A071075 -id:A071075 - 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.

A071076 Number of permutations that avoid the generalized pattern 123-4.

Original entry on oeis.org

1, 1, 2, 6, 23, 108, 598, 3815, 27532, 221708, 1970251, 19150132, 202064380, 2300071071, 28092017668, 366425723926, 5083645400819, 74745472084176, 1160974832572274, 18995175706664735, 326531476287842760, 5883736110875887560, 110893188848753125475

Offset: 0

Sergey Kitaev, May 26 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450
A. M. Baxter, Algorithms for Permutation Statistics, Ph. D. Dissertation, Rutgers University, May 2011.
Andrew M. Baxter and Lara K. Pudwell, Enumeration schemes for dashed patterns, arXiv preprint arXiv:1108.2642 [math.CO], 2011.
Sergey Kitaev, Partially Ordered Generalized Patterns, preprint.
Sergey Kitaev, Partially Ordered Generalized Patterns, Discrete Math. 298 (2005), no. 1-3, 212-229.
Yan Wang, Qi Fang, Shishuo Fu, Sergey Kitaev, and Haijun Li, Consecutive and quasi-consecutive patterns: des-Wilf classifications and generating functions, arXiv:2502.10128 [math.CO], 2025. See p. 9.

Cf. A049774, A071088, A071075, A071077.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1, add(
      `if`(t=1 and o>j, 0, b(u+j-1, o-j, t+1)), j=1..o)+
       add(b(u-j, o+j-1, 0), j=1..u))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Nov 14 2015

b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, Sum[If[t == 1 && o > j, 0, b[u + j - 1, o - j, t + 1]], {j, 1, o}] + Sum[b[u - j, o + j - 1, 0], {j, 1, u}]]; a[n_] := b[n, 0, 0]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 01 2016, after Alois P. Heinz *)

E.g.f.: exp(int(A(y), y=0..x)), where A(y) = (sqrt(3)/2)*exp(y/2)/cos((sqrt(3)/2)*y + Pi/6).

Let b(n) = A049774(n) = number of permutations of [n] that avoid the consecutive pattern 123. Then a(n) = Sum_{i = 0..n-1} binomial(n-1,i)*b(i)*a(n-1-i) with a(0) = b(0) = 1. [See the recurrence for A_n and B_n in the proof of Theorem 13 in Kitaev's papers.] -

More terms from Vladeta Jovovic, May 28 2002
Link added by Andrew Baxter, May 17 2011
Typos in formula corrected by Vaclav Kotesovec, Aug 23 2014

A071077 Number of permutations that avoid the generalized pattern 1234-5.

Original entry on oeis.org

1, 1, 2, 6, 24, 119, 705, 4857, 38142, 336291, 3289057, 35337067, 413698248, 5241768017, 71465060725, 1043175024243, 16231998346794, 268207096127991, 4690005160446721, 86528908665043683, 1679764981327051508, 34226671269330933413, 730361830628447403029

Offset: 0

Sergey Kitaev, May 26 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450
Sergey Kitaev, Partially Ordered Generalized Patterns, preprint.
Sergey Kitaev, Partially Ordered Generalized Patterns, Discrete Math. 298 (2005), no. 1-3, 212-229.

Cf. A071088, A071075, A071076, A117158.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1, add(
      `if`(t=2 and o>j, 0, b(u+j-1, o-j, t+1)), j=1..o)+
       add(b(u-j, o+j-1, 0), j=1..u))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Nov 14 2015

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, Sum[If[t == 2 && o>j, 0, b[u+j-1, o-j, t+1]], {j, 1, o}] + Sum[b[u-j, o+j-1, 0], {j, 1, u}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 25] (* Jean-François Alcover, Apr 23 2020, after Alois P. Heinz *)

E.g.f.: exp(int(A(y), y=0..x)), where A(y) = 1/(Sum_{i>=0} y^{4*i}/(4*i)! - Sum_{i>=0} y^{4*i+1}/(4*i+1)!).

Let b(n) = A117158(n) = number of permutations of [n] that avoid the consecutive pattern 1234. Then a(n) = Sum_{i = 0..n-1} binomial(n-1,i)*b(i)*a(n-1-i) with a(0) = b(0) = 1. [See the recurrence for A_n and B_n in the proof of Theorem 13 in Kitaev's papers.] - Petros Hadjicostas, Oct 31 2019

Corrected and extended by Vladeta Jovovic, May 28 2002

A071088 Number of permutations that avoid the generalized pattern 12345-6.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 719, 5022, 40064, 359400, 3580896, 39233867, 468818397, 6067548429, 84551873634, 1262188317534, 20095114167065, 339883289813330, 6086154606429378, 115025120586250896, 2288119443771888504, 47787869441095495395, 1045507132393256095282

Offset: 0

Sergey Kitaev, May 26 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450
Sergey Kitaev, Partially Ordered Generalized Patterns, preprint.
Sergey Kitaev, Partially Ordered Generalized Patterns, Discrete Math. 298 (2005), no. 1-3, 212-229.

Cf. A071075, A071076, A071077, A177523.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1, add(
      `if`(t=3 and o>j, 0, b(u+j-1, o-j, t+1)), j=1..o)+
       add(b(u-j, o+j-1, 0), j=1..u))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Nov 14 2015

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, Sum[If[t == 3 && o > j, 0, b[u+j-1, o-j, t+1]], {j, 1, o}] + Sum[b[u-j, o+j-1, 0], {j, 1, u}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 25] (* Jean-François Alcover, Nov 02 2020, after Alois P. Heinz *)

E.g.f.: exp(int(A(y), y=0..x)), where A(y) = 1/(Sum_{i>=0} y^{5*i}/(5*i)! - Sum_{i>=0} y^{5*i+1}/(5*i+1)!).

Let b(n) = A177523(n) = number of permutations of [n] that avoid the consecutive pattern 12345. Then a(n) = Sum_{i = 0..n-1} binomial(n-1,i)*b(i)*a(n-1-i) with a(0) = b(0) = 1. [See the recurrence for A_n and B_n in the proof of Theorem 13 in Kitaev's papers.] - Petros Hadjicostas, Nov 01 2019

More terms from Vladeta Jovovic, May 28 2002

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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A071076 Number of permutations that avoid the generalized pattern 123-4.

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A071077 Number of permutations that avoid the generalized pattern 1234-5.

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A071088 Number of permutations that avoid the generalized pattern 12345-6.

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

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Maple

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

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Maple

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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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Maple