cp's OEIS Frontend

A240306 Number of partitions p of n such that (maximal multiplicity of the parts of p) <= (number of distinct parts of p).

%I A240306 #5 Apr 14 2014 11:09:23
%S A240306 1,1,1,2,3,5,6,9,13,17,25,33,44,59,76,100,131,169,212,278,352,442,555,
%T A240306 703,871,1088,1342,1664,2046,2517,3064,3758,4574,5548,6718,8119,9797,
%U A240306 11784,14150,16935,20263,24179,28798,34237,40677,48122,57008,67291,79400
%N A240306 Number of partitions p of n such that (maximal multiplicity of the parts of p) <= (number of distinct parts of p).
%F A240306 a(n) = A240305(n) + A239964(n) for n >= 0.
%F A240306 a(n) + A240308(n) = A000041(n) for n >= 0.
%e A240306 a(6) counts these 6 partitions:  6, 51, 42, 411, 321, 2211.
%t A240306 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 A240306 t1 = Table[Count[f[n], p_ /; m[p] < d[p]], {n, 0, z}]  (* A240305 *)
%t A240306 t2 = Table[Count[f[n], p_ /; m[p] <= d[p]], {n, 0, z}] (* A240306 *)
%t A240306 t3 = Table[Count[f[n], p_ /; m[p] == d[p]], {n, 0, z}] (* A239964 *)
%t A240306 t4 = Table[Count[f[n], p_ /; m[p] >= d[p]], {n, 0, z}] (* A240308 *)
%t A240306 t5 = Table[Count[f[n], p_ /; m[p] > d[p]], {n, 0, z}]  (* A240309 *)
%Y A240306 Cf. A240305, A239964, A240308, A240309, A000041.
%K A240306 nonn,easy
%O A240306 0,4
%A A240306 _Clark Kimberling_, Apr 05 2014