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A345166
Number of separable integer partitions of n without an alternating permutation.
24
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
OFFSET
0,14
COMMENTS
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
LINKS
FORMULA
The Heinz numbers of these partitions are A345173 = A345171 /\ A335433.
a(n) = A325534(n) - A345170(n). - Andrew Howroyd, Jan 15 2024
EXAMPLE
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
MATHEMATICA
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}]
CROSSREFS
Allowing alternating permutations gives A325534, ranked by A335433.
Not requiring separability gives A345165, ranked by A345171.
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.
A000041 counts integer partitions.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions.
A005649 counts anti-run patterns.
A025047 counts alternating or wiggly compositions, also A025048, A025049.
A325535 counts inseparable partitions, ranked by A335448.
A344654 counts non-twin partitions w/o alt permutation, rank A344653.
A345162 counts normal partitions w/o alt permutation, complement A345163.
A345170 counts partitions w/ alt permutation, ranked by A345172.
Sequence in context: A030050 A018336 A194359 * A277006 A220355 A353954
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 13 2021
EXTENSIONS
a(26) onwards from Andrew Howroyd, Jan 15 2024
STATUS
approved