A383683 The number of possible values that can be obtained for the Shannon diversity index across all partitions of n.

1, 1, 2, 3, 5, 7, 11, 15, 21, 29, 39, 52, 68, 89, 117, 150, 192, 244, 309, 387, 485, 603, 749, 922, 1130, 1384, 1680, 2035, 2440, 2922, 3478, 4118, 4867, 5728, 6740, 7879, 9206, 10741, 12502, 14516, 16846, 19533, 22620, 26164, 30252, 34967, 40450, 46786

Offset: 0

Author

Noah A Rosenberg, May 05 2025

Keywords

• nonn

Comments

For a partition P of n into parts (n_1, n_2, ..., n_k), the Shannon diversity index is S(P) = -Sum_{i=1..k} (n_i/n)*log(n_i/n). a(n) is the number of distinct values that S(P) obtains across all possible partitions P of n.

Examples

			For n=0 through 7, each partition of n produces a distinct value of the Shannon diversity index, so that a(n) is equal to the number of partitions, A000041(n).
For n=8, partitions (2,2,2,2) and (4,1,1,1,1) both have the same Shannon diversity index, 2*log(2), so that a(8) = 21, one less than A000041(8).
		

Crossrefs

A000607 provides a lower bound for a(n).

Cf. A000041.