The Fractal Nature of Partitions

n

10

Rewrite the following partition of 21 as a dot product of frequencies and parts:

1+1+1+2+4+4+8=3×1+1×2+2×4+1×8=(3,1,0,2,0,0,0,1,0,0,0,…,0)·(1,2,3,4,…,21)

. The sum of the frequencies is 7.

Partitions are listed in reverse lexicographic order. For example, the five partitions of 4 are listed as

4,3+1,2+2,2+1+1,1+1+1+1

.

For each

n

, this Demonstration plots (from the bottom to the top) the means, medians, and sums of the nonzero frequencies of the parts of the partitions of

n

.