A378881 Irregular triangle listing permutations of {1,...,m} which are at maximum distance from the identity permutation under steps rotate left (L) or exchange first two elements (E).

1, 2, 1, 1, 3, 2, 3, 1, 2, 3, 2, 1, 2, 1, 4, 3, 2, 3, 1, 4, 2, 4, 3, 1, 1, 2, 5, 3, 4, 2, 1, 5, 3, 4, 3, 1, 6, 5, 4, 2, 2, 1, 7, 6, 3, 5, 4, 2, 1, 7, 6, 4, 3, 5, 3, 1, 8, 7, 6, 5, 4, 2, 2, 1, 9, 8, 7, 3, 6, 5, 4, 2, 1, 9, 8, 7, 5, 4, 3, 6, 3, 1, 10, 9, 8, 7, 6, 5, 4, 2

Offset: 1

Author

Kevin Ryde, Dec 09 2024

Keywords

• nonn
• tabf

Comments

Permutations are listed for successive m >= 1 and in lexicographic order for multiple permutations in each m.

The maximum distance is A039745(m) and there may be multiple permutations at that distance.

The number of permutations at the maximum distance is A186144(m). - Pontus von Brömssen, Dec 12 2024

Examples

			Triangle begins:
       k=1  2  3  4  5
  n=1:   1
  n=2:   2, 1
  n=3:   1, 3, 2
  n=4:   3, 1, 2
  n=5:   3, 2, 1
  n=6:   2, 1, 4, 3
  n=7:   2, 3, 1, 4
  n=8:   2, 4, 3, 1
  n=9:   1, 2, 5, 3, 4
  n=10:  2, 1, 5, 3, 4
For m=10 there is a single permutation at distance A039745(10) = 58, being row n=17,
  3,1, 10,9,8,7,6,5,4, 2
This shows a pattern seen in even m ranging 6 <= m <= 12 where elements 2 and 3 are exchanged in what would otherwise be decreasing elements (with wrap-around).
For m=11 there are two permutations at distance A039745(11) = 71, being rows n=18 and n=19,
  2,1, 11,10,9,8, 3,7,6,5,4
  2,1, 11,10,9,8, 6,5,4,3,7
                  \-------/
These show a pattern seen in odd m ranging 7 <= m <= 13 where the final (m-1)/2 elements are rotated left and right from what would otherwise be decreasing elements (with wrap-around).
		

Links

• Kevin Ryde, Table of n, a(n) for rows 1..22 (m=1..13)

Crossrefs

Cf. A039745, A186144.