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
KEYWORD
nonn,tabl
AUTHOR
Joerg Arndt, Jan 20 2024
STATUS
approved