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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.
3
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, 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, 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
OFFSET
1,5
COMMENTS
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}.
FORMULA
T(n, k) <= A373778(n, k).
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.
EXAMPLE
The triangle begins:
n| k: 1| 2| 3| 4| 5| 6| 7| 8| 9
====================================
[1] 1
[2] 1, 1
[3] 1, 2, 1
[4] 1, 2, 4, 1
[5] 1, 2, 6, 5, 1
[6] 1, 2, 6, 12, 6, 1
[7] 1, 2, 6, 17, 21, 7, 1
[8] 1, 2, 6, 22, 41, 28, 8, 1
[9] 1, 2, 6, 24, 69, 73, 36, 9, 1
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Thomas Scheuerle, Jun 22 2024
STATUS
approved