%I #4 Apr 22 2014 01:26:27

%S 1,1,2,3,4,6,9,13,17,24,31,45,57,77,98,129,166,219,271,350,439,556,

%T 689,879,1076,1347,1648,2051,2494,3079,3733,4583,5529,6727,8094,9814,

%U 11751,14158,16909,20295,24146,28856,34212,40719,48164,57081,67301,79534

%N Number of partitions p of n such that m(p) <= m(c(p)), where m = maximal multiplicity of parts, and c = conjugate.

%F a(n) - A240726(n) = A240728(n) for n >= 1.

%F a(n) + A240726(n) = A000041(n) for n >= 1.

%e a(7) counts these 9 partitions: 7, 61, 52, 511, 43, 421, 4111, 331, 322, of which the respective conjugates are 1111111, 211111, 22111, 31111, 2221, 3211, 4111, 322, 331.

%t z = 30; f[n_] := f[n] = IntegerPartitions[n]; c[p_] := Table[Count[#, _?(# >= i &)], {i, First[#]}] &[p]; m[p_] := Max[Map[Length, Split[p]]];

%t Table[Count[f[n], p_ /; m[p] < m[c[p]]], {n, 1, z}] (* A240726 *)

%t Table[Count[f[n], p_ /; m[p] <= m[c[p]]], {n, 1, z}] (* A240727 *)

%t Table[Count[f[n], p_ /; m[p] == m[c[p]]], {n, 1, z}] (* A240728 *)

%Y Cf. A240726, A240728, A240729, A000041.

%K nonn,easy

%O 1,3

%A _Clark Kimberling_, Apr 11 2014