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For some positive integer $r$, by an $r$-vector I will mean an $r$-tuple $(a_1,a_2,\dots,a_r)$ with $a_1,\dots,a_r$ nonnegative integers not all zero, and I will call it odd if $a_1,\dots,a_r$ are all odd. An odd partition of an $r$-vector $(a_1,a_2,\dots,a_r)$ is a multiset of odd $r$-vectors whose sum is equal to $(a_1,a_2,\dots,a_r)$. Note that when $r=1$, this is just the partition of a number into odd numbers.

Let $Q_r(a_1,\dots,a_r)$ be the number of odd partitions of the $r$-vector $(a_1,a_2,\dots,a_r)$. What is known about the sequence $Q_r(n,n,\dots,n)$, ($n=1,2,\dots$)? Especially is there any estimation or asymptotic for it? Of course the first case I'm interested in is $r=2$ since it is the classical theory of partitions for $r=1$.

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  • $\begingroup$ You depends on the asymptotical regime. For fixed $r$ and large $n$ standard techniques allows to get reasonable asymptotics. $\endgroup$
    – Fedor Petrov
    Commented Nov 22, 2015 at 16:05
  • $\begingroup$ Is it set or multiset, i.e., partition into odd or odd and distinct numbers? $\endgroup$
    – Max Alekseyev
    Commented Nov 23, 2015 at 15:42
  • $\begingroup$ @MaxAlekseyev It is a multiset, which means that the numbers are not necessarily distinct. $\endgroup$
    – Binzhou Xia
    Commented Nov 24, 2015 at 2:58
  • $\begingroup$ @BinzhouXia: I've corrected this in your question. $\endgroup$
    – Max Alekseyev
    Commented Nov 24, 2015 at 3:45

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