A369321 -id:A369321 - cp's OEIS frontend

A336070 Number of inversion sequences avoiding the vincular pattern 10-0 (or 10-1).

1, 1, 2, 6, 23, 106, 567, 3440, 23286, 173704, 1414102, 12465119, 118205428, 1199306902, 12958274048, 148502304614, 1798680392716, 22953847041950, 307774885768354, 4325220458515307, 63563589415836532, 974883257009308933, 15575374626562632462, 258780875395778033769, 4464364292401926006220

Offset: 0

Michael De Vlieger, Jul 07 2020

nonn

From Joerg Arndt, Jan 20 2024: (Start)
a(n) is the number of weak ascent sequences of length n.
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.
The number of length-n weak ascent sequences with maximal number of weak ascents is A000108(n).
(End)

			From _Joerg Arndt_, Jan 20 2024: (Start)
There are a(4) = 23 weak ascent sequences (dots for zeros):
   1:  [ . . . . ]
   2:  [ . . . 1 ]
   3:  [ . . . 2 ]
   4:  [ . . . 3 ]
   5:  [ . . 1 . ]
   6:  [ . . 1 1 ]
   7:  [ . . 1 2 ]
   8:  [ . . 1 3 ]
   9:  [ . . 2 . ]
  10:  [ . . 2 1 ]
  11:  [ . . 2 2 ]
  12:  [ . . 2 3 ]
  13:  [ . 1 . . ]
  14:  [ . 1 . 1 ]
  15:  [ . 1 . 2 ]
  16:  [ . 1 1 . ]
  17:  [ . 1 1 1 ]
  18:  [ . 1 1 2 ]
  19:  [ . 1 1 3 ]
  20:  [ . 1 2 . ]
  21:  [ . 1 2 1 ]
  22:  [ . 1 2 2 ]
  23:  [ . 1 2 3 ]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..400
Juan S. Auli and 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.
Beata Benyi, Anders Claesson, and Mark Dukes, Weak ascent sequences and related combinatorial structures, arXiv:2111.03159 [math.CO], (4-November-2021).

Cf. A000079, A000108, A000110, A022493, A091768, A102038, A113227, A263777, A328441, A336071, A336072.

Row sums of 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$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 23 2024

b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, Sum[b[n - 1, j, t + If[j >= i, 1, 0]], {j, 0, t + 1}]];
a[n_] := b[n, -1, -1];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jan 18 2025, 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)),", "); );
\\ print triangle a(n,k), see A369321:
\\ for (n=0, N, for(k=0,n, print1(a(n,k),", "); ); print(););
\\ Joerg Arndt, Jan 20 2024

a(0)=1 prepended and more terms from Joerg Arndt, Jan 20 2024

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 *)

A373115 Number weak ascent sequences of length 2n (prefixed with a zero) with exactly n weak ascents.

Original entry on oeis.org

1, 0, 0, 1, 35, 1661, 107002, 9047970, 972937247, 129603346139, 20934881571217, 4029204458109445, 910549073414709876, 238643240329544375336, 71772700696174395158056, 24544642886642172762170933, 9468192975202745545226891834, 4090995487728206638560153282674

Offset: 0

Alois P. Heinz, May 25 2024

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..70
Beata Benyi, Anders Claesson, and Mark Dukes, Weak ascent sequences and related combinatorial structures, arXiv:2111.03159 [math.CO], 2021-2022.

Cf. A357141, A369321.

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:
a:= n-> coeff(b(2*n, -1$2), x, n):
seq(a(n), n=0..17);

a(n) = A369321(2n,n).

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).

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A336070 Number of inversion sequences avoiding the vincular pattern 10-0 (or 10-1).

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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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A373115 Number weak ascent sequences of length 2n (prefixed with a zero) with exactly n weak ascents.

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Maple

Formula

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

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