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Number of partitions of n such that (least part) <= (multiplicity of greatest part).
6

%I #7 Apr 12 2014 16:21:22

%S 0,1,1,2,4,5,8,11,17,23,33,43,61,79,108,140,187,238,314,397,513,648,

%T 826,1032,1307,1622,2029,2508,3113,3821,4713,5754,7048,8569,10431,

%U 12618,15290,18413,22193,26628,31954,38184,45639,54340,64694,76780,91077,107732

%N Number of partitions of n such that (least part) <= (multiplicity of greatest part).

%C Also, for n >= 0, a(n) is the number of partitions of n such that (greatest part) >= (multiplicity of least part). For n >=1, a(n) is also the number of partitions of n such that (least part) >= (multiplicity of greatest part), as well as the number of partitions p of n such that min(p) = min(c(p)), where c = conjugate..

%F a(n) = A240178(n) + A240180(n), for n >= 0.

%F 2*a(n) + A240180(n) = A000041(n) for n >= 0.

%e a(6) counts these 8 partitions: 51, 411, 321, 3111, 222, 2211, 21111, 111111.

%t z = 60; f[n_] := f[n] = IntegerPartitions[n];

%t Table[Count[f[n], p_ /; Min[p] < Count[p, Max[p]]], {n, 0, z}] (* A240178 *)

%t Table[Count[f[n], p_ /; Min[p] <= Count[p, Max[p]]], {n, 0, z}] (* A240179 *)

%t Table[Count[f[n], p_ /; Min[p] == Count[p, Max[p]]], {n, 0, z}] (* A240180 *)

%t Table[Count[f[n], p_ /; Min[p] > Count[p, Max[p]]], {n, 0, z}] (* A240178, n>0 *)

%t Table[Count[f[n], p_ /; Min[p] >= Count[p, Max[p]]], {n, 0, z}] (* A240179, n>0 *)

%Y Cf. A240178, A240180, A000041.

%K nonn,easy

%O 0,4

%A _Clark Kimberling_, Apr 02 2014