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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).

%I A378881 #15 Dec 12 2024 15:20:00
%S A378881 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 A378881 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 A378881 6,5,4,2,1,9,8,7,5,4,3,6,3,1,10,9,8,7,6,5,4,2
%N 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).
%C A378881 Permutations are listed for successive m >= 1 and in lexicographic order for multiple permutations in each m.
%C A378881 The maximum distance is A039745(m) and there may be multiple permutations at that distance.
%C A378881 The number of permutations at the maximum distance is A186144(m). - _Pontus von Brömssen_, Dec 12 2024
%H A378881 Kevin Ryde, <a href="/A378881/b378881.txt">Table of n, a(n) for rows 1..22 (m=1..13)</a>
%e A378881 Triangle begins:
%e A378881        k=1  2  3  4  5
%e A378881   n=1:   1
%e A378881   n=2:   2, 1
%e A378881   n=3:   1, 3, 2
%e A378881   n=4:   3, 1, 2
%e A378881   n=5:   3, 2, 1
%e A378881   n=6:   2, 1, 4, 3
%e A378881   n=7:   2, 3, 1, 4
%e A378881   n=8:   2, 4, 3, 1
%e A378881   n=9:   1, 2, 5, 3, 4
%e A378881   n=10:  2, 1, 5, 3, 4
%e A378881 For m=10 there is a single permutation at distance A039745(10) = 58, being row n=17,
%e A378881   3,1, 10,9,8,7,6,5,4, 2
%e A378881 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 A378881 For m=11 there are two permutations at distance A039745(11) = 71, being rows n=18 and n=19,
%e A378881   2,1, 11,10,9,8, 3,7,6,5,4
%e A378881   2,1, 11,10,9,8, 6,5,4,3,7
%e A378881                   \-------/
%e A378881 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 A378881 Cf. A039745, A186144.
%K A378881 nonn,tabf
%O A378881 1,2
%A A378881 _Kevin Ryde_, Dec 09 2024