A200404 -id:A200404 - cp's OEIS frontend

A240885 Decimal expansion of the unique positive solution of Integral_{0..x} exp(-t^2/2) dt = 1.

1, 2, 7, 5, 5, 4, 7, 7, 3, 6, 4, 1, 7, 2, 1, 5, 3, 7, 8, 8, 0, 1, 3, 4, 3, 1, 9, 7, 4, 6, 7, 8, 5, 4, 7, 9, 0, 7, 3, 0, 7, 8, 1, 4, 3, 7, 4, 9, 4, 7, 2, 6, 1, 4, 3, 9, 4, 4, 8, 7, 3, 2, 6, 4, 6, 3, 1, 6, 4, 6, 9, 2, 5, 6, 4, 3, 0, 0, 8, 6, 4, 1, 6, 0, 4, 6, 2, 5, 2, 7, 5, 9, 5, 4, 2, 9, 3, 4, 5, 6

Offset: 1

Jean-François Alcover, Apr 14 2014

			1.2755477364172...

Steven Finch, Pattern-Avoiding Permutations [Internet archive]
Steven Finch, Pattern-Avoiding Permutations [Cached copy, with permission]

Cf. A052319, A071075, A111004, A200404, A263885.

RealDigits[Sqrt[2]*InverseErf[Sqrt[2/Pi]], 10, 100] // First

a=sqrt(Pi/2); b=a-1; c=1/sqrt(2); solve(x=1,2, a*erfc(c*x)-b) \\ Charles R Greathouse IV, Sep 02 2024

Solution to sqrt(Pi/2)*erf(x/sqrt(2)) = 1.

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

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

Original entry on oeis.org

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

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

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

cp's OEIS Frontend

A240885 Decimal expansion of the unique positive solution of Integral_{0..x} exp(-t^2/2) dt = 1.

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

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

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

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

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