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

Author

Michael De Vlieger, Jul 07 2020

Keywords

• nonn

Comments

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)

Examples

			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)
		

Links

• 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).

Crossrefs

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

Row sums of A369321.

Programs

Maple

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

Mathematica

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 *)

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)),", "); );
\\ 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

Extensions

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