A239497 Number of partitions p of n such that if h = min(p), then h is an (h,1)-separator of p; see Comments.

0, 0, 1, 1, 2, 3, 4, 5, 7, 9, 11, 15, 17, 22, 28, 34, 40, 52, 60, 75, 90, 109, 129, 160, 186, 225, 268, 321, 376, 455, 530, 632, 743, 878, 1028, 1219, 1416, 1667, 1947, 2281, 2648, 3103, 3593, 4189, 4857, 5638, 6516, 7564, 8715, 10080, 11614, 13394, 15392

Offset

1,5

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..53.

Example

a(7) counts these partitions: 61, 52, 43, 3211.

Mathematica

z = 35; t1 = Table[Count[IntegerPartitions[n],  p_ /; 2 Count[p, Min[p]] == Length[p]], {n, 1, z}]  (* A239497 *)
t2 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2 Min[p]] == Length[p]], {n, 1, z}] (* A239498 *)
t3 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p]] == Length[p]], {n, 1, z}] (* A118096 *)
t4 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Length[p]] == Length[p]], {n, 1, z}] (* A239500 *)
t5 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p] - Min[p]] == Length[p]], {n, 1, z}]  (* A239501 *)

Crossrefs

Cf. A239498, A118096, A239500, A239501, A239482.

Sequence in context: A076970 A064548 A291927 * A039850 A238486 A238482

Adjacent sequences: A239494 A239495 A239496 * A239498 A239499 A239500

Keyword

nonn,easy

Author

Clark Kimberling, Mar 24 2014

Status

approved