|
%I #43 Jul 08 2024 14:00:12
%S 1,1,1,1,2,1,1,2,4,1,1,2,6,5,1,1,2,6,12,6,1,1,2,6,17,21,7,1,1,2,6,22,
%T 41,28,8,1,1,2,6,24,69,73,36,9,1,1,2,6,24,94,156,113,45,10,1,1,2,6,24,
%U 109,273,291,162,55,11,1,1,2,6,24,118,408,614,477,220,66,12,1,1,2,6,24,120,526,1094,1127,699,286,78,13,1
%N Triangle T(n, k) read by rows: Maximum number of patterns of length k in a permutation from row n in A371822.
%C The row sums agree for n = 1..8 and 10..11 with A088532(n), where n = 11 was the last known value of A088532. The process described in A371822 gives in row 9 the permutation {6,1,9,4,7,2,5,8,3} but the closest optimal permutation would have been: {6,2,9,4,7,1,5,8,3}.
%F T(n, k) <= A373778(n, k).
%F Conjecture: T(n, n-2) = ceiling(n*(n-1)/2), for n > 6. This is expected because this triangle does asymptotically approximate the factorial numbers from the left to the right and Pascal's triangle from right to the left.
%e The triangle begins:
%e n| k: 1| 2| 3| 4| 5| 6| 7| 8| 9
%e ====================================
%e [1] 1
%e [2] 1, 1
%e [3] 1, 2, 1
%e [4] 1, 2, 4, 1
%e [5] 1, 2, 6, 5, 1
%e [6] 1, 2, 6, 12, 6, 1
%e [7] 1, 2, 6, 17, 21, 7, 1
%e [8] 1, 2, 6, 22, 41, 28, 8, 1
%e [9] 1, 2, 6, 24, 69, 73, 36, 9, 1
%Y Cf. A371822.
%Y Cf. A088532, A342474, A373778.
%K nonn,tabl,new
%O 1,5
%A _Thomas Scheuerle_, Jun 22 2024
|
