A097825 Triangle of permutations of [1..n] made by alternatively reversing terms counted from the left and right. (See comment.)

1, 2, 1, 2, 3, 1, 3, 1, 2, 4, 2, 1, 5, 4, 3, 1, 3, 2, 5, 6, 4, 6, 1, 2, 3, 5, 4, 7, 1, 7, 4, 5, 6, 8, 3, 2, 9, 7, 6, 1, 4, 5, 3, 2, 8, 7, 10, 8, 3, 6, 9, 5, 4, 1, 2, 5, 11, 10, 7, 6, 3, 8, 9, 1, 2, 4, 11, 10, 12, 1, 4, 5, 9, 6, 3, 2, 7, 8, 3, 6, 7, 11, 10, 5, 4, 9, 12, 13, 1, 8, 2, 14, 6, 7, 1, 2, 3, 13, 10, 9, 4, 5, 8, 11, 12

Offset

1,2

Comments

Start with [1..n]. Reverse the order of the leftmost 1 element. (trivial) Reverse the order of the rightmost 2 elements. Reverse the order of the leftmost 3 elements of the previous permutation. Reverse the order of the rightmost 4 elements of the previous permutation. ...until... Reverse the order of the rightmost n elements of the (n-1)th permutation if n is even. Or reverse the order of the leftmost n elements of the (n-1)th permutation if n is odd. (Of course, these options are the same thing, reversing the order of the entire permutation.)

Links

John Tyler Rascoe, Rows n = 1..148 of triangle, flattened

Example

For n=6, the reversals steps are:
[1,2,3,4,5,6]->[1,2,3,4,5,6]->[1,2,3,4,6,5]->[3,2,1,4,6,5]->[3,2,5,6,4,1]->[4,6,5,2,3,1]->[1,3,2,5,6,4].
Triangle begins:
      k=1  2  3  4  5  6
  n=1:  1,
  n=2:  2, 1,
  n=3:  2, 3, 1,
  n=4:  3, 1, 2, 4,
  n=5:  2, 1, 5, 4, 3,
  n=6:  1, 3, 2, 5, 6, 4,
  ...

Maple

p:=proc(n) local B, k, u, rev, w; with(linalg): u:=n->[seq(i, i=1..n)]; rev:=proc(a) [seq(a[nops(a)+1-i], i=1..nops(a))] end; w:=(m, n)->[seq(i, i=m..n)]; B[0]:=matrix(1, n, u(n)): if n mod 2 = 0 then for k from 1 to n/2 do B[2*k-1]:=concat(submatrix(B[2*k-2], [1], rev(u(2*k-1))), submatrix(B[2*k-2], [1], w(2*k, n))): B[2*k]:=concat(submatrix(B[2*k-1], [1], u(n-2*k)), submatrix(B[2*k-1], [1], rev(w(n+1-2*k, n)))) od else for k from 1 to (n-1)/2 do B[2*k-1]:=concat(submatrix(B[2*k-2], [1], rev(u(2*k-1))), submatrix(B[2*k-2], [1], w(2*k, n))): B[2*k]:=concat(submatrix(B[2*k-1], [1], u(n-2*k)), submatrix(B[2*k-1], [1], rev(w(n+1-2*k, n)))) od: B[n]:=concat(submatrix(B[n-1], [1], rev(u(n))), submatrix(B[n-1], [1], [])) fi end: for n from 1 to 12 do p(n) od; # supplies the sequence in triangular form # Emeric Deutsch, Nov 17 2004

Prog

(Python)
def A097825_row(n):
    c = list(range(1, n+1))
    for j in range(2, n):
        if j%2 == 0: c = c[:n-j]+c[:n-j-1:-1]
        else: c = c[j-1::-1]+c[j:]
    return(c[::-1])
for n in range(1, 15):
    print(n, A097825_row(n)) # John Tyler Rascoe, Apr 14 2023

Crossrefs

Cf. A097510 (column k=1), A097511 (main diagonal).

Sequence in context: A101161 A245049 A214261 * A002343 A082076 A231516

Adjacent sequences: A097822 A097823 A097824 * A097826 A097827 A097828

Keyword

easy,nonn,tabl

Author

Leroy Quet, Aug 26 2004

Extensions

More terms from Emeric Deutsch, Nov 17 2004

Name edited by John Tyler Rascoe, Apr 19 2023

Status

approved