%I #13 Jan 28 2022 01:14:34
%S 0,0,0,0,1,0,0,1,0,2,2,1,2,3,4,5,7,6,9,13,13,17,22,25,32,39,43,55,67,
%T 78,93,113,132,158,191,217,260,308,357,424,498,576,676,792,916,1069,
%U 1244,1436,1666,1934,2225,2573,2971,3410,3932,4524,5183,5951,6826
%N Number of partitions p of n such that if h = 2*min(p), then h is an (h,2)-separator of 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 = 0,1,2.
%e a(10) counts these partitions: 424, 23212.
%t z = 75; Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Min[p]] == Length[p] + 1], {n, 1, z}] (* A239729 *)
%t Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2 Min[p]] == Length[p] + 1], {n, 1, z}] (* A239481 *)
%t Table[Count[Rest[IntegerPartitions[n]], p_ /; 2 Count[p, Max[p]] == Length[p] + 1], {n, 1, z}] (* A239456 *)
%t Table[Count[Rest[IntegerPartitions[n]], p_ /; 2 Count[p, Length[p]] == Length[p] + 1], {n, 1, z}] (* A239499 *)
%t Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p] - Min[p]] == Length[p] + 1], {n, 1, z}] (*A239689 *)
%Y Cf. A239729, A239456, A239499, A239689.
%K nonn,easy
%O 1,10
%A _Clark Kimberling_, Mar 25 2014