A328441 -id:A328441 - cp's OEIS frontend

A328434 Number of inversion sequences of length n avoiding the consecutive patterns 101, 102, 201, and 210.

1, 1, 2, 6, 21, 81, 346, 1630, 8350, 45958, 269815, 1681285, 11071336, 76743040, 558062437, 4244853573, 33687390663, 278296576327, 2388351295760, 21254019548162, 195801111412320, 1864508416302520, 18326903140310011, 185711672802101781, 1937795878138303715

Offset: 0

Juan S. Auli, Oct 16 2019

nonn

A length n inversion sequence e_1e_2...e_n is a sequence of integers such that 0 <= e_i < i. The term a(n) counts the inversion sequences of length n with no entries e_i, e_{i+1}, e_{i+2} such that e_i > e_{i+1} != e_{i+2}. This is the same as the set of inversion sequences of length n avoiding the consecutive patterns 101, 102, 201, and 210.

			Note that a(4)=21. Indeed, of the 24 inversion sequences of length 4, the only ones that do not avoid the consecutive patterns 101, 102, 201, and 210 are 0101, 0102 and 0103.

Juan S. Auli and Sergi Elizalde, Consecutive patterns in inversion sequences II: avoiding patterns of relations, arXiv:1906.07365 [math.CO], 2019.

Cf. A328357, A328358, A328429, A328430, A328431, A328432, A328433, A328435, A328436, A328437, A328438, A328439, A328440, A328441, A328442.

# after Alois P. Heinz in A328357
b := proc(n, x, t) option remember; `if`(n=0, 1, add(
       `if`(t and i>x, 0, b(n-1, i, i<>x and x>-1)), i=0..n-1))
     end proc:
a := n -> b(n, -1, false):
seq(a(n), n = 0 .. 24);

b[n_, x_, t_] := b[n, x, t] = If[n == 0, 1, Sum[If[t && i > x, 0, b[n - 1, i, i != x && x > -1]], {i, 0, n - 1}]];
a[n_] := b[n, -1, False];
a /@ Range[0, 24] (* Jean-François Alcover, Mar 02 2020 after Alois P. Heinz in A328357 *)

A328435 Number of inversion sequences of length n avoiding the consecutive patterns 101, 102, and 201.

Original entry on oeis.org

1, 1, 2, 6, 21, 83, 368, 1814, 9837, 58095, 370499, 2534374, 18493023, 143280489, 1173971656, 10136279104, 91936857611, 873547634921, 8673546319685, 89796095349193, 967384904147690, 10825116242427973, 125613702370667158, 1509222589338456874, 18748890945849736182

Offset: 0

Juan S. Auli, Oct 17 2019

nonn

A length n inversion sequence e_1e_2...e_n is a sequence of integers such that 0 <= e_i < i. The term a(n) counts the inversion sequences of length n with no entries e_i, e_{i+1}, e_{i+2} such that e_i > e_{i+1} < e_{i+2}. This is the same as the set of inversion sequences of length n avoiding the consecutive patterns 101, 102, and 201.

			Note that a(4)=21. Indeed, of the 24 inversion sequences of length 4, the only ones that do not avoid the consecutive patterns 101, 102, and 201 are 0101, 0102, and 0103.

Juan S. Auli and Sergi Elizalde, Consecutive patterns in inversion sequences II: avoiding patterns of relations, arXiv:1906.07365 [math.CO], 2019.

Cf. A328357, A328358, A328429, A328430, A328431, A328432, A328433, A328434, A328436, A328437, A328438, A328439, A328440, A328441, A328442.

# after Alois P. Heinz in A328357
b := proc(n, x, t) option remember; `if`(n = 0, 1, add(
       `if`(t and x < i, 0, b(n - 1, i, i < x)), i = 0 .. n - 1))
     end proc:
a := n -> b(n, -1, false):
seq(a(n), n = 0 .. 24);

b[n_, x_, t_] := b[n, x, t] = If[n == 0, 1, Sum[If[t && x < i, 0, b[n - 1, i, i < x]], {i, 0, n - 1}]];
a[n_] := b[n, -1, False];
a /@ Range[0, 24] (* Jean-François Alcover, Mar 02 2020 after Alois P. Heinz in A328357 *)

A328436 Number of inversion sequences of length n avoiding the consecutive patterns 000 and 001.

Original entry on oeis.org

1, 1, 2, 3, 9, 37, 190, 1181, 8564, 70914, 659810, 6811371, 77232836, 953969548, 12747856402, 183218649413, 2818050980941, 46182485773217, 803323102085452, 14781372445602234, 286838921699435184, 5854404018902152208, 125367868007259046305, 2810511319383912299122

Offset: 0

Juan S. Auli, Oct 17 2019

nonn

A length n inversion sequence e_1e_2...e_n is a sequence of integers such that 0 <= e_i < i. The term a(n) counts the inversion sequences of length n with no entries e_i, e_{i+1}, e_{i+2} such that e_i = e_{i+1} <= e_{i+2}. This is the same as the set of inversion sequences of length n avoiding the consecutive patterns 000 and 001.

			The a(4)=9 length 4 inversion sequences avoiding the consecutive patterns 000 and 001 are 0100, 0110, 0120, 0101, 0121, 0102, 0122, 0103, and 0123.

Juan S. Auli and Sergi Elizalde, Consecutive patterns in inversion sequences II: avoiding patterns of relations, arXiv:1906.07365 [math.CO], 2019.

Cf. A328357, A328358, A328429, A328430, A328431, A328432, A328433, A328434, A328435, A328437, A328438, A328439, A328440, A328441, A328442.

# after Alois P. Heinz in A328357
b := proc(n, x, t) option remember; `if`(n = 0, 1, add(
       `if`(t and i = x, 0, b(n - 1, i, i <= x)), i = 0 .. n - 1))
     end proc:
a := n -> b(n, -1, false):
seq(a(n), n = 0 .. 24);

b[n_, x_, t_] := b[n, x, t] = If[n == 0, 1, Sum[If[t && i == x, 0, b[n - 1, i, i <= x]], {i, 0, n - 1}]];
a[n_] := b[n, -1, False];
a /@ Range[0, 24] (* Jean-François Alcover, Mar 02 2020 after Alois P. Heinz in A328357 *)

A328438 Number of inversion sequences of length n avoiding the consecutive patterns 000 and 011.

Original entry on oeis.org

1, 1, 2, 4, 13, 57, 304, 1937, 14315, 120264, 1131896, 11794453, 134774963, 1675630582, 22516745452, 325188337067, 5022796990606, 82620491929333, 1441894214312037, 26609607869036180, 517741915593936360, 10592513721179374467, 227325651424365263577, 5106351205789851629476

Offset: 0

Juan S. Auli, Oct 17 2019

nonn

A length n inversion sequence e_1e_2...e_n is a sequence of integers such that 0 <= e_i < i. The term a(n) counts the inversion sequences of length n with no entries e_i, e_{i+1}, e_{i+2} such that e_i <= e_{i+1} = e_{i+2}. This is the same as the set of inversion sequences of length n avoiding the consecutive patterns 000 and 011.

			The a(4)=13 length 4 inversion sequences avoiding the consecutive patterns 000 and 011 are 0100, 0010, 0020, 0120, 0101, 0021, 0121, 0102, 0012, 0103, 0013, 0023, and 0123.

Juan S. Auli and Sergi Elizalde, Consecutive patterns in inversion sequences II: avoiding patterns of relations, arXiv:1906.07365 [math.CO], 2019.

Cf. A328357, A328358, A328429, A328430, A328431, A328432, A328433, A328434, A328435, A328436, A328437, A328439, A328440, A328441, A328442.

# after Alois P. Heinz in A328357
b := proc(n, x, t) option remember; `if`(n = 0, 1, add(
       `if`(t and i <= x, 0, b(n - 1, i, i = x)), i = 0 .. n - 1))
     end proc:
a := n -> b(n, -1, false):
seq(a(n), n = 0 .. 24);

b[n_, x_, t_] := b[n, x, t] = If[n == 0, 1, Sum[If[t && i <= x, 0, b[n - 1, i, i == x]], {i, 0, n - 1}]];
a[n_] := b[n, -1, False];
a /@ Range[0, 24] (* Jean-François Alcover, Mar 02 2020 after Alois P. Heinz in A328357 *)

A328440 Number of inversion sequences of length n avoiding the consecutive patterns 000 and 100.

Original entry on oeis.org

1, 1, 2, 5, 18, 81, 448, 2920, 21955, 186981, 1779170, 18706222, 215364181, 2694650157, 36408144034, 528302958022, 8193953571315, 135277259197031, 2368556730208679, 43838335667451773, 855200666797199814, 17538187897491897945, 377199969925672569364, 8489656058119117230574

Offset: 0

Juan S. Auli, Oct 17 2019

nonn

A length n inversion sequence e_1e_2...e_n is a sequence of integers such that 0 <= e_i < i. The term a(n) counts the inversion sequences of length n with no entries e_i, e_{i+1}, e_{i+2} such that e_i >= e_{i+1} = e_{i+2}. This is the same as the set of inversion sequences of length n avoiding the consecutive patterns 000 and 100.

The term a(n) also counts the inversion sequences of length n with no entries e_i, e_{i+1}, e_{i+2} such that e_i = e_{i+1} >= e_{i+2}. This is the same as the set of inversion sequences of length n avoiding the consecutive patterns 000 and 110, see the Auli and Elizalde reference.

			The a(4)=18 length 4 inversion sequences avoiding the consecutive patterns 000 and 100 are 0010, 0110, 0020, 0120, 0101, 0011, 0021, 0121, 0102, 0012, 0112, 0022, 0122, 0103, 0013, 0113, 0023, and 0123.

Juan S. Auli and Sergi Elizalde, Consecutive patterns in inversion sequences II: avoiding patterns of relations, arXiv:1906.07365 [math.CO], 2019.

Cf. A328357, A328358, A328429, A328430, A328431, A328432, A328433, A328434, A328435, A328436, A328437, A328438, A328439, A328441, A328442.

# after Alois P. Heinz in A328357
b := proc(n, x, t) option remember; `if`(n = 0, 1, add(
       `if`(t and x <= i, 0, b(n - 1, i, i = x)), i = 0 .. n - 1))
     end proc:
a := n -> b(n, -1, false):
seq(a(n), n = 0 .. 24);

b[n_, x_, t_] := b[n, x, t] = If[n == 0, 1, Sum[If[t && x <= i, 0, b[n - 1, i, i == x]], {i, 0, n - 1}]];
a[n_] := b[n, -1, False];
a /@ Range[0, 24] (* Jean-François Alcover, Mar 02 2020 after Alois P. Heinz in A328357 *)

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

Original entry on oeis.org

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

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.

cp's OEIS Frontend

A328434 Number of inversion sequences of length n avoiding the consecutive patterns 101, 102, 201, and 210.

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A328435 Number of inversion sequences of length n avoiding the consecutive patterns 101, 102, and 201.

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A328436 Number of inversion sequences of length n avoiding the consecutive patterns 000 and 001.

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A328438 Number of inversion sequences of length n avoiding the consecutive patterns 000 and 011.

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A328440 Number of inversion sequences of length n avoiding the consecutive patterns 000 and 100.

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

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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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