A239511 Number of partitions p of n such that if h = 2*min(p), then h is an (h,0)-separator of p; see Comments.

1, 1, 1, 2, 1, 2, 3, 3, 4, 4, 5, 7, 9, 10, 11, 16, 17, 21, 26, 30, 38, 46, 53, 63, 76, 89, 106, 128, 149, 176, 210, 245, 287, 339, 392, 463, 542, 628, 733, 854, 989, 1150, 1336, 1542, 1782, 2063, 2373, 2736, 3155, 3620, 4162, 4783, 5476, 6275, 7185, 8210

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..56.

Example

a(9) counts these 4 partitions: 612, 513, 324, 31212.

Mathematica

z = 75; Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Min[p]] == Length[p] - 1], {n, 1, z}]  (* A239510 *)
Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2 Min[p]] == Length[p] - 1], {n, 1, z}]  (* A239511 *)
Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p]] == Length[p] - 1], {n, 1, z}]  (* A237828 *)
Table[Count[Rest[IntegerPartitions[n]], p_ /; 2 Count[p, Length[p]] == Length[p] - 1], {n, 1, z}]  (* A239513 *)
Table[Count[Rest[IntegerPartitions[n]], p_ /; 2 Count[p, Max[p] - Min[p]] == Length[p] - 1], {n, 1, z}] (* A239514 *)

Crossrefs

Cf. A239510, A237828, A239513, A239514, A239482.

Sequence in context: A087188 A102885 A323089 * A264052 A138585 A070048

Adjacent sequences: A239508 A239509 A239510 * A239512 A239513 A239514

Keyword

nonn,easy

Author

Clark Kimberling, Mar 24 2014

Status

approved