A240726 - OEIS

%I #5 Apr 22 2014 01:26:19

%S 0,1,1,2,3,5,6,9,13,18,25,32,44,58,78,102,131,166,219,277,353,446,566,

%T 696,882,1089,1362,1667,2071,2525,3109,3766,4614,5583,6789,8163,9886,

%U 11857,14276,17043,20437,24318,29049,34456,40970,48477,57453,67739,80009

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

%C Also, clearly, a(n) = number of partitions p of n such that m(p) > m(c(p)).

%F a(n) + A240728(n) = A240727(n) for n >= 1.

%F 2*a(n) + A240728(n) = A000041(n) for n >= 1.

%e a(7) counts these 6 partitions:  7, 61, 52, 511, 43, 421, of which the respective conjugates are 1111111, 211111, 22111, 31111, 2221, 321.

%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. A240727, A240728, A240729, A000041.

%K nonn,easy

%O 1,4

%A _Clark Kimberling_, Apr 11 2014