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.

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

Author

Alois P. Heinz, Aug 30 2022

Keywords

• nonn

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1802

Crossrefs

Column k=0 and also main diagonal of A356692.

Cf. A000142, A102407, A216837, A291683.

Programs

Maple

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);

Formula

a(n) = A356692(n,0) = A356692(n,n).

a(n) = 1 + A291683(n).

a(n) >= A102407(n) with equality only for n <= 7.