A378881 - OEIS

%I #15 Dec 12 2024 15:20:00

%S 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,

%T 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,

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

%N 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).

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

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

%C The number of permutations at the maximum distance is A186144(m). - _Pontus von Brömssen_, Dec 12 2024

%H Kevin Ryde, <a href="/A378881/b378881.txt">Table of n, a(n) for rows 1..22 (m=1..13)</a>

%e Triangle begins:

%e        k=1  2  3  4  5

%e   n=1:   1

%e   n=2:   2, 1

%e   n=3:   1, 3, 2

%e   n=4:   3, 1, 2

%e   n=5:   3, 2, 1

%e   n=6:   2, 1, 4, 3

%e   n=7:   2, 3, 1, 4

%e   n=8:   2, 4, 3, 1

%e   n=9:   1, 2, 5, 3, 4

%e   n=10:  2, 1, 5, 3, 4

%e For m=10 there is a single permutation at distance A039745(10) = 58, being row n=17,

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

%e 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).

%e For m=11 there are two permutations at distance A039745(11) = 71, being rows n=18 and n=19,

%e   2,1, 11,10,9,8, 3,7,6,5,4

%e   2,1, 11,10,9,8, 6,5,4,3,7

%e                   \-------/

%e 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).

%Y Cf. A039745, A186144.

%K nonn,tabf,new

%O 1,2

%A _Kevin Ryde_, Dec 09 2024