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%I #7 Jan 28 2022 01:01:50
%S 1,1,1,2,1,2,3,3,4,4,5,7,9,10,11,16,17,21,26,30,38,46,53,63,76,89,106,
%T 128,149,176,210,245,287,339,392,463,542,628,733,854,989,1150,1336,
%U 1542,1782,2063,2373,2736,3155,3620,4162,4783,5476,6275,7185,8210
%N Number of partitions p of n such that if h = 2*min(p), then h is an (h,0)-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(9) counts these 4 partitions: 612, 513, 324, 31212.
%t z = 75; Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Min[p]] == Length[p] - 1], {n, 1, z}] (* A239510 *)
%t Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2 Min[p]] == Length[p] - 1], {n, 1, z}] (* A239511 *)
%t Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p]] == Length[p] - 1], {n, 1, z}] (* A237828 *)
%t Table[Count[Rest[IntegerPartitions[n]], p_ /; 2 Count[p, Length[p]] == Length[p] - 1], {n, 1, z}] (* A239513 *)
%t Table[Count[Rest[IntegerPartitions[n]], p_ /; 2 Count[p, Max[p] - Min[p]] == Length[p] - 1], {n, 1, z}] (* A239514 *)
%Y Cf. A239510, A237828, A239513, A239514, A239482.
%K nonn,easy
%O 1,4
%A _Clark Kimberling_, Mar 24 2014
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