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

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}.

Links

Table of n, a(n) for n=1..91.

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

Cf. A371822.

Cf. A088532, A342474, A373778.

Sequence in context: A137855 A113143 A181802 * A373778 A110971 A136788

Adjacent sequences: A371820 A371821 A371822 * A371824 A371825 A371826

Keyword

nonn,tabl

Author

Thomas Scheuerle, Jun 22 2024

Status

approved