

A345195


Number of nonalternating antirun compositions of n.


17



0, 0, 0, 0, 0, 0, 2, 4, 10, 23, 49, 96, 192, 368, 692, 1299, 2403, 4400, 8029, 14556, 26253
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OFFSET

0,7


COMMENTS

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 does have the antirun permutations (2,3,2,1,2) and (2,1,2,3,2).
An antirun (separation or Carlitz composition) is a sequence with no adjacent equal parts.


LINKS



FORMULA



EXAMPLE

The a(9) = 23 antiruns:
(1,2,6) (1,2,4,2) (1,2,1,2,3)
(1,3,5) (1,2,5,1) (1,2,3,1,2)
(2,3,4) (1,3,4,1) (1,2,3,2,1)
(4,3,2) (1,4,3,1) (1,3,2,1,2)
(5,3,1) (1,5,2,1) (2,1,2,3,1)
(6,2,1) (2,1,2,4) (2,1,3,2,1)
(2,4,2,1) (3,2,1,2,1)
(3,1,2,3)
(3,2,1,3)
(4,2,1,2)


MATHEMATICA

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_, ___}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], sepQ[#]&&!wigQ[#]&]], {n, 0, 15}]


CROSSREFS

Nonantirun compositions are counted by A261983.
These compositions are ranked by A345169.
Nonalternating compositions are counted by A345192, ranked by A345168.
A032020 counts strict compositions.
A106356 counts compositions by number of maximal antiruns.
A345164 counts alternating permutations of prime indices, w/ twins A344606.
A345165 counts partitions w/o an alternating permutation, ranked by A345171.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
Cf. A005649, A008965, A114901, A178470, A333755, A344604, A344614, A344654, A344740, A345162, A345163, A348380, A348612, A348613.


KEYWORD

nonn,more


AUTHOR



STATUS

approved



