A combinatorial approach to partitions with parts in the gaps

Dennis Eichhorn

Acta Arithmetica (1998)

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

Abstract

top
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 ) .

How to cite

top

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

top
  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. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.