A240727 Number of partitions p of n such that m(p) <= m(c(p)), where m = maximal multiplicity of parts, and c = conjugate.

1, 1, 2, 3, 4, 6, 9, 13, 17, 24, 31, 45, 57, 77, 98, 129, 166, 219, 271, 350, 439, 556, 689, 879, 1076, 1347, 1648, 2051, 2494, 3079, 3733, 4583, 5529, 6727, 8094, 9814, 11751, 14158, 16909, 20295, 24146, 28856, 34212, 40719, 48164, 57081, 67301, 79534

Offset

1,3

Links

Table of n, a(n) for n=1..48.

Formula

a(n) - A240726(n) = A240728(n) for n >= 1.

a(n) + A240726(n) = A000041(n) for n >= 1.

Example

a(7) counts these 9 partitions:  7, 61, 52, 511, 43, 421, 4111, 331, 322, of which the respective conjugates are 1111111, 211111, 22111, 31111, 2221, 3211, 4111, 322, 331.

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. A240726, A240728, A240729, A000041.

Sequence in context: A097557 A175848 A240079 * A123648 A244393 A286929

Adjacent sequences: A240724 A240725 A240726 * A240728 A240729 A240730

Keyword

nonn,easy

Author

Clark Kimberling, Apr 11 2014

Status

approved