A344605 Number of alternating patterns of length n, including pairs (x,x).

1, 1, 3, 6, 22, 102, 562, 3618, 26586, 219798, 2018686, 20393790, 224750298, 2683250082, 34498833434, 475237879950, 6983085189454, 109021986683046, 1802213242949602, 31447143854808378, 577609702827987882, 11139837273501641502, 225075546284489412854

Offset: 0

Author

Gus Wiseman, May 27 2021

Keywords

• nonn

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.

Examples

			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)
		

Links

• Martin Ehrenstein, Table of n, a(n) for n = 0..25

Crossrefs

The version for permutations is A001250.

The version for compositions is A344604.

The version for permutations of prime indices is A344606.

A000670 counts patterns (ranked by A333217).

A003242 counts anti-run compositions.

A005649 counts anti-run patterns.

A019536 counts necklace patterns.

A025047 counts alternating or wiggly compositions, complement A345192.

A226316 counts patterns avoiding (1,2,3) (weakly: A052709).

A335515 counts patterns matching (1,2,3).

Cf. A000041, A006330, A049774, A102726, A103919, A124754, A128761, A335456, A335457, A335517, A344612, A344614, A344615, A348377.

Programs

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}]

Extensions

a(10) and beyond from Martin Ehrenstein, Jun 10 2021