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

0,6

Comments

Row sums give A000142.

Links

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

FindStat - Combinatorial Statistic Finder, The sum of the descent bottoms of a permutations.

Example

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;
  ...

Maple

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

Crossrefs

Cf. A000142, A263753.

Sequence in context: A257100 A152884 A360288 * A204984 A335341 A243473

Adjacent sequences: A263753 A263754 A263755 * A263757 A263758 A263759

Keyword

nonn,tabf

Author

Christian Stump, Oct 19 2015

Extensions

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

Status

approved