A137251 -id:A137251 - cp's OEIS frontend

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

Joerg Arndt, Jan 20 2024

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.

			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,
...

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.

Cf. A000108 (main diagonal), A336070 (row sums), A369322 (column sums).
T(2n,n) gives A373115.
Cf. A137251.

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

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 *)

\\ 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

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

A357141 Number of n X n triangular matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to 2n.

Original entry on oeis.org

1, 1, 6, 71, 1433, 44443, 1968580, 118159971, 9240555677, 913352224942, 111374887418013, 16428282185046946, 2883740893056526715, 594152447495864629867, 142006380268368661423424, 38973735372090120549251580, 12174162204364698538764222978, 4294569227671480526607187583713

Offset: 0

Alois P. Heinz, Sep 14 2022

nonn

Cf. A137251.

a(n) = A137251(2n,n).

A022494 Number of connected regular linearized chord diagrams of degree n.

Original entry on oeis.org

1, 1, 1, 2, 5, 16, 63, 293, 1561, 9321, 61436, 442134, 3446077, 28905485, 259585900, 2485120780, 25267283367, 271949606805, 3089330120711, 36943477086287, 463943009361687, 6105064699310785, 84011389289865102

Offset: 0

Alexander Stoimenow (stoimeno(AT)math.toronto.edu)

nonn

Gheorghe Coserea, Table of n, a(n) for n = 0..202
A. Stoimenow, Enumeration of chord diagrams and an upper bound for Vassiliev invariants, J. Knot Theory Ramifications, 7 (1998), no. 1, 93-114. [broken link], [DOI]
Don Zagier, Vassiliev invariants and a strange identity related to the Dedekind eta-function, Topology, vol.40, pp.945-960 (2001); see p.955.

Cf. A137251.

A137251_seq(N) = {
  my(x='x + O('x^(N+1)), t='t+O('t^(N+2)), q=1-x, z=1/t-1, p=vector(N+1));
  p[1]=1; for (n=1, #p-1, p[n+1] = p[n] * (1-q^n)/(1+z*q^n));
  apply(p->Vecrev(p), Vec((apply(p->Pol(p,'t), vecsum(p)/(1+z))-'t)/'t^2));
};
A022494_seq(N) = {
  my(s = 't+'t^2*'x*Ser(apply(v->Polrev(v,'t), A137251_seq(N))),
     r = Ser(vector(N+1, n, subst(polcoeff(s, n-1, 't), 'x, 'u + O('u^(N+1)))),'t));
  Vec(1+subst(Pol(t/serreverse(r) - 1,'t),'t,1));
};
A022494_seq(22) \\ Gheorghe Coserea, Nov 01 2017

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A357141 Number of n X n triangular matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to 2n.

Views

Author

Keywords

Crossrefs

Formula

A022494 Number of connected regular linearized chord diagrams of degree n.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI