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I was interested in frequency of numbers across all partitions of a particular number N.

say 5 = 5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1

frequencies of 1 to 5 are 11,3,2,1,1 respectively.

is there any formula describing their frequency distribution?

the plot of such a frequency distribution for 50 and 70 are

frequency of numbers across all partitions of 50

frequency of numbers across all partitions of 70

Ramprashanth
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  • Since for parts $m\gt N/2$ this is the number of partitions of $N-m$, there can't be a more closed form for this than for the partition function. – joriki Aug 10 '16 at 18:00
  • could you please elaborate? your explanation was not clear – Ramprashanth Aug 10 '16 at 22:07
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    Perhaps an example will help. You used $N=5$ as an example; then consider $m=3\gt5/2$: The frequency of $3$ is the number of partitions of $N-m=5-3=2$, since the partitions of $5$ with one part $3$ are in one-to-one correspondence with the partitions of $2$. – joriki Aug 10 '16 at 22:10

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