A369321 T(n,k) is the number of length-n weak ascent sequences (prefixed with a zero) with k weak ascents, triangle read by rows.

1, 0, 1, 0, 0, 2, 0, 0, 1, 5, 0, 0, 0, 9, 14, 0, 0, 0, 5, 59, 42, 0, 0, 0, 1, 92, 342, 132, 0, 0, 0, 0, 75, 1073, 1863, 429, 0, 0, 0, 0, 35, 1882, 10145, 9794, 1430, 0, 0, 0, 0, 9, 2131, 31345, 84977, 50380, 4862, 0, 0, 0, 0, 1, 1661, 64395, 417220, 658423, 255606, 16796

Offset: 0

Author

Joerg Arndt, Jan 20 2024

Keywords

• nonn
• tabl

Comments

A weak ascent sequence is a sequence [d(1), d(2), ..., d(n)] where d(1)=0, d(k)>=0, and d(k) <= 1 + asc([d(1), d(2), ..., d(k-1)]) and asc(.) counts the weak ascents d(j) >= d(j-1) of its argument.

Examples

			1,
0, 1,
0, 0, 2,
0, 0, 1, 5,
0, 0, 0, 9, 14,
0, 0, 0, 5, 59,   42,
0, 0, 0, 1, 92,  342,    132,
0, 0, 0, 0, 75, 1073,   1863,     429,
0, 0, 0, 0, 35, 1882,  10145,    9794,     1430,
0, 0, 0, 0,  9, 2131,  31345,   84977,    50380,     4862,
0, 0, 0, 0,  1, 1661,  64395,  417220,   658423,   255606,    16796,
0, 0, 0, 0,  0,  912,  95477, 1370141,  4818426,  4835924,  1285453,   58786,
0, 0, 0, 0,  0,  350, 107002, 3291589, 23507705, 50477693, 34184279, 6428798, 208012,
...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened
• Beata Benyi, Anders Claesson, and Mark Dukes, Weak ascent sequences and related combinatorial structures, arXiv:2111.03159 [math.CO], 2021-2022.

Crossrefs

Cf. A000108 (main diagonal), A336070 (row sums), A369322 (column sums).

T(2n,n) gives A373115.

Cf. A137251.

Programs

Maple

b:= proc(n, i, t) option remember; expand(`if`(n=0, 1, add(
      b(n-1, j, t+`if`(j>=i, 1, 0))*`if`(j>=i, x, 1), j=0..t+1)))
    end:
T:= (n, k)-> coeff(b(n, -1$2), x, k):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Jan 23 2024

Mathematica

b[n_, i_, t_] := b[n, i, t] = Expand[If[n == 0, 1, Sum[
   b[n - 1, j, t + If[j >= i, 1, 0]]*If[j >= i, x, 1], {j, 0, t + 1}]]];
T[n_, k_] := Coefficient[b[n, -1, -1], x, k];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, May 24 2024, after Alois P. Heinz *)

PARI

\\ see formula (5) on page 18 of the Benyi/Claesson/Dukes reference
N=40;
M=matrix(N, N, r, c, -1);  \\ memoization
a(n, k)=
{
    if ( n==0 && k==0, return(1) );
    if ( k==0, return(0) );
    if ( n==0, return(0) );
    if ( M[n, k] != -1 , return( M[n, k] ) );
    my( s );
    s = sum( i=0, n, sum( j=0, k-1,
         (-1)^j * binomial(k-j, i) * binomial(i, j) * a( n-i, k-j-1 )) );
    M[n, k] = s;
    return( s );
}
\\ for (n=0, N, print1( sum(k=1, n, a(n, k)), ", "); ); \\ A336070
for (n=0, N, for(k=0, n, print1(a(n, k), ", "); ); print(); );
\\ Joerg Arndt, Jan 20 2024

Formula

T(n,n) = A000108(n) (number of length-n weak ascent sequences with maximal number of weak ascents).