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

0,6

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.

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.

Formula

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

Example

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,
...

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 *)

Prog

(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

Crossrefs

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

T(2n,n) gives A373115.

Cf. A137251.

Sequence in context: A285638 A325667 A067310 * A122890 A309524 A331106

Adjacent sequences: A369318 A369319 A369320 * A369322 A369323 A369324

Keyword

nonn,tabl

Author

Joerg Arndt, Jan 20 2024

Status

approved