We show that the number of partitions of $n$ with alternating sum $k$ such that the multiplicity of each part is bounded by $2m+1$ equals the number of partitions of $n$ with $k$ odd parts such that the multiplicity of each even part is bounded by $m$. The first proof relies on two formulas with two parameters that are related to the four-parameter formulas of Boulet. We also give a combinatorial proof of this result by using Sylvester's bijection, which implies a stronger partition theorem. For $m=0$, our result reduces to Bessenrodt's refinement of Euler's partition theorem. If the alternating sum and the number of odd parts are not taken into account, we are led to a generalization of Euler's partition theorem, which can be deduced from a theorem of Andrews on equivalent upper bound sequences of multiplicities. Analogously, we show that the number of partitions of $n$ with alternating sum $k$ such that the multiplicity of each even part is bounded by $2m+1$ equals the number of partitions of $n$ with $k$ odd parts such that the multiplicity of each even part is also bounded by $2m+1$. We provide a combinatorial proof as well.
William Y.C. Chen
1
;
Ae Ja Yee
2
;
Albert J. W. Zhu
3
1
Center for Combinatorics, Nankai University
2
Department of Mathematics, The Pennsylvania State University
3
China Ship Development and Design Center
@article{10_37236_2318,
author = {William Y.C. Chen and Ae Ja Yee and Albert J. W. Zhu},
title = {Euler's partition theorem with upper bounds on multiplicities},
journal = {The electronic journal of combinatorics},
year = {2012},
volume = {19},
number = {3},
doi = {10.37236/2318},
zbl = {1253.05027},
url = {http://geodesic.mathdoc.fr/articles/10.37236/2318/}
}
TY - JOUR
AU - William Y.C. Chen
AU - Ae Ja Yee
AU - Albert J. W. Zhu
TI - Euler's partition theorem with upper bounds on multiplicities
JO - The electronic journal of combinatorics
PY - 2012
VL - 19
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/2318/
DO - 10.37236/2318
ID - 10_37236_2318
ER -
%0 Journal Article
%A William Y.C. Chen
%A Ae Ja Yee
%A Albert J. W. Zhu
%T Euler's partition theorem with upper bounds on multiplicities
%J The electronic journal of combinatorics
%D 2012
%V 19
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/2318/
%R 10.37236/2318
%F 10_37236_2318
William Y.C. Chen; Ae Ja Yee; Albert J. W. Zhu. Euler's partition theorem with upper bounds on multiplicities. The electronic journal of combinatorics, Tome 19 (2012) no. 3. doi: 10.37236/2318
