A240731 Number of partitions p of n such that m(p) = m(c(p)), where m = minimal multiplicity of parts, and c = conjugate.

1, 0, 1, 3, 5, 5, 11, 12, 22, 26, 42, 51, 79, 97, 138, 179, 241, 297, 410, 505, 666, 824, 1073, 1319, 1704, 2074, 2634, 3222, 4049, 4904, 6128, 7401, 9149, 11028, 13535, 16237, 19825, 23681, 28727, 34264, 41315, 49058, 58935, 69793, 83402, 98512, 117248

Offset

1,4

Links

Table of n, a(n) for n=1..47.

Formula

a(n) + 2*A240729(n) = A000041(n) for n >= 1.

a(n) + A240729(n) = A240730(n) for n >= 1.

Example

a(7) counts these 11 partitions:  61, 511, 43, 421, 4111, 331, 322, 3211, 31111, 2221, 211111, of which the respective conjugates are 5, 31111, 2221, 3211, 4111, 322, 331, 421, 511, 43, 61.

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. A240727, A240729, A240730, A000041.

Sequence in context: A098971 A093572 A317650 * A266505 A118132 A089167

Adjacent sequences: A240728 A240729 A240730 * A240732 A240733 A240734

Keyword

nonn,easy

Author

Clark Kimberling, Apr 11 2014

Status

approved