A239516 Number of partitions p of n that are separable by the 2*min(p); see Comments.

0, 0, 1, 1, 1, 3, 2, 4, 6, 6, 8, 12, 14, 16, 22, 27, 32, 41, 49, 60, 73, 88, 106, 130, 154, 184, 220, 262, 313, 373, 440, 520, 616, 723, 849, 1002, 1173, 1373, 1606, 1873, 2182, 2543, 2955, 3431, 3979, 4608, 5327, 6160, 7105, 8190, 9435, 10851, 12469, 14317

Offset

1,6

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

Let h = 2*min(p). The (h,0)-separable partition of 8 is 521; the (h,1)-separable partition is 3212; the (h,2)-separable partitions are 242, 21212. So, there are 1 + 1 + 2 = 4 h-separable partitions of 8.

Mathematica

z = 75; t1 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, Min[p]] <= Length[p] + 1], {n, 1, z}]  (* A239515 *)
t2 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, 2*Min[p]] <= Length[p] + 1], {n, 1, z}]  (* A239516 *)
t3 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, Max[p]] <= Length[p] + 1], {n, 1, z}]  (* A239517 *)
t4 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, Length[p]] <= Length[p] + 1], {n, 1, z}]  (* A239518 *)

Crossrefs

Cf. A239515, A239517, A239518, A239482.

Sequence in context: A059417 A078824 A014685 * A365533 A021312 A099258

Adjacent sequences: A239513 A239514 A239515 * A239517 A239518 A239519

Keyword

nonn,easy

Author

Clark Kimberling, Mar 26 2014

Status

approved