# 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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topDennis 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

top- [1] K. Alladi, Partition identities involving gaps and weights, Trans. Amer. Math. Soc. 349 (1997), 5001-5019. Zbl0893.11042
- [2] K. Alladi, Partition identities involving gaps and weights - II, Ramanujan J., to appear. Zbl0907.11037
- [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] D. Bowman, Partitions with numbers in their gaps, Acta Arith. 74 (1996), 97-105. Zbl0861.11058
- [5] L. Euler, Introductio in Analysis Infinitorum, Marcum-Michaelem Bousquet, Lousannae, 1748.

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