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A371823 Triangle T(n, k) read by rows: Maximum number of patterns of length k in a permutation from row n in A371822.

%I A371823 #43 Jul 08 2024 14:00:12
%S A371823 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 A371823 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 A371823 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 A371823 Triangle T(n, k) read by rows: Maximum number of patterns of length k in a permutation from row n in A371822.
%C A371823 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 A371823 T(n, k) <= A373778(n, k).
%F A371823 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 A371823 The triangle begins:
%e A371823    n| k: 1| 2| 3|  4|  5|  6|  7| 8| 9
%e A371823   ====================================
%e A371823   [1]    1
%e A371823   [2]    1, 1
%e A371823   [3]    1, 2, 1
%e A371823   [4]    1, 2, 4,  1
%e A371823   [5]    1, 2, 6,  5,  1
%e A371823   [6]    1, 2, 6, 12,  6,  1
%e A371823   [7]    1, 2, 6, 17, 21,  7,  1
%e A371823   [8]    1, 2, 6, 22, 41, 28,  8, 1
%e A371823   [9]    1, 2, 6, 24, 69, 73, 36, 9, 1
%Y A371823 Cf. A371822.
%Y A371823 Cf. A088532, A342474, A373778.
%K A371823 nonn,tabl
%O A371823 1,5
%A A371823 _Thomas Scheuerle_, Jun 22 2024