Juan S. Auli - cp's OEIS frontend

A328500 Number of inversion sequences of length n avoiding the consecutive pattern 102.

1, 1, 2, 6, 22, 96, 492, 2902, 19350, 143918, 1181540, 10614698, 103589738, 1091367634, 12346368424, 149276823258, 1921099070062, 26220186000950, 378308908684300, 5753387612678314, 91988260677198002, 1542570178562361018, 27072325866355742048

Offset: 0

Vaclav Kotesovec and Juan S. Auli, Oct 17 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..250
Juan S. Auli, Pattern Avoidance in Inversion Sequences, Ph. D. thesis, Dartmouth College, ProQuest Dissertations Publishing (2020), 27964164.
Juan S. Auli, Sergi Elizalde, Consecutive Patterns in Inversion Sequences, arXiv:1904.02694 [math.CO], 2019. See Table 3.

Cf. A049774, A071075, A200404.

b:= proc(n, j, t) option remember; `if`(n=0, 1, add(
      `if`(i<=j or i>=t, b(n-1, i, j), 0), i=1..n))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 18 2019

b[n_, j_, t_] := b[n, j, t] = If[n == 0, 1, Sum[If[i <= j || i >= t, b[n - 1, i, j], 0], {i, 1, n}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 25] (* Jean-François Alcover, Mar 01 2020, after Alois P. Heinz *)

a(n) ~ n! * c * d^n * n^alfa, where d = 1/A240885 = 1/(sqrt(2) * InverseErf(sqrt(2/Pi))), alfa = 0.294868853646259565..., c = 2.22826071050847602... - Vaclav Kotesovec, Oct 19 2019

A328507 Number of inversion sequences of length n avoiding the consecutive pattern 101.

Original entry on oeis.org

1, 1, 2, 6, 23, 109, 619, 4113, 31352, 269841, 2589026, 27404677, 317265161, 3988181568, 54099618419, 787705115000, 12253696410675, 202831037178017, 3559585021719875, 66018657264425355, 1290284788431977106, 26505045303122642171, 570918508059059670322

Offset: 0

Vaclav Kotesovec and Juan S. Auli, Oct 17 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..250
Juan S. Auli, Pattern Avoidance in Inversion Sequences, Ph. D. thesis, Dartmouth College, ProQuest Dissertations Publishing (2020), 27964164.
Juan S. Auli, Sergi Elizalde, Consecutive Patterns in Inversion Sequences, arXiv:1904.02694 [math.CO], 2019. See Table 4.

Cf. A049774, A052169, A071075, A200404.

b:= proc(n, j, t) option remember; `if`(n=0, 1, add(
      `if`(i<=j or i<>t, b(n-1, i, j), 0), i=1..n))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 18 2019

b[n_, j_, t_] := b[n, j, t] = If[n == 0, 1, Sum[If[i <= j || i != t, b[n-1, i, j], 0], {i, 1, n}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 25] (* Jean-François Alcover, Mar 19 2020, after Alois P. Heinz *)

a(n) ~ n! * c / sqrt(n), where c = 2.48988835987151440021135203237... - Vaclav Kotesovec, Oct 19 2019

A328504 Number of inversion sequences of length n avoiding the consecutive pattern 010.

Original entry on oeis.org

1, 1, 2, 5, 17, 76, 417, 2701, 20199, 171329, 1624851, 17036586, 195685618, 2443572835, 32959210808, 477542545691, 7396931591165, 121976733648960, 2133460758692093, 39450254899737811, 768950119933799815, 15757352298761474101, 338663233082663363407

Offset: 0

Vaclav Kotesovec and Juan S. Auli, Oct 17 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..250
Juan S. Auli, Pattern Avoidance in Inversion Sequences, Ph. D. thesis, Dartmouth College, ProQuest Dissertations Publishing (2020), 27964164.
Juan S. Auli, Sergi Elizalde, Consecutive Patterns in Inversion Sequences, arXiv:1904.02694 [math.CO], 2019. See Table 4.

Cf. A049774, A052169, A071075, A200404.

b:= proc(n, j, t) option remember; `if`(n=0, 1, add(
      `if`(i>=j or i<>t, b(n-1, i, j), 0), i=1..n))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 18 2019

b[n_, j_, t_] := b[n, j, t] = If[n == 0, 1, Sum[If[i >= j || i != t, b[n - 1, i, j], 0], {i, 1, n}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 25] (* Jean-François Alcover, Mar 12 2020, after Alois P. Heinz *)

a(n) ~ n! * c / sqrt(n), where c = 1.410641128930866501817126119... - Vaclav Kotesovec, Oct 19 2019

A328501 Number of inversion sequences of length n avoiding the consecutive pattern 201.

Original entry on oeis.org

1, 1, 2, 6, 24, 118, 684, 4548, 34036, 282696, 2577936, 25589100, 274539856, 3164909164, 39006958856, 511759353776, 7120140764224, 104703385864788, 1622530610142744, 26425922582118000, 451264786489454168, 8062192403534869432, 150395837509736576208

Offset: 0

Vaclav Kotesovec and Juan S. Auli, Oct 17 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..465
Juan S. Auli, Pattern Avoidance in Inversion Sequences, Ph. D. thesis, Dartmouth College, ProQuest Dissertations Publishing (2020), 27964164.
Juan S. Auli, Sergi Elizalde, Consecutive Patterns in Inversion Sequences, arXiv:1904.02694 [math.CO], 2019. See Table 3.

Cf. A049774, A071075, A200404.

a(n) ~ n! * c * d^n * n^alfa, where d = 1/A240885 = 1/(sqrt(2) * InverseErf(sqrt(2/Pi))) = 0.783976931203547499124248654869812535747328200022..., alfa = 1.9218908815253415257398764962146978742409244378248756048362586275529..., c = 0.05831456121798260255226478044037424484656774525125436523149657... - Vaclav Kotesovec, Oct 18 2019

A328441 Number of inversion sequences of length n avoiding the consecutive pattern 100.

Original entry on oeis.org

1, 1, 2, 6, 23, 109, 618, 4098, 31173, 267809, 2565520, 27120007, 313616532, 3938508241, 53381045786, 776672993274, 12074274033482, 199746500391688, 3503656507826887, 64951437702821877, 1268898555348831913, 26055882443142671038, 561050228044941209930, 12641053014560238560492, 297439800300471548183778

Offset: 0

Juan S. Auli, Oct 17 2019

nonn

A length n inversion sequence e_1, e_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}. That is, a(n) counts the inversion sequences of length n avoiding the consecutive pattern 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}. That is, a(n) also counts the inversion sequences of length n avoiding the consecutive pattern 110, see the Auli and Elizalde links.

			Note that a(4)=23. Indeed, of the 24 inversion sequences of length 4, the only one that does not avoid the consecutive pattern 100 is 0100.
Similarly, 0110 is the only inversion sequence of length 4 that does not avoid the consecutive pattern 110.

Vaclav Kotesovec, Table of n, a(n) for n = 0..448
Juan S. Auli, Pattern Avoidance in Inversion Sequences, Ph. D. thesis, Dartmouth College, ProQuest Dissertations Publishing (2020), 27964164.
Juan S. Auli and Sergi Elizalde, Consecutive Patterns in Inversion Sequences, arXiv:1904.02694 [math.CO], 2019.
Juan S. Auli and Sergi Elizalde, Consecutive patterns in inversion sequences II: avoiding patterns of relations, arXiv:1906.07365 [math.CO], 2019.
Juan S. Auli and Sergi Elizalde, Wilf equivalences between vincular patterns in inversion sequences, arXiv:2003.11533 [math.CO], 2020.

Cf. A328357, A328358, A328429, A328430, A328431, A328432, A328433, A328434, A328435, A328436, A328437, A328438, A328439, A328440, A328442

# after Alois P. Heinz in A328357
b := proc(n, x, t) local i; 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);

i100[1] = 1; i100[2] = 2; i100[n_] := i100[n] = Sum[s100[n, k], {k, 0, n - 1}]; s100[n_, k_] := s100[n, k] = i100[n - 1] - Sum[s100[n - 2, j], {j, k + 1, n - 3}]; Flatten[{1, Table[i100[m], {m, 1, 25}]}] (* Vaclav Kotesovec, Oct 18 2019 *)

a(n) ~ n! * c / sqrt(n), where c = 2.428754692682297906864850201408427747198... - Vaclav Kotesovec, Oct 18 2019

A328442 Number of inversion sequences of length n avoiding the consecutive pattern 210.

Original entry on oeis.org

1, 1, 2, 6, 24, 118, 684, 4554, 34192, 285558, 2624496, 26315990, 285828324, 3342566724, 41869664320, 559265742918, 7934746600620, 119162454310392, 1888417811354292, 31492626988890798, 551302582228438512, 10107905106374914860, 193700015975819881008, 3872391687779493752340, 80623321999146782133372

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}. That is, a(n) counts the inversion sequences of length n avoiding the consecutive pattern 210.

			Note that a(5)=118. Indeed, of the 120 inversion sequences of length 5, the only ones that do not avoid the consecutive patterns 210 are 00210 and 01210.

Vaclav Kotesovec, Table of n, a(n) for n = 0..460
Juan S. Auli, Pattern Avoidance in Inversion Sequences, Ph. D. thesis, Dartmouth College, ProQuest Dissertations Publishing (2020), 27964164.
Juan S. Auli and Sergi Elizalde, Consecutive Patterns in Inversion Sequences, arXiv:1904.02694 [math.CO], 2019.
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, A328440, A328441.

# 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, x < i)), i = 0 .. n - 1))
     end proc:
a := n -> b(n, n, 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, x < i]], {i, 0, n - 1}]];
a[n_] := b[n, n, False];
a /@ Range[0, 24] (* Jean-François Alcover, Mar 02 2020, after Alois P. Heinz in A328357 *)

a(n) ~ n! * c * (3^(3/2)/(2*Pi))^n * n^(2*Pi/3^(3/2)), where c = 0.24427562500895080639039917229089... - Vaclav Kotesovec, Oct 19 2019

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 *)

A328439 Number of inversion sequences of length n avoiding the consecutive pattern 011.

Original entry on oeis.org

1, 1, 2, 5, 17, 75, 407, 2621, 19524, 165090, 1561900, 16345264, 187452475, 2337729329, 31497068553, 455930417721, 7056447326642, 116279714536838, 2032547040624336, 37563420959431569, 731810131489893185, 14989602024463575408, 322032777284323744894, 7240745954488939549295

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}. That is, a(n) counts the inversion sequences of length n avoiding the consecutive pattern 011.

			The a(4)=17 length 4 inversion sequences avoiding the consecutive pattern 011 are 0000, 0100, 0010, 0020, 0120, 0001, 0101, 0021, 0121, 0002, 0102, 0012, 0003, 0103, 0013, 0023, and 0123.

Vaclav Kotesovec, Table of n, a(n) for n = 0..448
Juan S. Auli, Pattern Avoidance in Inversion Sequences, Ph. D. thesis, Dartmouth College, ProQuest Dissertations Publishing (2020), 27964164.
Juan S. Auli and Sergi Elizalde, Consecutive Patterns in Inversion Sequences, arXiv:1904.02694 [math.CO], 2019.
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, 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 *)

a(n) ~ n! * c / sqrt(n), where c = 1.3306953765239857433314976921138998977998... - Vaclav Kotesovec, Oct 19 2019

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 *)

A328437 Number of inversion sequences of length n avoiding the consecutive pattern 001.

Original entry on oeis.org

1, 1, 2, 4, 11, 42, 210, 1292, 9352, 77505, 722294, 7470003, 84854788, 1049924370, 14052654158, 202271440732, 3115338658280, 51118336314648, 890201500701303, 16397264064993185, 318505677099378561, 6506565509515408206, 139449260758011488550, 3128599281190613701180

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}. That is, a(n) counts the inversion sequences of length n avoiding the consecutive pattern 001.

			The a(4)=11 length 4 inversion sequences avoiding the consecutive pattern 001 are 0000, 0100, 0110, 0120, 0101, 0111, 0121, 0102, 0122, 0103, and 0123.

Vaclav Kotesovec, Table of n, a(n) for n = 0..448
Juan S. Auli, Pattern Avoidance in Inversion Sequences, Ph. D. thesis, Dartmouth College, ProQuest Dissertations Publishing (2020), 27964164.
Juan S. Auli and Sergi Elizalde, Consecutive Patterns in Inversion Sequences, arXiv:1904.02694 [math.CO], 2019.
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, 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 *)

a(n) ~ n! * c / sqrt(n), where c = 0.549342310436989831962783548104445992522... - Vaclav Kotesovec, Oct 18 2019

cp's OEIS Frontend

User: Juan S. Auli

A328500 Number of inversion sequences of length n avoiding the consecutive pattern 102.

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A328507 Number of inversion sequences of length n avoiding the consecutive pattern 101.

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A328504 Number of inversion sequences of length n avoiding the consecutive pattern 010.

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A328501 Number of inversion sequences of length n avoiding the consecutive pattern 201.

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A328441 Number of inversion sequences of length n avoiding the consecutive pattern 100.

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A328442 Number of inversion sequences of length n avoiding the consecutive pattern 210.

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

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A328439 Number of inversion sequences of length n avoiding the consecutive pattern 011.

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