A113227 -id:A113227 - 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

A374545 Number of length n inversion sequences avoiding the patterns 100 and 101.

Original entry on oeis.org

1, 1, 2, 6, 22, 93, 439, 2267, 12628, 75119, 473610, 3146376, 21923158, 159610880, 1210421617, 9536525715, 77885388296, 658112437816, 5743744103287, 51701114086088, 479340725109862, 4572111104329002, 44818226981293308, 451062570619242508, 4656687307250419533

Offset: 0

Benjamin Testart, Jul 12 2024

nonn

Benjamin Testart, Table of n, a(n) for n = 0..580
Benjamin Testart, Completing the enumeration of inversion sequences avoiding one or two patterns of length 3, arXiv:2407.07701 [math.CO], 2024.

Cf. A113227, A263780, A279564, A279570, A374544.

A193593 Augmentation of the triangle A193592. See Comments.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 6, 10, 6, 1, 10, 31, 40, 23, 1, 15, 75, 166, 187, 105, 1, 21, 155, 530, 958, 993, 549, 1, 28, 287, 1415, 3786, 5988, 5865, 3207, 1, 36, 490, 3311, 12441, 28056, 40380, 37947, 20577, 1, 45, 786, 7000, 35469, 109451, 217720, 292092

Offset: 0

Clark Kimberling, Jul 31 2011

For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.

Regarding A193592, (column 1)=A014616, (column 2)=A090809, (right edge)=A113227.

			First 5 rows:
1
1...1
1...3...2
1...6...10...6
1...10..31...40...23
Rows reversed as in Callan's n-edge increasing ordered trees with outdegree k:
 1
0      1
0      1      1
0      2      3      1
0      6     10      6      1
0     23     40     31     10      1
0    105    187    166     75     15      1
0    549    993    958    530    155     21     1
0   3207   5865   5988   3786   1415    287    28    1
0  20577  37947  40380  28056  12441   3311   490   36   1
0 143239 265901 292092 217720 109451  35469  7000  786  45 1

D. Callan, A bijection to count (1-23-4)-avoiding permutations, arXiv:1008.2375 (rows reversed)

Cf. A193091, A193592, A113227 (row sums and diagonal), A090809 (3rd col).

p[n_, 0] := 1; p[n_, k_] := n + 1 - k /; k > 0;
Table[p[n, k], {n, 0, 5}, {k, 0, n}] (* A193592 *)
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 12}]]  (* A193593 *)
Flatten[Table[v[n], {n, 0, 10}]]

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.

A374546 Number of length n inversion sequences avoiding the patterns 110 and 201.

Original entry on oeis.org

1, 1, 2, 6, 23, 103, 512, 2739, 15464, 90964, 552570, 3444473, 21930271, 142105583, 934612523, 6225356918, 41923198109, 285028459807, 1954160675724, 13497387857456, 93843199637004, 656325719024695, 4614695142311098, 32602630763193546, 231343227544240096

Offset: 0

Benjamin Testart, Jul 13 2024

nonn

Benjamin Testart, Table of n, a(n) for n = 0..850
Benjamin Testart, Completing the enumeration of inversion sequences avoiding one or two patterns of length 3, arXiv:2407.07701 [math.CO], 2024.

Cf. A113227, A263777, A279570, A360052, A374543.

A374547 Number of length n inversion sequences avoiding the patterns 101 and 210.

Original entry on oeis.org

1, 1, 2, 6, 23, 103, 513, 2763, 15816, 95109, 595873, 3865396, 25837360, 177275162, 1244621410, 8918587896, 65085848546, 482852672433, 3635811839722, 27749901618649, 214430320809516, 1675833302839457, 13234389150662469, 105525920568921495, 848962775349348810

Offset: 0

Benjamin Testart, Jul 13 2024

nonn

Benjamin Testart, Table of n, a(n) for n = 0..800
Benjamin Testart, Completing the enumeration of inversion sequences avoiding one or two patterns of length 3, arXiv:2407.07701 [math.CO], 2024.

Cf. A113227, A263777, A360052, A374542, A374543.

A374551 Number of length n inversion sequences avoiding the patterns 101 and 120.

Original entry on oeis.org

1, 1, 2, 6, 22, 90, 397, 1859, 9145, 46882, 248899, 1361845, 7650233, 43990073, 258287170, 1545382674, 9406108839, 58155261970, 364774287298, 2318661708634, 14921368078303, 97132997631387, 639118975577638, 4247742637567555, 28499086448886207

Offset: 0

Benjamin Testart, Jul 17 2024

nonn

Benjamin Testart, Table of n, a(n) for n = 0..250
Benjamin Testart, Completing the enumeration of inversion sequences avoiding one or two patterns of length 3, arXiv:2407.07701 [math.CO], 2024.

Cf. A113227, A263778, A374545, A374547, A374548.

cp's OEIS Frontend

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

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A374545 Number of length n inversion sequences avoiding the patterns 100 and 101.

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A193593 Augmentation of the triangle A193592. See Comments.

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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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A374546 Number of length n inversion sequences avoiding the patterns 110 and 201.

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A374547 Number of length n inversion sequences avoiding the patterns 101 and 210.

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A374551 Number of length n inversion sequences avoiding the patterns 101 and 120.

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