A116480 Maximum number of subpartitions for any partition of n.

1, 2, 3, 5, 7, 10, 14, 19, 26, 33, 42, 56, 75, 94, 118, 145, 181, 230, 286, 356, 428, 522, 633, 774, 915, 1125, 1341, 1621, 1935, 2351

Offset: 0

Author

Franklin T. Adams-Watters, Mar 19 2006

Keywords

• more
• nonn

Comments

The sequence grows roughly as an exponential in the square root of n. a(n) <= 1 + Sum_{0<=kA000108) subpartitions; m ~ sqrt(2n) and the Catalan numbers grow exponentially. Through n=30, there is either a unique partition with the maximum number of subpartitions, or a unique pair of conjugate partitions, except for n=10, where there is a 3-way between [5,3,1^2] and its conjugate [4,2^2,1^2] and two self-conjugate partitions: [4,3,2,1] and [5,2,1^3]. It is not clear what the limiting shape of the maximum partition is. The minimum number of subpartitions is n+1, for the conjugate partitions [n] and [1^n].

Examples

			The 5 partitions of 4 are [4], [3,1], [2^2], [2,1^2], [1^4]; these have respectively 5,7,6,7 and 5 subpartitions, so a(4) = 7, the largest of these.
		

Crossrefs

Cf. A115728, A115729, A000041, A000108.