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%I #3 Mar 31 2012 13:21:30
%S 1,2,3,5,7,10,14,19,26,33,42,56,75,94,118,145,181,230,286,356,428,522,
%T 633,774,915,1125,1341,1621,1935,2351
%N Maximum number of subpartitions for any partition of n.
%C 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,m-1,...,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 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].
%e 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.
%Y Cf. A115728, A115729, A000041, A000108.
%K more,nonn
%O 0,2
%A _Franklin T. Adams-Watters_, Mar 19 2006
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