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A239511
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Number of partitions p of n such that if h = 2*min(p), then h is an (h,0)-separator of p; see Comments.
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4
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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, 128, 149, 176, 210, 245, 287, 339, 392, 463, 542, 628, 733, 854, 989, 1150, 1336, 1542, 1782, 2063, 2373, 2736, 3155, 3620, 4162, 4783, 5476, 6275, 7185, 8210
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OFFSET
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1,4
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COMMENTS
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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.
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LINKS
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EXAMPLE
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a(9) counts these 4 partitions: 612, 513, 324, 31212.
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MATHEMATICA
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z = 75; Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Min[p]] == Length[p] - 1], {n, 1, z}] (* A239510 *)
Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2 Min[p]] == Length[p] - 1], {n, 1, z}] (* A239511 *)
Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p]] == Length[p] - 1], {n, 1, z}] (* A237828 *)
Table[Count[Rest[IntegerPartitions[n]], p_ /; 2 Count[p, Length[p]] == Length[p] - 1], {n, 1, z}] (* A239513 *)
Table[Count[Rest[IntegerPartitions[n]], p_ /; 2 Count[p, Max[p] - Min[p]] == Length[p] - 1], {n, 1, z}] (* A239514 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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