A240179 Number of partitions of n such that (least part) <= (multiplicity of greatest part).

0, 1, 1, 2, 4, 5, 8, 11, 17, 23, 33, 43, 61, 79, 108, 140, 187, 238, 314, 397, 513, 648, 826, 1032, 1307, 1622, 2029, 2508, 3113, 3821, 4713, 5754, 7048, 8569, 10431, 12618, 15290, 18413, 22193, 26628, 31954, 38184, 45639, 54340, 64694, 76780, 91077, 107732

Offset: 0

Author

Clark Kimberling, Apr 02 2014

Keywords

• nonn
• easy

Comments

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..

Examples

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

Crossrefs

Cf. A240178, A240180, A000041.

Programs

Mathematica

z = 60; f[n_] := f[n] = IntegerPartitions[n];
Table[Count[f[n], p_ /; Min[p] < Count[p, Max[p]]], {n, 0, z}]  (* A240178 *)
Table[Count[f[n], p_ /; Min[p] <= Count[p, Max[p]]], {n, 0, z}] (* A240179 *)
Table[Count[f[n], p_ /; Min[p] == Count[p, Max[p]]], {n, 0, z}] (* A240180 *)
Table[Count[f[n], p_ /; Min[p] > Count[p, Max[p]]], {n, 0, z}]  (* A240178, n>0 *)
Table[Count[f[n], p_ /; Min[p] >= Count[p, Max[p]]], {n, 0, z}] (* A240179, n>0 *)

Formula

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

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