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

0,9

Comments

Row sums give A000142.

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

Links

Alois P. Heinz, Rows n = 0..23, flattened

FindStat - Combinatorial Statistic Finder, The sum of the descent tops of a permutation

Example

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;
  ...

Maple

b:= proc(s) option remember; (n-> `if`(n=0, 1, expand(
      add(b(s minus {j})*`if`(j<n, x^n, 1), j=s))))(nops(s))
    end:
T:= n-> (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 *)

Crossrefs

Cf. A000142, A263756.

Sequence in context: A395033 A317595 A385860 * A303877 A112743 A230427

Adjacent sequences: A263750 A263751 A263752 * A263754 A263755 A263756

Keyword

nonn,tabf

Author

Christian Stump, Oct 19 2015

Extensions

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

Status

approved