OFFSET
1,1
COMMENTS
A multiset 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 Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
EXAMPLE
The terms together with their prime indices begin:
270: {1,2,2,2,3}
378: {1,2,2,2,4}
594: {1,2,2,2,5}
702: {1,2,2,2,6}
918: {1,2,2,2,7}
1026: {1,2,2,2,8}
1242: {1,2,2,2,9}
1566: {1,2,2,2,10}
1620: {1,1,2,2,2,2,3}
1674: {1,2,2,2,11}
1750: {1,3,3,3,4}
1998: {1,2,2,2,12}
2214: {1,2,2,2,13}
2268: {1,1,2,2,2,2,4}
2322: {1,2,2,2,14}
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
wigQ[y_]:=Or[Length[y]==0, Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
sepQ[y_]:=!MatchQ[y, {___, x_, x_, ___}];
Select[Range[1000], Select[Permutations[primeMS[#]], wigQ]=={}&&!Select[Permutations[primeMS[#]], sepQ]=={}&]
CROSSREFS
The partitions with these Heinz numbers are counted by A345166.
Permutations of this type are ranked by A345169.
Numbers with a factorization of this type are counted by A348609.
A000041 counts integer partitions.
A003242 counts anti-run compositions.
A344606 counts alternating permutations of prime indices with twins.
A344740 counts twins and partitions with an alternating permutation.
A345164 counts alternating permutations of prime factors.
A345165 counts partitions without an alternating permutation.
A345170 counts partitions with an alternating permutation.
A348379 counts factorizations with an alternating permutation.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 13 2021
STATUS
approved