A240178 Number of partitions of n such that (least part) < (multiplicity of greatest part).

0, 0, 1, 1, 1, 2, 3, 4, 5, 7, 9, 13, 16, 22, 27, 36, 44, 59, 71, 93, 114, 144, 176, 223, 268, 336, 407, 502, 605, 744, 891, 1088, 1301, 1574, 1879, 2265, 2687, 3224, 3822, 4557, 5384, 6399, 7535, 8921, 10481, 12354, 14481, 17022, 19888, 23307, 27178, 31745

Offset

0,6

Comments

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..51.

Formula

a(n) = A240179(n) - A240180(n), for n >= 0.

Example

a(6) counts these 3 partitions:  222, 2211, 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. A240179, A240180.

Sequence in context: A250553 A026450 A280451 * A039863 A036802 A333265

Adjacent sequences: A240175 A240176 A240177 * A240179 A240180 A240181

Keyword

nonn,easy

Author

Clark Kimberling, Apr 02 2014

Status

approved