A239516 - OEIS

%I #8 Jan 28 2022 00:56:41

%S 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,

%T 184,220,262,313,373,440,520,616,723,849,1002,1173,1373,1606,1873,

%U 2182,2543,2955,3431,3979,4608,5327,6160,7105,8190,9435,10851,12469,14317

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

%C 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.

%e 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.

%t z = 75; t1 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, Min[p]] <= Length[p] + 1], {n, 1, z}]  (* A239515 *)

%t t2 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, 2*Min[p]] <= Length[p] + 1], {n, 1, z}]  (* A239516 *)

%t t3 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, Max[p]] <= Length[p] + 1], {n, 1, z}]  (* A239517 *)

%t t4 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, Length[p]] <= Length[p] + 1], {n, 1, z}]  (* A239518 *)

%Y Cf. A239515, A239517, A239518, A239482.

%K nonn,easy

%O 1,6

%A _Clark Kimberling_, Mar 26 2014