A239729 Number of partitions p of n such that if h = min(p), then h is an (h,2)-separator of p; see Comments.

1, 1, 1, 2, 2, 2, 4, 4, 5, 7, 8, 10, 14, 16, 19, 25, 31, 36, 47, 55, 67, 83, 99, 119, 146, 173, 208, 250, 298, 352, 424, 500, 593, 703, 829, 974, 1154, 1350, 1585, 1859, 2175, 2537, 2968, 3452, 4019, 4672, 5425, 6283, 7290, 8421, 9735, 11240, 12962, 14921

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, or 2.

Links

Table of n, a(n) for n=1..54.

Example

a(9) counts these 5 partitions: 612, 513, 414, 423, 31212.

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. A239481, A239456, A239499, A239689.

Sequence in context: A035372 A035576 A272397 * A005859 A274143 A166271

Adjacent sequences: A239726 A239727 A239728 * A239730 A239731 A239732

Keyword

nonn,easy

Author

Clark Kimberling, Mar 25 2014

Status

approved