A333310 Triangle read by rows: T(n,k) is the number of permutations sigma of [n] such that sigma(1) = k and sigma(j)/j > sigma(j+1)/(j+1) for 1 <= j <= n-1.

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 2, 0, 1, 2, 2, 1, 3, 0, 1, 3, 5, 2, 3, 5, 0, 1, 3, 6, 5, 3, 4, 8, 0, 1, 4, 8, 12, 8, 5, 9, 13, 0, 1, 4, 12, 20, 18, 8, 11, 13, 21, 0, 1, 5, 18, 29, 42, 21, 22, 19, 27, 38, 0, 1, 5, 23, 44, 69, 48, 33, 30, 33, 38, 64

Offset

1,13

Comments

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

Links

Seiichi Manyama, Rows n = 1..18, flattened

Mathematics.StackExchange, Why are the numbers of two different permutations the same?, Mar 07 2020.

Example

Triangle begins:
n\k  | 1  2  3   4   5   6   7   8   9  10  11  12
-----+--------------------------------------------
   1 | 1;
   2 | 0, 1;
   3 | 0, 1, 1;
   4 | 0, 1, 1,  1;
   5 | 0, 1, 2,  1,  2;
   6 | 0, 1, 2,  2,  1,  3;
   7 | 0, 1, 3,  5,  2,  3,  5;
   8 | 0, 1, 3,  6,  5,  3,  4,  8;
   9 | 0, 1, 4,  8, 12,  8,  5,  9, 13;
  10 | 0, 1, 4, 12, 20, 18,  8, 11, 13, 21;
  11 | 0, 1, 5, 18, 29, 42, 21, 22, 19, 27, 38;
  12 | 0, 1, 5, 23, 44, 69, 48, 33, 30, 33, 38, 64;

Crossrefs

Row sums give A309807.

Cf. A332954.

Sequence in context: A114002 A114004 A306518 * A049986 A218797 A137289

Adjacent sequences: A333307 A333308 A333309 * A333311 A333312 A333313

Keyword

nonn,tabl

Author

Seiichi Manyama, Mar 14 2020

Status

approved