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
Andrew Howroyd, Table of n, a(n) for n = 0..1000
FORMULA
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
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.
A003242 counts anti-run compositions.
A005649 counts anti-run patterns.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 13 2021
EXTENSIONS
a(26) onwards from Andrew Howroyd, Jan 15 2024
STATUS
approved