A345164 Number of alternating permutations of the multiset of prime factors of n.

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 0, 0, 2, 0, 1, 1, 4, 1, 0, 2, 2, 2, 2, 1, 2, 2, 0, 1, 4, 1, 1, 1, 2, 1, 0, 0, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 4, 1, 2, 1, 0, 2, 4, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 2, 4, 1, 0, 0, 2, 1, 4, 2, 2, 2

Offset

1,6

Comments

First differs from A335452 at a(30) = 4, A335452(30) = 6. The anti-runs (2,3,5) and (5,3,2) are not alternating.

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 permutation, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).

Links

Table of n, a(n) for n=1..87.

Example

The a(n) alternating permutations of prime indices for n = 180, 210, 300, 420, 900:
  (12132)  (1324)  (13132)  (12143)  (121323)
  (21213)  (1423)  (13231)  (13142)  (132312)
  (21312)  (2143)  (21313)  (13241)  (213132)
  (23121)  (2314)  (23131)  (14132)  (213231)
  (31212)  (2413)  (31213)  (14231)  (231213)
           (3142)  (31312)  (21314)  (231312)
           (3241)           (21413)  (312132)
           (3412)           (23141)  (323121)
           (4132)           (24131)
           (4231)           (31214)
                            (31412)
                            (34121)
                            (41213)
                            (41312)

Mathematica

wigQ[y_]:=Or[Length[y]==0, Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[Permutations[Flatten[ConstantArray@@@FactorInteger[n]]], wigQ]], {n, 30}]

Crossrefs

Counting all permutations gives A008480.

Dominated by A335452 (number of separations of prime factors).

Including twins (x,x) gives A344606.

Positions of zeros are A345171, counted by A345165.

Positions of nonzero terms are A345172.

A000041 counts integer partitions.

A001250 counts alternating permutations.

A003242 counts anti-run compositions.

A025047 counts alternating or wiggly compositions, also A025048, A025049.

A325534 counts separable partitions, ranked by A335433.

A325535 counts inseparable partitions, ranked by A335448.

A344604 counts alternating compositions with twins.

A344654 counts non-twin partitions w/o alternating permutation, rank: A344653.

A344740 counts twins and partitions w/ alternating permutation, rank: A344742.

A345166 counts separable partitions w/o alternating permutation, rank: A345173.

A345170 counts partitions with a alternating permutation.

Cf. A001222, A071321, A071322, A316523, A316524, A333489, A335126, A344605, A344614, A344616, A344652, A345163, A345168.

Sequence in context: A263251 A318370 A339742 * A050326 A335452 A355817

Adjacent sequences: A345161 A345162 A345163 * A345165 A345166 A345167

Keyword

nonn

Author

Gus Wiseman, Jun 13 2021

Status

approved