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%I #11 Jan 28 2022 01:00:48
%S 0,0,1,0,0,2,0,1,2,0,3,3,2,2,3,5,4,8,4,5,9,6,13,10,11,15,14,17,16,20,
%T 21,26,29,30,33,36,35,41,47,47,61,61,66,71,73,85,88,98,102,114,122,
%U 131,148,154,163,182,188,205,220,231,249,271,293,306,338,359
%N Number of partitions p of n such that if h = max(p) - min(p), then h is an (h,1)-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(11) counts these partitions: 4313, 4232, 321212.
%t z = 35; t1 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Min[p]] == Length[p]], {n, 1, z}] (* A239497 *)
%t t2 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2 Min[p]] == Length[p]], {n, 1, z}] (* A239498 *)
%t t3 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p]] == Length[p]], {n, 1, z}] (* A118096 *)
%t t4 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Length[p]] == Length[p]], {n, 1, z}] (* A239500 *)
%t t5 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p] - Min[p]] == Length[p]], {n, 1, z}] (* A239501 *)
%Y Cf. A239497, A239498, A239499, A239500, A239482.
%K nonn,easy
%O 1,6
%A _Clark Kimberling_, Mar 24 2014
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