Number of Partitions (with Distinct Parts) Having Largest (Least) Parts from a given Set
Abstract
We study partitions where the largest or smallest part is drawn from a prescribed subset and occurs with fixed multiplicity. Extending Euler's results, we establish new identities for partitions with parts in power-of-2 closures of finite indivisible sets. We further derive explicit asymptotic formulas and congruence relations, yielding new structural insights.
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References
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Archibald, M., Blecher, A., Brennan, C., Knopfmacher, A., and Mansour, T.,
Partitions with fixed differences between largest and smallest parts,
\emph{Quaestiones Mathematicae}, \textbf{37}(2), 149--158, 2014.
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Archibald, M., Blecher, A., Brennan, C., Knopfmacher, A., and Mansour, T.,
Partitions according to multiplicities and part sizes,
\emph{Australasian Journal of Combinatorics}, \textbf{66}(1), 104--119, 2016.
\bibitem{andrews2004}
Andrews, G. E. and Eriksson, K.,
\emph{Integer Partitions},
Cambridge University Press, Cambridge, 2004.
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Partitions with fixed differences between largest and smallest parts,
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Interpolated denumerants and Lambert series,
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Some elementary partition inequalities and their implications,
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\emph{Number Theory in the Spirit of Ramanujan},
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Binner, D. S. and Rattan, A.,
On conjectures concerning the smallest part and missing parts of integer partitions,
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Binner, D. S. and Rattan, A.,
A comparison of integer partitions based on smallest part,
\emph{Electronic Journal of Combinatorics}, \textbf{29}(4), P4.12, 2022.
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Erd\H{o}s, P.,
On an elementary proof of some asymptotic formulas in the theory of partitions,
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Euler, L.,
De partitione numerorum,
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Mahanta, P. J. and Saikia, M. P.,
Extensions of some results of Jovovic and Dhar,
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Nathanson, M. B.,
Partition with parts in a finite set,
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Netto, E.,
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Partition bijections, a survey,
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R\o dseth, \O. J. and Sellers, J. A.,
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Sertöz, S. and Özlük, A. E.,
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Stanley, R. P.,
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Wright, E. M.,
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Archibald, M., Blecher, A., Brennan, C., Knopfmacher, A., and Mansour, T.,
Partitions with fixed differences between largest and smallest parts,
\emph{Quaestiones Mathematicae}, \textbf{37}(2), 149--158, 2014.
\bibitem{archibald2016}
Archibald, M., Blecher, A., Brennan, C., Knopfmacher, A., and Mansour, T.,
Partitions according to multiplicities and part sizes,
\emph{Australasian Journal of Combinatorics}, \textbf{66}(1), 104--119, 2016.
\bibitem{andrews2004}
Andrews, G. E. and Eriksson, K.,
\emph{Integer Partitions},
Cambridge University Press, Cambridge, 2004.
\bibitem{beck}
Beck, M.,
Partitions with fixed differences between largest and smallest parts,
Available at \url{https://matthbeck.github.io/papers/fixeddiffpartslides.pdf}.
\bibitem{bell1943}
Bell, E. T.,
Interpolated denumerants and Lambert series,
\emph{American Journal of Mathematics}, \textbf{65}, 382--386, 1943.
\bibitem{berkovich2017}
Berkovich, A. and Uncu, A. K.,
Some elementary partition inequalities and their implications,
arXiv:1708.01957 [math.CO], 2017.
\bibitem{berndt2006}
Berndt, B. C.,
\emph{Number Theory in the Spirit of Ramanujan},
AMS Press, Providence, RI, 2006.
\bibitem{binner2020}
Binner, D. S. and Rattan, A.,
On conjectures concerning the smallest part and missing parts of integer partitions,
arXiv:2006.15287 [math.CO], 2020.
\bibitem{binner2022}
Binner, D. S. and Rattan, A.,
A comparison of integer partitions based on smallest part,
\emph{Electronic Journal of Combinatorics}, \textbf{29}(4), P4.12, 2022.
\bibitem{euler1753}
Euler, L.,
De partitione numerorum, Introductio in Analysin Infinitorum, Caput XVI (1753),
in: Leonardi Euleri Opera Omnia,
edited by A. Kratzer and F. Rudio,
Teubner, Leipzig, 1911.
\bibitem{david2015}
David Christopher, A. and Christober, D.,
On Asymptotic Formula of the Partition Function $p_A(n)$,
\emph{INTEGERS}, \textbf{15}, A32, 2015.
\bibitem{erdos1942}
Erd\H{o}s, P.,
On an elementary proof of some asymptotic formulas in the theory of partitions,
\emph{Annals of Mathematics (2)}, \textbf{43}, 437--450, 1942.
\bibitem{euler1751}
Euler, L.,
De partitione numerorum,
\emph{Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae}, \textbf{3}, 125--169, 1751.
\bibitem{mahanta2023}
Mahanta, P. J. and Saikia, M. P.,
Extensions of some results of Jovovic and Dhar,
arXiv:2301.09849 [math.CO], 2023.
\bibitem{nathanson2000}
Nathanson, M. B.,
Partition with parts in a finite set,
\emph{Proceedings of the American Mathematical Society}, \textbf{128}, 1269--1273, 2000.
\bibitem{netto1927}
Netto, E.,
\emph{Lehrbuch der Combinatorik},
Teubner, Leipzig, 1927.
\bibitem{pak2006}
Pak, I.,
Partition bijections, a survey,
\emph{Ramanujan Journal}, \textbf{12}, 5--75, 2006.
\bibitem{polya1925}
P\'olya, G. and Szeg\H{o}, G.,
\emph{Aufgaben und Lehrs\"atze aus der Analysis},
Springer-Verlag, Berlin, 1925.
(English translation: Problems and Theorems in Analysis, Springer-Verlag, New York, 1972.)
\bibitem{hardy1918}
Hardy, G. H. and Ramanujan, S.,
Asymptotic formulae in combinatory analysis,
\emph{Proceedings of the London Mathematical Society (2)}, \textbf{17}(1), 75--115, 1918.
\bibitem{rodseth2006}
R\o dseth, \O. J. and Sellers, J. A.,
Partition with parts in a finite set,
\emph{International Journal of Number Theory}, \textbf{2}, 455--?, 2006.
\bibitem{sertoz1991}
Sertöz, S. and Özlük, A. E.,
On the number of representations of an integer by a linear form,
\emph{Istanbul Univ. Fen Fak. Mat. Fiz. Astron. Derg.}, \textbf{50}, 1991.
\bibitem{stanley1997}
Stanley, R. P.,
\emph{Enumerative Combinatorics, Volume 1},
Cambridge University Press, Cambridge, 1997.
\bibitem{wright1961}
Wright, E. M.,
A simple proof of a known result in partitions,
\emph{American Mathematical Monthly}, \textbf{68}, 144--145, 1961.
Published
2026-03-21
Section
Research Articles
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