A336070 -id:A336070 - 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).

A336071 Number of inversion sequences avoiding the vincular pattern 1-01 (or 1-10).

Original entry on oeis.org

1, 2, 6, 23, 107, 584, 3655, 25790, 202495, 1750763

Offset: 1

Michael De Vlieger, Jul 07 2020

Juan S. Auli, Sergi Elizalde, Wilf equivalences between vincular patterns in inversion sequences, arXiv:2003.11533 [math.CO], 2020. See p. 5, Table 1. Gives terms 1-10.

Cf. A000079, A000110, A022493, A091768, A102038, A113227, A263777, A328441, A336070, A336072.

A336072 Number of inversion sequences avoiding the vincular pattern 2-01 (or 2-10).

Original entry on oeis.org

1, 2, 6, 24, 118, 680, 4460, 32634, 262536, 2296532

Offset: 1

Michael De Vlieger, Jul 07 2020

Juan S. Auli, Sergi Elizalde, Wilf equivalences between vincular patterns in inversion sequences, arXiv:2003.11533 [math.CO], 2020. See p. 5, Table 1. Gives terms 1-10.

Cf. A000079, A000110, A022493, A091768, A102038, A113227, A263777, A328441, A336070, A336071.

A369322 a(n) is the number of weak ascent sequences (of any length) with n weak ascents.

Original entry on oeis.org

1, 1, 3, 20, 285, 8498, 521549, 65149296, 16446593964, 8354292354562, 8517018874559019, 17400156347544892896, 71175200852044807325678, 582639858848549658827324726, 9542182685892187892079287210803, 312611431819035281373960038697247872

Offset: 0

Joerg Arndt, Jan 20 2024

nonn

Column sums of A369321.

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.

Alois P. Heinz, Table of n, a(n) for n = 0..50
Beata Benyi, Anders Claesson, Mark Dukes, Weak ascent sequences and related combinatorial structures, arXiv:2111.03159 [math.CO], (4-November-2021).

Cf. A369321, A336070.

b:= proc(n, i, t, k) option remember;
     `if`(k<0, 0,  `if`(n=0, `if`(k=0, 1, 0), add((d->
        b(n-1, j, t+d, k-d))(`if`(j>=i, 1, 0)), j=0..t+1)))
    end:
a:= n-> add(b(j, -1$2, n), j=n..n*(n+1)/2):
seq(a(n), n=0..15);  # Alois P. Heinz, Jan 25 2024

b[n_, i_, t_, k_] := b[n, i, t, k] = If[k < 0, 0, If[n == 0, If[k == 0, 1, 0], Sum[Function[d, b[n-1, j, t+d, k-d]][If[j >= i, 1, 0]], {j, 0, t+1}]]];
a[n_] := Sum[b[j, -1, -1, n], {j, n, n*(n+1)/2}];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Jul 07 2025, after Alois P. Heinz *)

A369476 Total number of weak ascents in all length-n weak ascent sequences.

Original entry on oeis.org

0, 1, 4, 17, 83, 461, 2873, 19846, 150418, 1240398, 11051017, 105740309, 1081101474, 11758967146, 135544030566, 1650178088102, 21155166649234, 284821038726404, 4017445572746953, 59238368957321225, 911319667593472401, 14600491369553323529, 243205500769215401307

Offset: 0

Alois P. Heinz, Jan 23 2024

nonn

See A369321 for definitions.

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. A336070, A369321.

b:= proc(n, i, t) option remember; `if`(n=0, 1,
      add(b(n-1, j, t+`if`(j>=i, 1, 0)), j=0..t+1))
    end:
a:= n-> b(n+1, -1$2)-b(n, -1$2):
seq(a(n), n=0..23);

a(n) = Sum_{k=0..n} k * A369321(n,k).

a(n) = A336070(n+1) - A336070(n).

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.

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A336071 Number of inversion sequences avoiding the vincular pattern 1-01 (or 1-10).

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A336072 Number of inversion sequences avoiding the vincular pattern 2-01 (or 2-10).

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A369322 a(n) is the number of weak ascent sequences (of any length) with n weak ascents.

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A369476 Total number of weak ascents in all length-n weak ascent sequences.

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