A263753 Triangle read by rows: T(n,k) (n>=0, k>=0) is the number of permutations of n with sum of descent tops equal to k.

1, 1, 1, 0, 1, 1, 0, 1, 3, 0, 1, 1, 0, 1, 3, 7, 1, 3, 7, 0, 1, 1, 0, 1, 3, 7, 16, 3, 14, 17, 32, 3, 7, 15, 0, 1, 1, 0, 1, 3, 7, 16, 34, 14, 32, 69, 72, 129, 32, 68, 70, 118, 7, 15, 31, 0, 1, 1, 0, 1, 3, 7, 16, 34, 77, 32, 100, 149, 274, 292, 496, 220, 388, 536

Offset: 0

Author

Christian Stump, Oct 19 2015

Keywords

• nonn
• tabf

Comments

Row sums give A000142.

Row lengths are given by A000217 for n>=1. - Omar E. Pol, Oct 25 2015

Examples

			Triangle begins:
  1;
  1;
  1,0,1;
  1,0,1,3,0,1;
  1,0,1,3,7,1,3,7,0,1;
  1,0,1,3,7,16,3,14,17,32,3,7,15,0,1;
  1,0,1,3,7,16,34,14,32,69,72,129,32,68,70,118,7,15,31,0,1;
  ...
		

Links

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

Crossrefs

Cf. A000142, A263756.

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^n, 1], {j, s}]]]];
T[n_] := With[{p = b[Range[n]]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 0, 9}] // Flatten (* Jean-François Alcover, Apr 23 2025, after Alois P. Heinz *)

Extensions

One term prepended and one term corrected by Alois P. Heinz, Oct 25 2015