A239456 Number of partitions p of n such that if h = max(p), then h is an (h,2)-separator of p; see Comments.

0, 0, 0, 0, 1, 0, 1, 2, 1, 1, 4, 2, 3, 4, 4, 5, 7, 5, 7, 9, 9, 9, 14, 11, 13, 16, 17, 19, 22, 20, 25, 28, 29, 30, 38, 37, 41, 45, 48, 51, 60, 59, 67, 73, 76, 82, 93, 94, 103, 111, 121, 127, 142, 143, 158, 171, 180, 191, 211, 218, 236, 252, 270, 284, 309, 320

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..66.

Example

a(11) counts these 4 partitions: 515, 434, 31313, 2121212.

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, A239499, A239689.

Sequence in context: A127420 A129033 A054090 * A122517 A256098 A177276

Adjacent sequences: A239453 A239454 A239455 * A239457 A239458 A239459

Keyword

nonn,easy

Author

Clark Kimberling, Mar 25 2014

Status

approved