# A combinatorial approach to partitions with parts in the gaps

Acta Arithmetica (1998)

• Volume: 85, Issue: 2, page 119-133
• ISSN: 0065-1036

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## Abstract

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Many links exist between ordinary partitions and partitions with parts in the “gaps”. In this paper, we explore combinatorial explanations for some of these links, along with some natural generalizations. In particular, if we let ${p}_{k,m}\left(j,n\right)$ be the number of partitions of n into j parts where each part is ≡ k (mod m), 1 ≤ k ≤ m, and we let $p{*}_{k,m}\left(j,n\right)$ be the number of partitions of n into j parts where each part is ≡ k (mod m) with parts of size k in the gaps, then $p{*}_{k,m}\left(j,n\right)={p}_{k,m}\left(j,n\right)$.

## How to cite

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Dennis Eichhorn. "A combinatorial approach to partitions with parts in the gaps." Acta Arithmetica 85.2 (1998): 119-133. <http://eudml.org/doc/207157>.

@article{DennisEichhorn1998,
abstract = {Many links exist between ordinary partitions and partitions with parts in the “gaps”. In this paper, we explore combinatorial explanations for some of these links, along with some natural generalizations. In particular, if we let $p^_\{k,m\}(j,n)$ be the number of partitions of n into j parts where each part is ≡ k (mod m), 1 ≤ k ≤ m, and we let $p*_\{k,m\}(j,n)$ be the number of partitions of n into j parts where each part is ≡ k (mod m) with parts of size k in the gaps, then $p*_\{k,m\}(j,n)=p_\{k,m\}(j,n)$.},
author = {Dennis Eichhorn},
journal = {Acta Arithmetica},
keywords = {partitions; compositions},
language = {eng},
number = {2},
pages = {119-133},
title = {A combinatorial approach to partitions with parts in the gaps},
url = {http://eudml.org/doc/207157},
volume = {85},
year = {1998},
}

TY - JOUR
AU - Dennis Eichhorn
TI - A combinatorial approach to partitions with parts in the gaps
JO - Acta Arithmetica
PY - 1998
VL - 85
IS - 2
SP - 119
EP - 133
AB - Many links exist between ordinary partitions and partitions with parts in the “gaps”. In this paper, we explore combinatorial explanations for some of these links, along with some natural generalizations. In particular, if we let $p^_{k,m}(j,n)$ be the number of partitions of n into j parts where each part is ≡ k (mod m), 1 ≤ k ≤ m, and we let $p*_{k,m}(j,n)$ be the number of partitions of n into j parts where each part is ≡ k (mod m) with parts of size k in the gaps, then $p*_{k,m}(j,n)=p_{k,m}(j,n)$.
LA - eng
KW - partitions; compositions
UR - http://eudml.org/doc/207157
ER -

## References

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1. [1] K. Alladi, Partition identities involving gaps and weights, Trans. Amer. Math. Soc. 349 (1997), 5001-5019. Zbl0893.11042
2. [2] K. Alladi, Partition identities involving gaps and weights - II, Ramanujan J., to appear. Zbl0907.11037
3. [3] K. Alladi, Weighted partition identities and applications, in: Analytic Number Theory, Proceedings of a Conference in Honor of Heini Halberstam, Vol. 1 (Allerton Park, Ill., 1995), Progr. Math. 138, Birkhäuser, Boston, 1996, 1-15.
4. [4] D. Bowman, Partitions with numbers in their gaps, Acta Arith. 74 (1996), 97-105. Zbl0861.11058
5. [5] L. Euler, Introductio in Analysis Infinitorum, Marcum-Michaelem Bousquet, Lousannae, 1748.

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