

A345194


Number of alternating patterns of length n.


27



1, 1, 2, 6, 22, 102, 562, 3618, 26586, 219798, 2018686, 20393790, 224750298, 2683250082, 34498833434, 475237879950, 6983085189454, 109021986683046, 1802213242949602, 31447143854808378, 577609702827987882, 11139837273501641502, 225075546284489412854
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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 and ranked by A333217.
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 alternating pattern is necessarily an antirun (A005649).
The version with twins (A344605) is identical to this sequence except with a(2) = 3 instead of 2.
Conjecture: Also the number of weakly up/down patterns of length n, where a sequence is weakly up/down if it is alternately weakly increasing and weakly decreasing, starting with an increase. For example, the a(0) = 1 through a(3) = 6 weakly up/down patterns are:
() (1) (1,1) (1,1,1)
(2,1) (1,1,2)
(2,1,1)
(2,1,2)
(2,1,3)
(3,1,2)
(End)


LINKS



FORMULA



EXAMPLE

The a(0) = 1 through a(3) = 6 alternating patterns:
() (1) (1,2) (1,2,1)
(2,1) (1,3,2)
(2,1,2)
(2,1,3)
(2,3,1)
(3,1,2)


MATHEMATICA

wigQ[y_]:=Or[Length[y]==0, Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]1];
allnorm[n_]:=If[n<=0, {{}}, Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n], wigQ]], {n, 0, 6}]


PROG

(PARI)
F(p, x) = {sum(k=0, p, (1)^((k+1)\2)*binomial((p+k)\2, k)*x^k)}
R(n, k) = {Vec(if(k==1, x, 2*F(k2, x)/F(k1, x)2(k2)*x) + O(x*x^n))}
seq(n)= {concat([1], sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(1)^(rk)) ))} \\ Andrew Howroyd, Feb 04 2022


CROSSREFS

The version with twins (x,x) is A344605.
The version for perms of prime indices is A345164, complement A350251.
The complement is counted by A350252.
A349055 = normal multisets w/ alternating permutation, complement A349050.


KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



