A239951 Number of partitions of n such that (number of distinct parts) > least part.

0, 0, 0, 1, 2, 4, 6, 10, 14, 22, 30, 44, 59, 84, 109, 151, 195, 261, 335, 440, 558, 723, 909, 1160, 1452, 1829, 2272, 2839, 3503, 4336, 5326, 6542, 7984, 9756, 11842, 14376, 17382, 20985, 25255, 30355, 36372, 43528, 51960, 61925, 73645, 87460, 103648, 122650

Offset

0,5

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Formula

a(n) + A239949(n) = A000041(n) for n >= 0.

Example

a(6) counts these 6 partitions:  51, 411, 321, 3111, 2211, 21111.

Maple

b:= proc(n, i, d) option remember; `if`(n=0, 1, `if`(i<=d, 0,
      add(b(n-i*j, i-1, d+`if`(j=0, 0, 1)), j=0..n/i)))
    end:
a:= n-> combinat[numbpart](n) -b(n$2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 02 2014

Mathematica

z = 50; d[p_] := d[p] = Length[DeleteDuplicates[p]]; f[n_] := f[n] = IntegerPartitions[n];
Table[Count[f[n], p_ /; d[p] < Min[p]], {n, 0, z}]  (*A239948*)
Table[Count[f[n], p_ /; d[p] <= Min[p]], {n, 0, z}] (*A239949*)
Table[Count[f[n], p_ /; d[p] == Min[p]], {n, 0, z}] (*A239950*)
Table[Count[f[n], p_ /; d[p] > Min[p]], {n, 0, z}]  (*A239951*)
Table[Count[f[n], p_ /; d[p] >= Min[p]], {n, 0, z}] (*A239952*)
b[n_, i_, d_] := b[n, i, d] = If[n==0, 1, If[i<=d, 0, Sum[b[n-i*j, i-1, d + If[j==0, 0, 1]], {j, 0, n/i}]]]; a[n_] := PartitionsP[n] - b[n, n, 0]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Nov 17 2015, after Alois P. Heinz *)

Crossrefs

Cf. A239948, A239949, A239950, A239952.

Sequence in context: A371719 A333315 A307889 * A077625 A027383 A364671

Adjacent sequences: A239948 A239949 A239950 * A239952 A239953 A239954

Keyword

nonn,easy

Author

Clark Kimberling, Mar 30 2014

Status

approved