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Élie Mosaki
Partitions sans petites parts (II)
Journal de théorie des nombres de Bordeaux, 20 no. 2 (2008), p. 431-464, doi: 10.5802/jtnb.636
Article PDF | Reviews MR 2477513 | Zbl pre05543171

Résumé - Abstract

Let $r(n,m)$ denote the number of partitions of $n$ into parts, each of which is at least $m$, and $R(n,m)=r(n-m,m)$ the number of partitions of $n$ with smallest part $m$. In a precedent paper (see [9]) the asymptotics for $r(n,m)$ is obtained uniformly for $1\le m=O(\sqrt{n})$; we complete this asymptotics uniformly for $1\le m=(n\log ^{-3}n)$. To prolong the results until $m\le n$, we give an estimate for $r(n,m)$ which holds for $n^{2/3}\le m\le n$, by use of the relation $r(n,m)=\sum _{t=1}^{\lfloor n/m\rfloor }P(n-(m-1)t,t)$, $P(i,t)$ denoting the number of partitions of $i$ into exactly $t$ parts. We also give an elementary combinatorial proof for the decrease of $R(n,m)$ in terms of $m$, $m\le n-1$.

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