A332954 Triangle read by rows: T(n,k) is the number of permutations sigma of [n] such that sigma(j)/(j+k) > sigma(j+1)/(j+k+1) for 1 <= j <= n-1.

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 6, 3, 2, 1, 1, 1, 9, 5, 3, 2, 1, 1, 1, 19, 8, 5, 3, 2, 1, 1, 1, 30, 13, 7, 5, 3, 2, 1, 1, 1, 60, 21, 12, 7, 5, 3, 2, 1, 1, 1, 108, 38, 17, 11, 7, 5, 3, 2, 1, 1, 1

Offset: 0

Author

Seiichi Manyama, Mar 04 2020

Keywords

• nonn
• tabl

Comments

Conjecture: T(2*n+4,n) = A052955(n+2). This is true for n <= 10.

T(n+1,k) is equal to the number of permutations sigma of [n] such that sigma(j)/(j+k) >= sigma(j+1)/(j+k+1) for 1 <= j <= n-1.

Examples

			Triangle begins:
n\k  |   0   1   2   3   4  5  6  7  8  9 10 11
-----+-----------------------------------------
   0 |   1;
   1 |   1,  1;
   2 |   1,  1,  1;
   3 |   2,  1,  1,  1;
   4 |   3,  2,  1,  1,  1;
   5 |   6,  3,  2,  1,  1, 1;
   6 |   9,  5,  3,  2,  1, 1, 1;
   7 |  19,  8,  5,  3,  2, 1, 1, 1;
   8 |  30, 13,  7,  5,  3, 2, 1, 1, 1;
   9 |  60, 21, 12,  7,  5, 3, 2, 1, 1, 1;
  10 | 108, 38, 17, 11,  7, 5, 3, 2, 1, 1, 1;
  11 | 222, 64, 31, 16, 11, 7, 5, 3, 2, 1, 1, 1;
		

Links

• Seiichi Manyama, Rows n = 0..18, flattened
• Mathematics.StackExchange, Why are the numbers of two different permutations the same?, Mar 07 2020.

Crossrefs

T(n,0) gives A309807.

Cf. A052955.