A350138 - OEIS

%I #11 Jan 13 2024 14:59:26

%S 0,0,0,2,32,338,3560,40058,492664,6647666,98210192,1581844994,

%T 27642067000,521491848218,10572345303576,229332715217954,

%U 5301688511602448,130152723055769810,3381930236770946120,92738693031618794378,2676532576838728227352

%N Number of non-weakly alternating patterns of length n.

%C 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.

%C We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.

%C Conjecture: The directed cases, which count non-weakly up/down or non-weakly down/up patterns, are both equal to the strong case: A350252.

%H Andrew Howroyd, <a href="/A350138/b350138.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = A000670(n) - A349058(n).

%e The a(4) = 32 patterns:

%e   (1,1,2,3)  (2,1,1,2)  (3,1,1,2)  (4,1,2,3)

%e   (1,2,2,1)  (2,1,1,3)  (3,1,2,3)  (4,2,1,3)

%e   (1,2,3,1)  (2,1,2,3)  (3,1,2,4)  (4,3,1,2)

%e   (1,2,3,2)  (2,1,3,4)  (3,2,1,1)  (4,3,2,1)

%e   (1,2,3,3)  (2,3,2,1)  (3,2,1,2)

%e   (1,2,3,4)  (2,3,3,1)  (3,2,1,3)

%e   (1,2,4,3)  (2,3,4,1)  (3,2,1,4)

%e   (1,3,2,1)  (2,4,3,1)  (3,3,2,1)

%e   (1,3,3,2)             (3,4,2,1)

%e   (1,3,4,2)

%e   (1,4,3,2)

%t allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];

%t whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];

%t Table[Length[Select[Join@@Permutations/@allnorm[n],!whkQ[#]&&!whkQ[-#]&]],{n,0,6}]

%o (PARI)

%o 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}

%o 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

%Y The unordered version is A274230, complement A052955.

%Y The strong case of compositions is A345192, ranked by A345168.

%Y The strict case is A348615, complement A001250.

%Y For compositions we have A349053, complement A349052, ranked by A349057.

%Y The complement is counted by A349058.

%Y The version for partitions is A349061, complement A349060.

%Y The version for permutations of prime indices: A349797, complement A349056.

%Y The version for ordered factorizations is A350139, complement A349059.

%Y The strong case is A350252, complement A345194. Also the directed case?

%Y A003242 = Carlitz compositions, complement A261983, ranked by A333489.

%Y A005649 = anti-run patterns, complement A069321.

%Y A025047/A129852/A129853 = alternating compositions, ranked by A345167.

%Y A345163 = normal partitions w/ alternating permutation, complement A345162.

%Y A345170 = partitions w/ alternating permutation, complement A345165.

%Y A349055 = normal multisets w/ alternating permutation, complement A349050.

%Y Cf. A049774, A096441, A336103, A344605, A344614, A344615, A344740, A348610, A349794.

%K nonn

%O 0,4

%A _Gus Wiseman_, Dec 24 2021

%E a(9) onwards from _Andrew Howroyd_, Jan 13 2024