A241827 Number of partitions p = [x(1), ..., x(k)], where x(1) >= x(2) >= ... >= x(k), of n such that min(x(i) - x(i-1)) > number of distinct parts of p.

1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 8, 8, 10, 11, 13, 14, 17, 18, 21, 23, 26, 28, 32, 34, 38, 41, 45, 48, 53, 57, 62, 67, 73, 79, 86, 93, 101, 110, 119, 129, 140, 152, 164, 178, 192, 208, 224, 242, 260, 281, 301, 324, 347, 373, 398, 427, 455, 487

Offset

0,6

Comments

For the partition [n] of n, "min(x(i) - x(i-1))" is (as in the Mathematica program) interpreted as n.

Links

Table of n, a(n) for n=0..62.

Formula

a(n) = A241826(n) - A241825(n).

a(n) + A241823(n) + A241825(n) = A000041(n) for n >= 0.

Example

a(6) counts these 2 partitions:  6, 51.

Mathematica

z = 30; f[n_] := f[n] = IntegerPartitions[n]; d[p_] := d[p] = Length[DeleteDuplicates[p]]; g1[p_] := Min[-Differences[p]]
Table[Count[f[n], p_ /; g1[p] < d[p]], {n, 0, z}]  (* A241823 *)
Table[Count[f[n], p_ /; g1[p] <= d[p]], {n, 0, z}] (* A241824 *)
Table[Count[f[n], p_ /; g1[p] == d[p]], {n, 0, z}] (* A241825 *)
Table[Count[f[n], p_ /; g1[p] >= d[p]], {n, 0, z}] (* A241826 *)
Table[Count[f[n], p_ /; g1[p] > d[p]], {n, 0, z}]  (* A241827 *)

Crossrefs

Cf. A241823, A241824, A241826, A241826, A000041.

Sequence in context: A264594 A026829 A025154 * A096401 A264593 A026828

Adjacent sequences: A241824 A241825 A241826 * A241828 A241829 A241830

Keyword

nonn,easy

Author

Clark Kimberling, Apr 30 2014

Status

approved