A239467 Number of 1-separable partitions of n; see Comments.

0, 0, 1, 2, 3, 4, 6, 8, 11, 14, 19, 24, 31, 39, 50, 62, 78, 96, 120, 147, 181, 220, 270, 327, 397, 478, 578, 693, 833, 994, 1189, 1414, 1683, 1994, 2365, 2792, 3297, 3880, 4568, 5359, 6287, 7354, 8602, 10036, 11704, 13618, 15841, 18387, 21332, 24702, 28591

Offset

1,4

Comments

Suppose that p is a partition of n into 2 or more parts and that h is a part of p. Then p is (h,0)-separable if there is an ordering x, h, x, h, ..., h, x of the parts of p, where each x represents any part of p except h. Here, the number of h's on the ends of the ordering is 0. Similarly, p is (h,1)-separable if there is an ordering x, h, x, h, ..., x, h, where the number of h's on the ends is 1; next, p is (h,2)-separable if there is an ordering h, x, h, ..., x, h. Finally, p is h-separable if it is (h,i)-separable for i = 0,1,2.

Links

Table of n, a(n) for n=1..51.

Example

(1,0)-separable partitions of 7: 421, 313;
(1,1)-separable partitions of 7: 61, 3121;
(1,2)-separable partitions of 7: 151, 12121;
1-separable partitions of 7: 421, 313, 61, 3121, 151, 12121, so that a(7) = 6.

Mathematica

z = 55; t1 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, 1] <= Length[p] + 1], {n, 1, z}] (* A239467 *)
t2 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, 2] <= Length[p] + 1], {n, 1, z}] (* A239468 *)
t3 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, 3] <= Length[p] + 1], {n, 1, z}] (* A239469 *)
t4 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, 4] <= Length[p] + 1], {n, 1, z}] (* A239470 *)
t5 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, 5] <= Length[p] + 1], {n, 1, z}] (* A239472 *)

Crossrefs

Cf. A239468, A239469, A239470, A239471.

Sequence in context: A233693 A003412 A038348 * A035945 A094707 A353864

Adjacent sequences: A239464 A239465 A239466 * A239468 A239469 A239470

Keyword

nonn,easy

Author

Clark Kimberling, Mar 20 2014

Status

approved