A239514 Number of partitions p of n such that if h = max(p) - min(p), then h is an (h,0)-separator of p; see Comments.

0, 0, 0, 0, 1, 1, 0, 2, 2, 3, 2, 4, 2, 7, 6, 7, 6, 10, 7, 14, 12, 18, 12, 22, 18, 23, 23, 31, 29, 42, 33, 45, 42, 54, 49, 68, 62, 78, 76, 95, 87, 110, 102, 124, 128, 150, 141, 178, 174, 203, 203, 237, 228, 272, 269, 308, 318, 360, 356, 422, 420, 472, 482

Offset

1,8

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..63.

Formula

a(12) counts these partitions: 615, 642, 43131, 3121212.

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, A239511, A237828, A239513, A239482.

Sequence in context: A318470 A175501 A144370 * A304464 A087050 A341285

Adjacent sequences: A239511 A239512 A239513 * A239515 A239516 A239517

Keyword

nonn,easy

Author

Clark Kimberling, Mar 24 2014

Status

approved