A216837
Number of permutations p of {1,...,n} such that at most one element of {p(1),...,p(i-1)} is between p(i) and p(i+1) for all i from 1 to n-1.
Original entry on oeis.org
1, 1, 2, 6, 20, 72, 268, 1020, 3936, 15332, 60112, 236780, 935848, 3708236, 14721912, 58533264, 232991656, 928261480, 3700935760, 14763921580, 58924038816, 235258847064, 939576469152, 3753419774180, 14997257109992, 59933657096280, 239547378220840
Offset: 0
a(4) = 20 = 4! - 4, because 4 permutations of {1,...,4} do not satisfy the condition: 2314, 2341, 3214, 3241.
Cf.
A174700,
A174701,
A174702,
A174703,
A174704,
A174705,
A174706,
A174707,
A174708,
A185030,
A356692.
-
b:= proc(u, o) option remember; `if`(u+o=0, 1,
add(b(sort([o-j, u+j-1])[]), j=1..min(2, o))+
add(b(sort([u-j, o+j-1])[]), j=1..min(2, u)))
end:
a:= n-> `if`(n=0, 1, add(b(sort([j-1, n-j])[]), j=1..n)):
seq(a(n), n=0..35);
-
b[u_, o_] := b[u, o] = If[u+o == 0, 1, Sum[b[Sequence @@ Sort[{o-j, u+j-1}]], {j, 1, Min[2, o]}] + Sum[b[Sequence @@ Sort[{u-j, o+j-1}]], {j, 1, Min[2, u]}]]; a[n_] := If[n == 0, 1, Sum[b[Sequence @@ Sort[{j-1, n-j}]], {j, 1, n}]]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Feb 05 2015, after Alois P. Heinz *)
A356832
Number of permutations p of [n] such that at most one element of {p(1),...,p(i-1)} is between p(i) and p(i+1) for all i < n and n = 0 or p(n) < 3.
Original entry on oeis.org
1, 1, 2, 4, 10, 26, 72, 206, 608, 1834, 5636, 17578, 55516, 177192, 570700, 1852572, 6055080, 19910730, 65823752, 218654100, 729459552, 2443051214, 8210993364, 27685671844, 93625082140, 317470233150, 1079183930828, 3676951654520, 12554734605496, 42952566314236
Offset: 0
a(0) = 1: (), the empty permutation.
a(1) = 1: 1.
a(2) = 2: 12, 21.
a(3) = 4: 132, 231, 312, 321.
a(4) = 10: 1342, 1432, 2431, 3142, 3412, 3421, 4132, 4231, 4312, 4321.
a(5) = 26: 13542, 14532, 15342, 15432, 24531, 25431, 31542, 35142, 35412, 35421, 41532, 42531, 45132, 45231, 45312, 45321, 51342, 51432, 52431, 53142, 53412, 53421, 54132, 54231, 54312, 54321.
Column k=0 and also main diagonal of
A356692.
-
b:= proc(u, o) option remember; `if`(u+o=0, 1,
add(b(sort([o-j, u+j-1])[]), j=1..min(2, o))+
add(b(sort([u-j, o+j-1])[]), j=1..min(2, u)))
end:
a:= n-> b(0, n):
seq(a(n), n=0..30);
# second Maple program:
b:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
`if`(n=0, 1, add(b(n-1, j), j=k-2..k+1)))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..30);
A356853
Number of permutations p of [2n+1] such that at most one element of {p(1),...,p(i-1)} is between p(i) and p(i+1) for all i <= 2n and p(2n+1) = n+1.
Original entry on oeis.org
1, 2, 20, 216, 2720, 36228, 503216, 7171404, 104142520, 1533200656, 22811374568, 342216338652, 5168324302672, 78483423004680, 1197266739443160, 18335055482658748, 281714880491273736, 4340894020114398672, 67055152953864109240, 1038097819961270208088
Offset: 0
a(0) = 1: 1.
a(1) = 2: 132, 312.
a(2) = 20: 12453, 12543, 14253, 14523, 15243, 15423, 21453, 21543, 25143, 25413, 41253, 41523, 45123, 45213, 51243, 51423, 52143, 52413, 54123, 54213.
a(3) = 216: 1235764, 1236754, 1237564, 1237654, ..., 7651234, 7651324, 7652134, 7653124.
-
b:= proc(u, o) option remember; `if`(u+o=0, 1,
add(b(sort([o-j, u+j-1])[]), j=1..min(2, o))+
add(b(sort([u-j, o+j-1])[]), j=1..min(2, u)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..20);
# second Maple program:
b:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
`if`(n=0, 1, add(b(n-1, j), j=k-2..k+1)))
end:
a:= n-> b(2*n, n):
seq(a(n), n=0..20);
Showing 1-3 of 3 results.