A240729 Number of partitions p of n such that m(p) < m(c(p)), where m = minimal multiplicity of parts, and c = conjugate.

0, 1, 1, 1, 1, 3, 2, 5, 4, 8, 7, 13, 11, 19, 19, 26, 28, 44, 40, 61, 63, 89, 91, 128, 127, 181, 188, 248, 258, 350, 357, 474, 497, 641, 674, 870, 906, 1167, 1229, 1537, 1634, 2058, 2163, 2691, 2866, 3523, 3753, 4603, 4883, 5969, 6372, 7676, 8226

Offset

1,6

Links

Table of n, a(n) for n=1..53.

Formula

a(n) + A240731(n) = A240730(n) for n >= 1.

2*a(n) + 2*A240730(n) = A000041(n) for n >= 1.

Example

a(7) counts these 2 partitions:  7, 52, of which the respective conjugates are 1111111, 22111.

Mathematica

z = 30; f[n_] := f[n] = IntegerPartitions[n]; c[p_] := Table[Count[#, _?(# >= i &)], {i, First[#]}] &[p];  m[p_] := Min[Map[Length, Split[p]]];
Table[Count[f[n], p_ /; m[p] < m[c[p]]], {n, 1, z}]  (* A240729 *)
Table[Count[f[n], p_ /; m[p] <= m[c[p]]], {n, 1, z}] (* A240730 *)
Table[Count[f[n], p_ /; m[p] == m[c[p]]], {n, 1, z}] (* A240731 *)

Crossrefs

Cf. A240726, A240730, A240731, A000041.

Sequence in context: A348542 A318460 A276582 * A339371 A258259 A240182

Adjacent sequences: A240726 A240727 A240728 * A240730 A240731 A240732

Keyword

nonn,easy

Author

Clark Kimberling, Apr 11 2014

Status

approved