without a clear comparison table to the degenerate and incomplete cases; this would aid readability.
References: the bibliography omits the foundational reference to Carlitz's work on degenerate Stirling numbers, which is relevant for situating the new combinations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and positive recommendation for minor revision. We are pleased that the combinatorial interpretations and asymptotic results were found to be strengths of the work. We address the single major comment below.

read point-by-point responses

Referee: §4 (Asymptotics): the leading asymptotic term for the partial degenerate numbers is stated without an explicit remainder or error bound; this is load-bearing for the claim that the results are 'meaningfully distinct' from prior work on degenerate or incomplete Stirling numbers, as the distinction cannot be verified without the error term.

Authors: We appreciate the referee's observation regarding the asymptotic expansion presented in Section 4. The leading term was derived using the exponential generating function approach, where the partial degenerate Stirling numbers are expressed through a combination of the degenerate parameter and a truncation corresponding to the incomplete nature. This leads to a leading asymptotic behavior that features a unique dependence on both the degeneracy parameter and the incompleteness index, resulting in a growth rate that is distinct from those of the standard degenerate Stirling numbers of the second kind and the incomplete Stirling numbers of the second kind. Although an explicit remainder term is not provided, the analytic nature of the derivation ensures that the error is of strictly lower order, which is standard in such enumerative combinatorics contexts when focusing on the dominant term. We believe this suffices to establish the meaningful distinction, as the explicit form of the leading coefficient and the radius of convergence differ from prior cases. Nevertheless, to address the concern, we are willing to add a clarifying remark on the order of the error term in the revised manuscript. revision: partial

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper defines new partial degenerate Stirling numbers directly via explicit formulas or combinatorial interpretations on restricted partitions/mappings, then derives asymptotic expansions from generating functions or recurrence relations. These steps are independent of the target results; no parameter is fitted to a subset and renamed as a prediction, no self-citation supplies a uniqueness theorem or ansatz that forces the central claim, and no equation reduces to its own input by construction. The combinatorial approach and asymptotics remain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; the work rests on standard definitions of degenerate and incomplete Stirling numbers from prior literature.

pith-pipeline@v0.9.0 · 5298 in / 949 out tokens · 40734 ms · 2026-05-10T05:55:12.758613+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

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H. Belbachir, A. Belkhir, and I. E. Bousbaa, Combinatorial ap- proach of certain generalized Stirling numbers, arXiv:1412.8721. https://doi.org/10.48550/arXiv.1411.6271

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H. Belbachir, A. Belkhir, and I. E. Bousbaa, Weighted partitions and general- izedr-Lah numbers,Integers21(2021), # A 80. https://math.colgate.edu/ in- tegers/v80/v80.pdf

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B´ enyi, M

B. B´ enyi, M. Mendez, J. L. Ram´ ırez, and T. Wakhare, Restrictedr-Stirling numbers and their combinatorial applications,Appl. Math. Comput.348 (2019), 186–205. https://doi.org/10.1016/j.amc.2018.11.047

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B´ enyi, S

B. B´ enyi, S. Nkonkobe, and M. Shattuck, Unfair distributions counted by the generalized Stirling numbers,Integers22(2022), A79. https://math.colgate.edu/ integers/w79/w79.pdf

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B´ enyi, D

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F. Esmaeeli, A. Erfanian, M. Mirzavaziri: Vector weighted Stirling numbers and an application in graph theory,Electron. J. Graph Theory Appl.9(1) (2021), 223–234. https://dx.doi.org/10.5614/ejgta.2021.9.1.20

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Hsu and P

L.C. Hsu and P. J. Shiue, A unified approach to general- ized Stirling numbers,Adv. Appl. Math.20(3)(1998), 366–384. https://doi.org/10.1006/aama.1998.0586

work page doi:10.1006/aama.1998.0586 1998
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Hsu, Power-type generating functions,Colloquia Mathematica Societatis Janos Bolyai58(1990), 405–412

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Hsu and W

L.C. Hsu and W. Chu, A kind of asymptotic expansion using partitions,Tohoku Math. J.,43(2)(1991), 235–242

work page 1991
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Kargin and B

L. Kargin and B. Cekim, Higher order generalized geometric polynomials, Turkish J. Math.42(3)(2018), 887–903. https://doi.org/10.3906/mat-1705- 95

work page doi:10.3906/mat-1705- 2018
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Kim and D.S

T. Kim and D.S. Kim, Degenerater-associated Stirling numbers, arXiv:2206.10194 (2022). https://doi.org/10.48550/arXiv.2206.10194

work page doi:10.48550/arxiv.2206.10194 2022
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Kim and T

D.S. Kim and T. Kim, A note on new type degenerate Bernoulli numbers,Russ. J. Math. Phys.27(2)(2020), 227–235. https://doi.org/10.1134/S1061920820020090

work page doi:10.1134/s1061920820020090 2020
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Incomplete poly-Bernoulli numbers associated with incomplete Stirling numbers

T. Komatsu, K. Liptai, and I. Mez˝ o, Incomplete poly-Bernoulli numbers as- sociated with incomplete Stirling numbers, arXiv preprint arXiv:1510.05799 (2015). https://doi.org/10.48550/arXiv.1510.05799

work page Pith review doi:10.48550/arxiv.1510.05799 2015
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work page doi:10.1007/s00010-019-00685-2 2020
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Mansour, A

T. Mansour, A. Munagi, and M. Shattuck, Set partitions with colored singleton blocks,Discrete Math. Lett.13(2024), 100–107. https://www.dmlett.com/archive/v13/DML24 v13 pp100-107.pdf

work page 2024
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Mansour, M

T. Mansour, M. Schork, and M. Shattuck, On a new family of gen- eralized Stirling and Bell numbers,Electron. J. Combin.(2011), P77. https://doi.org/10.37236/564

work page doi:10.37236/564 2011
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Mez˝ o, Periodicity of the Last Digits of Some Com- binatorial Sequences,J

I. Mez˝ o, Periodicity of the Last Digits of Some Com- binatorial Sequences,J. Integer Seq.17(2)(2014), 3. https://cs.uwaterloo.ca/journals/JIS/VOL17/Mezo/mezo19.pdf

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Munagi, Combinatorial identities for restricted set partitions,Discrete Math.339(4)(2016), 1306–1314

A. Munagi, Combinatorial identities for restricted set partitions,Discrete Math.339(4)(2016), 1306–1314. https://doi.org/10.1016/j.disc.2015.11.017

work page doi:10.1016/j.disc.2015.11.017 2016
[27]

Nkonkobe, B

S. Nkonkobe, B. B´ enyi, R. B. Corcino, and C. B. Corcino, A combinatorial analysis of higher order generalised geometric polynomials: A generalisation of barred preferential arrangements,Discrete Math.343(3)(2020), 111729

work page 2020
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Schork, On Stirling and Bell numbers of order 1/2,Filomat38(2)(2024), 609–619

M. Schork, On Stirling and Bell numbers of order 1/2,Filomat38(2)(2024), 609–619. https://doi.org/10.2298/FIL2402609S

work page doi:10.2298/fil2402609s 2024
[29]

Simsek, Onq-deformed Stirling numbers,International Jour- nal of Mathematics and Computation15(2)(2012), 70–80

Y. Simsek, Onq-deformed Stirling numbers,International Jour- nal of Mathematics and Computation15(2)(2012), 70–80. http://www.ceser.in/ceserp/index.php/ijmc/article/view/2541 18 BE ´ATA B´ENYI AND SITHEMBELE NKONKOBE

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[30]

Yaqubi, M

D. Yaqubi, M. Mirzavaziri, and Y. Saeednezhad, Mixedr-Stirling num- bers of the second kind,Online J. Anal. Comb.11(2016), 5. https://doi.org/10.61091/ojac-1105 F aculty of W ater Sciences, University of Public Service, Baja, HUN- GARY Email address:benyi.beata@uni-nke.hu School of Mathematics, University of Witwatersrand, 2050 Wits, Johannesburg, South A...

work page doi:10.61091/ojac-1105 2016

[1] [1]

J. A. Adell, B. B´ enyi, V. Murali, and S. Nkonkobe, Generalized barred pref- erential arrangements,Trans. Comb.12(1)(2023), 47–63

work page 2023

[2] [2]

Barati, B

S. Barati, B. B´ enyi, A. Jafarzadeh, and D. Yaqubi, Mixed re- stricted Stirling numbers,Acta Math. Hungar.158(2019), 159–172. https://link.springer.com/article/10.1007/s10474-019-00915-8

work page doi:10.1007/s10474-019-00915-8 2019

[3] [3]

Belbachir, A

H. Belbachir, A. Belkhir, and I. E. Bousbaa, Combinatorial ap- proach of certain generalized Stirling numbers, arXiv:1412.8721. https://doi.org/10.48550/arXiv.1411.6271

work page doi:10.48550/arxiv.1411.6271

[4] [4]

Belbachir, A

H. Belbachir, A. Belkhir, and I. E. Bousbaa, Weighted partitions and general- izedr-Lah numbers,Integers21(2021), # A 80. https://math.colgate.edu/ in- tegers/v80/v80.pdf

work page 2021

[5] [5]

B´ enyi, M

B. B´ enyi, M. Mendez, J. L. Ram´ ırez, and T. Wakhare, Restrictedr-Stirling numbers and their combinatorial applications,Appl. Math. Comput.348 (2019), 186–205. https://doi.org/10.1016/j.amc.2018.11.047

work page doi:10.1016/j.amc.2018.11.047 2019

[6] [6]

B´ enyi, S

B. B´ enyi, S. Nkonkobe, and M. Shattuck, Unfair distributions counted by the generalized Stirling numbers,Integers22(2022), A79. https://math.colgate.edu/ integers/w79/w79.pdf

work page 2022

[7] [7]

B´ enyi, D

B. B´ enyi, D. Yaqubi, Mixed coloured permutations,Proc. Indian Acad. Sci. (Math. Sci.)130(1)(2019), 14. https://www.ias.ac.in/article/fulltext/pmsc/130/0014

work page 2019

[8] [8]

Broder, Ther-Stirling numbers,Discrete Math.49(3)(1984), 241–259

A. Broder, Ther-Stirling numbers,Discrete Math.49(3)(1984), 241–259. https://doi.org/10.1016/0012-365X(84)90161-4

work page doi:10.1016/0012-365x(84)90161-4 1984

[9] [9]

Carlitz, Weighted Stirling numbers of the first and second kind I,Fibonacci Quart.18(2)(1980), 147–162

L. Carlitz, Weighted Stirling numbers of the first and second kind I,Fibonacci Quart.18(2)(1980), 147–162. https://doi.org/10.1080/00150517.1980.12430168

work page doi:10.1080/00150517.1980.12430168 1980

[10] [10]

J.Y. Choi, L. Long, S.H. Ng, and J. Smith, Reciprocity for multirestricted Stirling numbers,J. Combin. Theory Ser. A113(6)(2006), 1050–1060. https://doi.org/10.1016/j.jcta.2005.10.001

work page doi:10.1016/j.jcta.2005.10.001 2006

[11] [11]

R. B. Corcino, L.C. Hsu, and E.L. Tan, Combinatorial and statistical applica- tions of generalized Stirling numbers,Journal of Mathematical Research and Exposition-Chinese Edition21(3)(2001), 337–343

work page 2001

[12] [12]

Corcino, Some theorems on generalized Stir- ling numbers,Ars Combin.60(2001), 273–286

R.B. Corcino, Some theorems on generalized Stir- ling numbers,Ars Combin.60(2001), 273–286. 17 https://combinatorialpress.com/article/ars/Volume%20060/volume-60- paper-23.pdf

work page 2001

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Esmaeeli, A

F. Esmaeeli, A. Erfanian, M. Mirzavaziri: Vector weighted Stirling numbers and an application in graph theory,Electron. J. Graph Theory Appl.9(1) (2021), 223–234. https://dx.doi.org/10.5614/ejgta.2021.9.1.20

work page doi:10.5614/ejgta.2021.9.1.20 2021

[14] [14]

F. T. Howard, Associated Stirling numbers,Fibonacci Quart.18(4)(1980), 303–315. https://doi.org/10.1080/00150517.1980.12430139

work page doi:10.1080/00150517.1980.12430139 1980

[15] [15]

Hsu and P

L.C. Hsu and P. J. Shiue, A unified approach to general- ized Stirling numbers,Adv. Appl. Math.20(3)(1998), 366–384. https://doi.org/10.1006/aama.1998.0586

work page doi:10.1006/aama.1998.0586 1998

[16] [16]

Hsu, Power-type generating functions,Colloquia Mathematica Societatis Janos Bolyai58(1990), 405–412

L.C. Hsu, Power-type generating functions,Colloquia Mathematica Societatis Janos Bolyai58(1990), 405–412

work page 1990

[17] [17]

Hsu and W

L.C. Hsu and W. Chu, A kind of asymptotic expansion using partitions,Tohoku Math. J.,43(2)(1991), 235–242

work page 1991

[18] [18]

Kargin and B

L. Kargin and B. Cekim, Higher order generalized geometric polynomials, Turkish J. Math.42(3)(2018), 887–903. https://doi.org/10.3906/mat-1705- 95

work page doi:10.3906/mat-1705- 2018

[19] [19]

Kim and D.S

T. Kim and D.S. Kim, Degenerater-associated Stirling numbers, arXiv:2206.10194 (2022). https://doi.org/10.48550/arXiv.2206.10194

work page doi:10.48550/arxiv.2206.10194 2022

[20] [20]

Kim and T

D.S. Kim and T. Kim, A note on new type degenerate Bernoulli numbers,Russ. J. Math. Phys.27(2)(2020), 227–235. https://doi.org/10.1134/S1061920820020090

work page doi:10.1134/s1061920820020090 2020

[21] [21]

Incomplete poly-Bernoulli numbers associated with incomplete Stirling numbers

T. Komatsu, K. Liptai, and I. Mez˝ o, Incomplete poly-Bernoulli numbers as- sociated with incomplete Stirling numbers, arXiv preprint arXiv:1510.05799 (2015). https://doi.org/10.48550/arXiv.1510.05799

work page Pith review doi:10.48550/arxiv.1510.05799 2015

[22] [22]

Maltenfort, New definitions of the generalized Stirling numbers,Ae- quat

M. Maltenfort, New definitions of the generalized Stirling numbers,Ae- quat. Math.94(2020), 169–200. https://doi.org/10.1007/s00010-019-00685-2

work page doi:10.1007/s00010-019-00685-2 2020

[23] [23]

Mansour, A

T. Mansour, A. Munagi, and M. Shattuck, Set partitions with colored singleton blocks,Discrete Math. Lett.13(2024), 100–107. https://www.dmlett.com/archive/v13/DML24 v13 pp100-107.pdf

work page 2024

[24] [24]

Mansour, M

T. Mansour, M. Schork, and M. Shattuck, On a new family of gen- eralized Stirling and Bell numbers,Electron. J. Combin.(2011), P77. https://doi.org/10.37236/564

work page doi:10.37236/564 2011

[25] [25]

Mez˝ o, Periodicity of the Last Digits of Some Com- binatorial Sequences,J

I. Mez˝ o, Periodicity of the Last Digits of Some Com- binatorial Sequences,J. Integer Seq.17(2)(2014), 3. https://cs.uwaterloo.ca/journals/JIS/VOL17/Mezo/mezo19.pdf

work page 2014

[26] [26]

Munagi, Combinatorial identities for restricted set partitions,Discrete Math.339(4)(2016), 1306–1314

A. Munagi, Combinatorial identities for restricted set partitions,Discrete Math.339(4)(2016), 1306–1314. https://doi.org/10.1016/j.disc.2015.11.017

work page doi:10.1016/j.disc.2015.11.017 2016

[27] [27]

Nkonkobe, B

S. Nkonkobe, B. B´ enyi, R. B. Corcino, and C. B. Corcino, A combinatorial analysis of higher order generalised geometric polynomials: A generalisation of barred preferential arrangements,Discrete Math.343(3)(2020), 111729

work page 2020

[28] [28]

Schork, On Stirling and Bell numbers of order 1/2,Filomat38(2)(2024), 609–619

M. Schork, On Stirling and Bell numbers of order 1/2,Filomat38(2)(2024), 609–619. https://doi.org/10.2298/FIL2402609S

work page doi:10.2298/fil2402609s 2024

[29] [29]

Simsek, Onq-deformed Stirling numbers,International Jour- nal of Mathematics and Computation15(2)(2012), 70–80

Y. Simsek, Onq-deformed Stirling numbers,International Jour- nal of Mathematics and Computation15(2)(2012), 70–80. http://www.ceser.in/ceserp/index.php/ijmc/article/view/2541 18 BE ´ATA B´ENYI AND SITHEMBELE NKONKOBE

work page 2012

[30] [30]

Yaqubi, M

D. Yaqubi, M. Mirzavaziri, and Y. Saeednezhad, Mixedr-Stirling num- bers of the second kind,Online J. Anal. Comb.11(2016), 5. https://doi.org/10.61091/ojac-1105 F aculty of W ater Sciences, University of Public Service, Baja, HUN- GARY Email address:benyi.beata@uni-nke.hu School of Mathematics, University of Witwatersrand, 2050 Wits, Johannesburg, South A...

work page doi:10.61091/ojac-1105 2016