cp's OEIS Frontend

A240308 Number of partitions p of n such that (maximal multiplicity of the parts of p) >= (number of distinct parts of p).

%I A240308 #4 Apr 14 2014 11:09:51
%S A240308 0,1,2,2,4,5,8,10,15,18,28,35,48,63,85,106,141,180,229,294,374,468,
%T A240308 591,741,925,1149,1421,1751,2163,2648,3239,3944,4813,5825,7062,8518,
%U A240308 10286,12340,14835,17739,21223,25287,30155,35787,42522,50296,59556,70243,82902
%N A240308 Number of partitions p of n such that (maximal multiplicity of the parts of p) >= (number of distinct parts of p).
%F A240308 a(n) = A239964(n) + A240308(n) for n >= 0.
%F A240308 a(n) + A240305(n) = A000041(n) for n >= 0.
%e A240308 a(6) counts these 8 partitions:  6, 411, 33, 3111, 222, 2211, 21111, 111111.
%t A240308 z = 60; f[n_] := f[n] = IntegerPartitions[n]; m[p_] := Max[Map[Length, Split[p]]]  (* maximal multiplicity *); d[p_] := d[p] = Length[DeleteDuplicates[p]] (* number of distinct terms *)
%t A240308 t1 = Table[Count[f[n], p_ /; m[p] < d[p]], {n, 0, z}]  (* A240305 *)
%t A240308 t2 = Table[Count[f[n], p_ /; m[p] <= d[p]], {n, 0, z}] (* A240306 *)
%t A240308 t3 = Table[Count[f[n], p_ /; m[p] == d[p]], {n, 0, z}] (* A239964 *)
%t A240308 t4 = Table[Count[f[n], p_ /; m[p] >= d[p]], {n, 0, z}] (* A240308 *)
%t A240308 t5 = Table[Count[f[n], p_ /; m[p] > d[p]], {n, 0, z}]  (* A240309 *)
%Y A240308 Cf. A240305, A240306, A239964, A240309, A000041.
%K A240308 nonn,easy
%O A240308 0,3
%A A240308 _Clark Kimberling_, Apr 05 2014