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
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
Programs
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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).
Comments