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, 116, 149, 189, 240, 298, 373, 455, 562, 690, 837, 1014, 1227, 1480, 1772, 2110, 2516, 2980, 3522, 4147, 4879, 5729, 6688, 7797, 9082, 10546, 12225, 14114, 16303, 18771, 21585, 24760, 28355, 32456, 37042, 42230, 48091, 54612

Offset

0,3

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.

Links

Table of n, a(n) for n=0..50.

Bradley K. Moon and Noah A. Rosenberg, Integer Sequences for Diversity Statistics, J. Int. Seq. 29(1) (2026), 26.1.5. See pp. 1, 6.

Example

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(n) provides a lower bound for a(n).

Cf. A000041.

Sequence in context: A332745 A042953 A023028 * A246579 A232480 A332638

Adjacent sequences: A383680 A383681 A383682 * A383684 A383685 A383686

Keyword

nonn

Author

Noah A Rosenberg, May 05 2025

Extensions

Data corrected by Noah A Rosenberg, Dec 02 2025

Status

approved