A239481 Number of partitions p of n such that if h = 2*min(p), then h is an (h,2)-separator of p; see Comments.

0, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 1, 2, 3, 4, 5, 7, 6, 9, 13, 13, 17, 22, 25, 32, 39, 43, 55, 67, 78, 93, 113, 132, 158, 191, 217, 260, 308, 357, 424, 498, 576, 676, 792, 916, 1069, 1244, 1436, 1666, 1934, 2225, 2573, 2971, 3410, 3932, 4524, 5183, 5951, 6826

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 = 0,1,2.

Links

Table of n, a(n) for n=1..59.

Example

a(10) counts these partitions: 424, 23212.

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, A239456, A239499, A239689.

Sequence in context: A366804 A259910 A072549 * A200114 A120652 A330608

Adjacent sequences: A239478 A239479 A239480 * A239482 A239483 A239484

Keyword

nonn,easy

Author

Clark Kimberling, Mar 25 2014

Status

approved