OFFSET
0,3
COMMENTS
We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670. A sequence is alternating (cf. A025047) including pairs (x,x) if there are no adjacent triples (..., x, y, z, ...) where x <= y <= z or x >= y >= z. These sequences avoid the weak consecutive patterns (1,2,3) and (3,2,1).
An alternating pattern of length > 2 is necessarily an anti-run (A005649).
The version without pairs (x,x) is identical to this sequence except a(2) = 2 instead of 3.
LINKS
Martin Ehrenstein, Table of n, a(n) for n = 0..25
EXAMPLE
The a(0) = 1 through a(4) = 22 patterns:
() (1) (1,1) (1,2,1) (1,2,1,2)
(1,2) (1,3,2) (1,2,1,3)
(2,1) (2,1,2) (1,3,1,2)
(2,1,3) (1,3,2,3)
(2,3,1) (1,3,2,4)
(3,1,2) (1,4,2,3)
(2,1,2,1)
(2,1,3,1)
(2,1,3,2)
(2,1,4,3)
(2,3,1,2)
(2,3,1,3)
(2,3,1,4)
(2,4,1,3)
(3,1,2,1)
(3,1,3,2)
(3,1,4,2)
(3,2,3,1)
(3,2,4,1)
(3,4,1,2)
(4,1,3,2)
(4,2,3,1)
MATHEMATICA
allnorm[n_]:=If[n<=0, {{}}, Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n], !MatchQ[#, {___, x_, y_, z_, ___}/; x<=y<=z||x>=y>=z]&]], {n, 0, 6}]
CROSSREFS
The version for permutations is A001250.
The version for compositions is A344604.
The version for permutations of prime indices is A344606.
A003242 counts anti-run compositions.
A005649 counts anti-run patterns.
A019536 counts necklace patterns.
A335515 counts patterns matching (1,2,3).
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 27 2021
EXTENSIONS
a(10) and beyond from Martin Ehrenstein, Jun 10 2021
STATUS
approved