OFFSET
0,4
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.
We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.
Conjecture: The directed cases, which count non-weakly up/down or non-weakly down/up patterns, are both equal to the strong case: A350252.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..200
EXAMPLE
The a(4) = 32 patterns:
(1,1,2,3) (2,1,1,2) (3,1,1,2) (4,1,2,3)
(1,2,2,1) (2,1,1,3) (3,1,2,3) (4,2,1,3)
(1,2,3,1) (2,1,2,3) (3,1,2,4) (4,3,1,2)
(1,2,3,2) (2,1,3,4) (3,2,1,1) (4,3,2,1)
(1,2,3,3) (2,3,2,1) (3,2,1,2)
(1,2,3,4) (2,3,3,1) (3,2,1,3)
(1,2,4,3) (2,3,4,1) (3,2,1,4)
(1,3,2,1) (2,4,3,1) (3,3,2,1)
(1,3,3,2) (3,4,2,1)
(1,3,4,2)
(1,4,3,2)
MATHEMATICA
allnorm[n_]:=If[n<=0, {{}}, Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1]];
whkQ[y_]:=And@@Table[If[EvenQ[m], y[[m]]<=y[[m+1]], y[[m]]>=y[[m+1]]], {m, 1, Length[y]-1}];
Table[Length[Select[Join@@Permutations/@allnorm[n], !whkQ[#]&&!whkQ[-#]&]], {n, 0, 6}]
PROG
(PARI)
R(n, k)={my(v=vector(k, i, 1), u=vector(n)); for(r=1, n, if(r%2==0, my(s=v[k]); forstep(i=k, 2, -1, v[i] = s - v[i-1]); v[1] = s); for(i=2, k, v[i] += v[i-1]); u[r]=v[k]); u}
seq(n)= {concat([0], vector(n, i, 1) + sum(k=1, n, (vector(n, i, k^i) - 2*R(n, k))*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) ) )} \\ Andrew Howroyd, Jan 13 2024
CROSSREFS
The complement is counted by A349058.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 24 2021
EXTENSIONS
a(9) onwards from Andrew Howroyd, Jan 13 2024
STATUS
approved