A294219 Number T(n,k) of ascent sequences of length n where the maximum of 0 and all letter multiplicities equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 9, 4, 1, 0, 1, 26, 20, 5, 1, 0, 1, 82, 97, 30, 6, 1, 0, 1, 276, 496, 191, 42, 7, 1, 0, 1, 1014, 2686, 1259, 310, 56, 8, 1, 0, 1, 4006, 15481, 8784, 2416, 470, 72, 9, 1, 0, 1, 17046, 94843, 65012, 19787, 4141, 677, 90, 10, 1

Offset: 0

Author

Alois P. Heinz, Oct 25 2017

Keywords

• nonn
• tabl

Examples

			T(4,1) = 1: 0123.
T(4,2) = 9: 0011, 0012, 0101, 0102, 0110, 0112, 0120, 0121, 0122.
T(4,3) = 4: 0001, 0010, 0100, 0111.
T(4,4) = 1: 0000.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,     1;
  0, 1,     3,     1;
  0, 1,     9,     4,     1;
  0, 1,    26,    20,     5,     1;
  0, 1,    82,    97,    30,     6,    1;
  0, 1,   276,   496,   191,    42,    7,   1;
  0, 1,  1014,  2686,  1259,   310,   56,   8,  1;
  0, 1,  4006, 15481,  8784,  2416,  470,  72,  9,  1;
  0, 1, 17046, 94843, 65012, 19787, 4141, 677, 90, 10, 1;
  ...
		

Crossrefs

Columns k=0-1 give: A000007, A057427.

Row sums give A022493.

Cf. A294220.

Programs

Maple

b:= proc(n, i, t, p, k) option remember; `if`(n=0, 1,
      add(`if`(coeff(p, x, j)=k, 0, b(n-1, j, t+
          `if`(j>i, 1, 0), p+x^j, k)), j=1..t+1))
    end:
A:= (n, k)-> b(n, 0$3, k):
T:= (n, k)-> A(n, k)-`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=0..n), n=0..10);

Mathematica

b[n_, i_, t_, p_, k_] := b[n, i, t, p, k] = If[n == 0, 1, Sum[If[ Coefficient[p, x, j] == k, 0, b[n - 1, j, t + If[j > i, 1, 0], p + x^j, k]], {j, t + 1}]];
A[n_, k_] :=  b[n, 0, 0, 0, k];
T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 29 2020, after Maple *)

Formula

T(n,k) = A294220(n,k) - A294220(n,k-1) for k>0, T(n,0) = A294220(n,k) = A000007(n).