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%I #7 Jan 28 2022 01:13:55
%S 1,1,1,2,2,2,4,4,5,7,8,10,14,16,19,25,31,36,47,55,67,83,99,119,146,
%T 173,208,250,298,352,424,500,593,703,829,974,1154,1350,1585,1859,2175,
%U 2537,2968,3452,4019,4672,5425,6283,7290,8421,9735,11240,12962,14921
%N Number of partitions p of n such that if h = 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, or 2.
%e a(9) counts these 5 partitions: 612, 513, 414, 423, 31212.
%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. A239481, A239456, A239499, A239689.
%K nonn,easy
%O 1,4
%A _Clark Kimberling_, Mar 25 2014
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