A239689 Number of partitions p of n such that if h = max(p) - min(p), then h is an (h,2)-separator of p; see Comments.

0, 0, 0, 1, 0, 0, 1, 1, 0, 2, 0, 1, 3, 2, 1, 4, 0, 4, 5, 5, 4, 5, 5, 7, 7, 10, 6, 15, 10, 13, 13, 14, 16, 20, 18, 26, 24, 29, 28, 34, 27, 39, 40, 47, 46, 57, 55, 65, 63, 75, 72, 83, 85, 101, 101, 117, 115, 136, 134, 157, 156, 170, 181, 200, 197, 229, 230

Offset

1,10

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, or 2.

Links

Table of n, a(n) for n=1..67.

Example

a(13) counts these 3 partitions: 24232, 2321212, 121212121.

Mathematica

z = 75; Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Min[p]] == Length[p] + 1], {n, 1, z}] (* A239729 *)
Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2 Min[p]] == Length[p] + 1], {n, 1, z}] (* A239481 *)
Table[Count[Rest[IntegerPartitions[n]], p_ /; 2 Count[p, Max[p]] == Length[p] + 1], {n, 1, z}] (* A239456 *)
Table[Count[Rest[IntegerPartitions[n]], p_ /; 2 Count[p, Length[p]] == Length[p] + 1], {n, 1, z}] (* A239499 *)
Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p] - Min[p]] == Length[p] + 1], {n, 1, z}] (* A239689 *)

Crossrefs

Cf. A239729, A239481, A239456, A239499.

Sequence in context: A197119 A124377 A144190 * A177802 A022827 A302764

Adjacent sequences: A239686 A239687 A239688 * A239690 A239691 A239692

Keyword

nonn,easy

Author

Clark Kimberling, Mar 25 2014

Status

approved