A263756 Triangle read by rows: T(n,k) (n>=0, k>=0) is the number of permutations of n with sum of descent bottoms equal to k.

1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 3, 8, 3, 1, 1, 1, 15, 7, 34, 18, 14, 18, 8, 3, 1, 1, 1, 31, 15, 122, 72, 69, 147, 83, 71, 33, 45, 18, 8, 3, 1, 1, 1, 63, 31, 406, 252, 263, 822, 544, 554, 399, 613, 351, 307, 160, 102, 96, 45, 18, 8, 3, 1, 1, 1, 127, 63, 1298, 828

Offset: 0

Author

Christian Stump, Oct 19 2015

Keywords

• nonn
• tabf

Comments

Row sums give A000142.

Examples

			Triangle begins:
  1;
  1;
  1,1;
  1,3,1,1;
  1,7,3,8,3,1,1;
  1,15,7,34,18,14,18,8,3,1,1;
  1,31,15,122,72,69,147,83,71,33,45,18,8,3,1,1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..23, flattened
• FindStat - Combinatorial Statistic Finder, The sum of the descent bottoms of a permutations.

Crossrefs

Cf. A000142, A263753.

Programs

Maple

b:= proc(s) option remember; (n-> `if`(n=0, 1, expand(
      add(b(s minus {j})*`if`(j (p-> seq(coeff(p, x, i), i=0..degree(p)))(b({$1..n})):
seq(T(n), n=0..9);  # Alois P. Heinz, Oct 25 2015, revised, Jan 31 2023

Mathematica

b[s_] := b[s] = With[{n = Length[s]}, If[n == 0, 1, Expand[       Sum[b[s~Complement~{j}]*If[j < n, x^j, 1], {j, s}]]]];
T[n_] := CoefficientList[b[Range[n]], x];
Table[T[n], {n, 0, 9}] // Flatten (* Jean-François Alcover, May 26 2023, after Alois P. Heinz *)

Extensions

Two terms (for rows 0 and 1) prepended and more terms from Alois P. Heinz, Oct 25 2015