

A116480


Maximum number of subpartitions for any partition of n.


1



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
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OFFSET

0,2


COMMENTS

The sequence grows roughly as an exponential in the square root of n. a(n) <= 1 + Sum_{0<=k<n} p(k) is a trivial upper bound and that grows as specified. A lower bound comes from [m,m1,...,1], which has C_{m+1} (Catalan numbers A000108) 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 3way between [5,3,1^2] and its conjugate [4,2^2,1^2] and two selfconjugate 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].


LINKS



EXAMPLE

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



KEYWORD

more,nonn


AUTHOR



STATUS

approved



