A240726 Number of partitions p of n such that m(p) < m(c(p)), where m = maximal multiplicity of parts, and c = conjugate.

0, 1, 1, 2, 3, 5, 6, 9, 13, 18, 25, 32, 44, 58, 78, 102, 131, 166, 219, 277, 353, 446, 566, 696, 882, 1089, 1362, 1667, 2071, 2525, 3109, 3766, 4614, 5583, 6789, 8163, 9886, 11857, 14276, 17043, 20437, 24318, 29049, 34456, 40970, 48477, 57453, 67739, 80009

Offset

1,4

Comments

Also, clearly, a(n) = number of partitions p of n such that m(p) > m(c(p)).

Links

Table of n, a(n) for n=1..49.

Formula

a(n) + A240728(n) = A240727(n) for n >= 1.

2*a(n) + A240728(n) = A000041(n) for n >= 1.

Example

a(7) counts these 6 partitions:  7, 61, 52, 511, 43, 421, of which the respective conjugates are 1111111, 211111, 22111, 31111, 2221, 321.

Mathematica

z = 30; f[n_] := f[n] = IntegerPartitions[n]; c[p_] := Table[Count[#, _?(# >= i &)], {i, First[#]}] &[p];  m[p_] := Max[Map[Length, Split[p]]];
Table[Count[f[n], p_ /; m[p] < m[c[p]]], {n, 1, z}] (* A240726 *)
Table[Count[f[n], p_ /; m[p] <= m[c[p]]], {n, 1, z}] (* A240727 *)
Table[Count[f[n], p_ /; m[p] == m[c[p]]], {n, 1, z}] (* A240728 *)

Crossrefs

Cf. A240727, A240728, A240729, A000041.

Sequence in context: A166048 A240306 A094873 * A240203 A018126 A087900

Adjacent sequences: A240723 A240724 A240725 * A240727 A240728 A240729

Keyword

nonn,easy

Author

Clark Kimberling, Apr 11 2014

Status

approved