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

0,4

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..

Links

Table of n, a(n) for n=0..47.

Formula

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

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

Example

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

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 *)

Crossrefs

Cf. A240178, A240180, A000041.

Sequence in context: A237754 A182564 A288523 * A295032 A280017 A080136

Adjacent sequences: A240176 A240177 A240178 * A240180 A240181 A240182

Keyword

nonn,easy

Author

Clark Kimberling, Apr 02 2014

Status

approved