A240728 - OEIS

%I #4 Apr 22 2014 01:26:36

%S 1,0,1,1,1,1,3,4,4,6,6,13,13,19,20,27,35,53,52,73,86,110,123,183,194,

%T 258,286,384,423,554,624,817,915,1144,1305,1651,1865,2301,2633,3252,

%U 3709,4538,5163,6263,7194,8604,9848,11795,13507,16086

%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) = A240727(n) for n >= 1.

%F a(n) + 2*A240726(n) = A000041(n) for n >= 1.

%e a(7) counts these 3 partitions:  4111, 331, 322, of which the respective conjugates are 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, A240727, A240729, A000041.

%K nonn,easy

%O 1,7

%A _Clark Kimberling_, Apr 11 2014