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A345166
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Number of separable integer partitions of n without an alternating permutation.
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24
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0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 3, 5, 6, 7, 10, 14, 18, 21, 27, 35, 42, 54, 65, 78, 95, 117, 140, 170, 202, 239, 286, 343, 401, 476, 562, 660, 775, 910, 1056, 1241, 1444, 1678, 1948, 2267, 2615, 3031, 3502, 4036, 4647, 5356, 6143, 7068, 8101, 9274, 10613, 12151, 13856
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OFFSET
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0,14
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COMMENTS
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A partition is separable if it has an anti-run permutation (no adjacent parts equal).
A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it has the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).
The partitions counted by this sequence are those with 2m-1 parts with m being the multiplicity of a part which is neither the smallest or largest part. For example, 4322221 is such a partition since the multiplicity of 2 is 4, the total number of parts is 7, and 2 is neither the smallest or largest part. - Andrew Howroyd, Jan 15 2024
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LINKS
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FORMULA
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EXAMPLE
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The a(10) = 1 through a(16) = 6 partitions:
32221 42221 52221 62221 43331 43332 53332
3222211 72221 53331 63331
4222211 82221 92221
3322221 4322221
5222211 6222211
322222111
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MATHEMATICA
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wigQ[y_]:=Or[Length[y]==0, Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[IntegerPartitions[n], Select[Permutations[#], !MatchQ[#, {___, x_, x_, ___}]&]!={}&&Select[Permutations[#], wigQ]=={}&]], {n, 0, 15}]
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CROSSREFS
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Permutations of this type are ranked by A345169.
The Heinz numbers of these partitions are A345173.
Numbers with a factorization of this type are A348609.
A003242 counts anti-run compositions.
A345162 counts normal partitions w/o alt permutation, complement A345163.
Cf. A000070, A103919, A335126, A344604, A344607, A344615, A344740, A344742, A345164, A345166, A345168, A345192, A348379.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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