A368948 First track of a certain single track permutation.

0, 1, 2, 2, 1, 0, 1, 0, 0, 1, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 0, 1, 2, 2, 1, 0, 1, 0, 0, 1, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 2, 2, 1, 0, 0, 1, 0, 1, 2, 2, 1, 0, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 1, 0, 0, 1, 0, 1, 2, 2, 1, 0

Offset

0,3

Comments

Generate permutations of [0,1,2,3,...,n] by counting in the (rising) factorial number system and let j be its track incremented, then do the following swaps corresponding to j:

j: swaps (elements in permutation, or positions in inverse permutation)

0: (0,1)

1: (1,2)

2: (0,1) (2,3) once every 3! = 6 times

3: (1,2) (3,4)

4: (0,1) (2,3) (4,5) once every 5! = 120 times

5: (1,2) (3,4) (5,6)

6: (0,1) (2,3) (4,5) (6,7) once every 7! = 5040 times

j: (m, m+1) (m+2, m+3) ... (j, j+1) where m = j%2

(the average number of swaps is 1 + 1/3!*(1 + 1/5!*(1 + 1/7!*(...))) = 1.16805...).

The resulting permutations have the single track property: all tracks (columns under "permutation" in the example) are cyclic shifts of the first track.

Links

Joerg Arndt, Table of n, a(n) for n = 0..5039

Example

In the following dots denote zeros.
         permutation     inverse       factorial    j: swaps
                         permutation   number
   0:    [ . 1 2 3 ]    [ . 1 2 3 ]    [ . . . ]
   1:    [ 1 . 2 3 ]    [ 1 . 2 3 ]    [ 1 . . ]    0: (0,1)
   2:    [ 2 . 1 3 ]    [ 1 2 . 3 ]    [ . 1 . ]    1: (1,2)
   3:    [ 2 1 . 3 ]    [ 2 1 . 3 ]    [ 1 1 . ]    0: (0,1)
   4:    [ 1 2 . 3 ]    [ 2 . 1 3 ]    [ . 2 . ]    1: (1,2)
   5:    [ . 2 1 3 ]    [ . 2 1 3 ]    [ 1 2 . ]    0: (0,1)
   6:    [ 1 3 . 2 ]    [ 2 . 3 1 ]    [ . . 1 ]    2: (0,1) (2,3)
   7:    [ . 3 1 2 ]    [ . 2 3 1 ]    [ 1 . 1 ]    0: (0,1)
   8:    [ . 3 2 1 ]    [ . 3 2 1 ]    [ . 1 1 ]    1: (1,2)
   9:    [ 1 3 2 . ]    [ 3 . 2 1 ]    [ 1 1 1 ]    0: (0,1)
  10:    [ 2 3 1 . ]    [ 3 2 . 1 ]    [ . 2 1 ]    1: (1,2)
  11:    [ 2 3 . 1 ]    [ 2 3 . 1 ]    [ 1 2 1 ]    0: (0,1)
  12:    [ 3 2 1 . ]    [ 3 2 1 . ]    [ . . 2 ]    2: (0,1) (2,3)
  13:    [ 3 2 . 1 ]    [ 2 3 1 . ]    [ 1 . 2 ]    0: (0,1)
  14:    [ 3 1 . 2 ]    [ 2 1 3 . ]    [ . 1 2 ]    1: (1,2)
  15:    [ 3 . 1 2 ]    [ 1 2 3 . ]    [ 1 1 2 ]    0: (0,1)
  16:    [ 3 . 2 1 ]    [ 1 3 2 . ]    [ . 2 2 ]    1: (1,2)
  17:    [ 3 1 2 . ]    [ 3 1 2 . ]    [ 1 2 2 ]    0: (0,1)
  18:    [ 2 . 3 1 ]    [ 1 3 . 2 ]    [ . . 3 ]    2: (0,1) (2,3)
  19:    [ 2 1 3 . ]    [ 3 1 . 2 ]    [ 1 . 3 ]    0: (0,1)
  20:    [ 1 2 3 . ]    [ 3 . 1 2 ]    [ . 1 3 ]    1: (1,2)
  21:    [ . 2 3 1 ]    [ . 3 1 2 ]    [ 1 1 3 ]    0: (0,1)
  22:    [ . 1 3 2 ]    [ . 1 3 2 ]    [ . 2 3 ]    1: (1,2)
  23:    [ 1 . 3 2 ]    [ 1 . 3 2 ]    [ 1 2 3 ]    0: (0,1)

Crossrefs

Cf. A159880, A123400.

Sequence in context: A359440 A377942 A347687 * A368579 A287331 A179769

Adjacent sequences: A368945 A368946 A368947 * A368949 A368950 A368951

Keyword

nonn

Author

Joerg Arndt, Jan 10 2024

Status

approved