A117156 -id:A117156 - cp's OEIS frontend

A177475 Number of permutations of order n avoiding the consecutive pattern 131'2.

1, 1, 2, 6, 20, 81, 390, 2161, 13678, 96983, 764368, 6630898, 62748250, 643442919, 7104914398, 84062375725, 1060919238874, 14226075039395, 201982580807466, 3027049675655253, 47753241018325280, 790998083929598213, 13726222157931958274, 249018700470309832015, 4714071198944211367704

Offset: 0

Signy Olafsdottir (signy06(AT)ru.is), May 09 2010

nonn

To avoid 131'2 means not to have four consecutive letters such that if the third letter is removed, then in the obtained 3 letter word the smallest letter is the first one, and the largest letter is the second one.

S. Kitaev, Introduction to partially ordered patterns, Discrete Applied Mathematics 155 (2007), 929-944.
S. Kitaev, Introduction to partially ordered patterns.

Cf. A117156, A177470, A177471, A177472, A177473, A177476, A177477, A177478, A177479, A177480, A177481, A177482, A177483, A177484, A376694.

Edited and terms a(0),a(10)-a(24) added by Max Alekseyev, Oct 02 2024

A177476 Number of partitions of order n avoiding the consecutive pattern 231'1.

Original entry on oeis.org

1, 1, 2, 6, 20, 83, 402, 2245, 14192, 100650, 792508, 6859260, 64772648, 662630653, 7301841444, 86212535179, 1085834949064, 14530898302390, 205897508769218, 3079580500287978, 48485072137150344, 801518797091165406, 13881049047327393608, 251325130816997882224, 4748240560493406374592

Offset: 0

Signy Olafsdottir (signy06(AT)ru.is), May 09 2010

nonn

To avoid 231'1 means not to have four consecutive letters such that if the third letter is removed, then in the obtained 3 letter word the smallest letter is the last one, and the largest letter is the second one.

S. Kitaev, Introduction to partially ordered patterns, Discrete Applied Mathematics 155 (2007), 929-944.

Cf. A117156, A177470, A177471, A177472, A177473, A177475, A177477, A177478, A177479, A177480, A177481, A177482, A177483, A177484, A376694.

ok[{x_, y_, , z}] := Not[x>z && y>z && y>x]; a[n_] := Length@ Select[ Permutations@ Range@ n, AllTrue[ Partition[#, 4, 1], ok] &]; a /@ Range[0, 9]

a(0), a(10)-a(14) from Alois P. Heinz, Mar 10 2020
a(15)-a(16) from Giovanni Resta, Mar 11 2020
a(17)-a(24) from Max Alekseyev, Oct 02 2024

A177480 Number of permutations of order n avoiding the consecutive pattern egfh with e

Original entry on oeis.org

1, 1, 2, 6, 20, 84, 412, 2300, 14676, 104536, 825660, 7168860, 67826340, 695174208, 7671602644, 90700227700, 1143825611348, 15325929083336, 217429459642252, 3256039887793868, 51325896829151684, 849518895902379696, 14730333827970237220, 267028337196612514596, 5051094767395355339476

Offset: 0

Signy Olafsdottir (signy06(AT)ru.is), May 09 2010

nonn

To avoid egfh means not to have four consecutive letters such that the first and the second letters are less than the third and the fourth letters.

S. Kitaev, Introduction to partially ordered patterns, Discrete Applied Mathematics 155 (2007), 929-944.

Cf. A117156, A177470, A177471, A177472, A177473, A177475, A177476, A177477, A177478, A177479, A177481, A177482, A177483, A177484, A376694.

ok[{e_, f_, g_, h_}] := e > g || e > h || f > g || f > h; a[n_] := Length@ Select[ Permutations[Range@n], AllTrue[ Partition[#, 4, 1], ok] &]; a /@ Range[0, 9] (* Giovanni Resta, Mar 11 2020 *)

a(0), a(10)-a(14) from Alois P. Heinz, Mar 10 2020
a(15)-a(16) from Giovanni Resta, Mar 11 2020
Edited and a(17)-a(24) added by Max Alekseyev, Oct 01 2024

A177481 Number of permutations of order n avoiding the consecutive pattern efgh with e

Original entry on oeis.org

1, 1, 2, 6, 20, 80, 404, 2368, 15488, 114480, 948992, 8625672, 85223792, 913869056, 10567326528, 130796711016, 1726077013456, 24213357455936, 359694651093152, 5638959807231240, 93052021458248400, 1612444736747193696, 29271726199933801536, 555518182602741687432, 11001032351303890637648

Offset: 0

Signy Olafsdottir (signy06(AT)ru.is), May 09 2010

nonn

To avoid efgh means not to have four consecutive letters such that the first and the third letters are less than the second and the fourth letters.

S. Kitaev, Introduction to partially ordered patterns, Discrete Applied Mathematics 155 (2007), 929-944.

Cf. A117156, A177470, A177471, A177472, A177473, A177475, A177476, A177477, A177478, A177479, A177480, A177482, A177483, A177484, A376694.

ok[{e_,f_,g_,h_}] := e>f || e>h || g>f || g>h; a[n_] := Length@ Select[ Permutations[ Range@ n], AllTrue[ Partition[#, 4, 1], ok] &]; Array[a, 9, 0] (* Giovanni Resta, Mar 11 2020 *)

a(0), a(10)-a(16) from Giovanni Resta, Mar 11 2020

Edited and a(17)-a(24) added by Max Alekseyev, Oct 01 2024

A201692 Number of permutations that avoid the consecutive pattern 1423.

Original entry on oeis.org

1, 1, 2, 6, 23, 110, 631, 4218, 32221, 276896, 2643883, 27768955, 318174363, 3949415431, 52794067318, 756137263377, 11551672922816, 187507250145806, 3222662529113641, 58464560588277289, 1116469710152742025, 22386721651323946628, 470259350616967829363

Offset: 0

N. J. A. Sloane, Dec 03 2011

Alois P. Heinz, Table of n, a(n) for n = 0..250 (terms n = 0..60 from Ray Chandler)
A. Baxter, B. Nakamura, and D. Zeilberger. Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes
V. Dotsenko and A. Khoroshkin, Shuffle algebras, homology, and consecutive pattern avoidance, arXiv preprint arXiv:1109.2690, 2011

Cf. A113228, A113229, A117156, A117158, A117226, A201693.

c := proc(n,l)
    if n = 1 then
        if l = 0 then
            1;
        else
            0;
        end if;
    elif n= 2 or n = 3 then
        0;
    else
        a := 0 ;
        for k from 1 to (n-2)/2 do
            a := a+procname(n-2*k-1,l-k)*binomial(n-k-2,k) ;
        end do:
        a ;
    end if;
end proc:
A201693 := proc(nmax)
    g := 1-t ;
    for n from 2 to nmax do
        for l from 0 to n/2 do
            g := g-c(n,l)*t^n*(-1)^l/n! ;
        end do:
    end do:
    taylor(1/g,t=0,nmax) ;
end proc:
nmax := 25 ;
egf := A201693(nmax) ;
for n from 0 to nmax-1 do
    printf("%d,",coeftayl(egf,t=0,n)*n!) ;
end do: # R. J. Mathar, Dec 04 2011
# second Maple program:
b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      add(b(u-j, o+j-1, `if`(0 b(n, 0$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Nov 07 2013

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, Sum[b[u-j, o+j-1, If[0Jean-François Alcover, Mar 18 2014, after Alois P. Heinz *)

The reference gives an e.g.f. There is an associated triangle of numbers c_{n,l} that should be added to the OEIS if it is not already present.

a(n) ~ c * d^n * n!, where d = 0.95482605094987833345080179991528996596888600981..., c = 1.1567436851576902067739566662625378535625602... . - Vaclav Kotesovec, Sep 11 2014

Definition corrected by N. J. A. Sloane, Mar 15 2015

A201693 Number of permutations that avoid the consecutive pattern 2413.

Original entry on oeis.org

1, 1, 2, 6, 23, 110, 632, 4237, 32465, 279828, 2679950, 28232972, 324470844, 4039771856, 54165468774, 778128659247, 11923645252411, 194131328012012, 3346615262190736, 60897160676005110, 1166446154857250412, 23459656378909613446, 494290181112325561351

Offset: 0

N. J. A. Sloane, Dec 03 2011

Alois P. Heinz, Table of n, a(n) for n = 0..140 (first 61 terms from Ray Chandler)
A. Baxter, B. Nakamura, and D. Zeilberger. Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes
V. Dotsenko and A. Khoroshkin, Shuffle algebras, homology, and consecutive pattern avoidance, arXiv preprint arXiv:1109.2690, 2011

Cf. A113228, A113229, A117156, A117158, A117226, A201692.

The reference gives a g.f. There is an associated triangle of numbers c_{n,l} that should be added to the OEIS if it is not already present.

More terms from Ray Chandler, Dec 06 2011

A231166 Number of permutations of [n] avoiding simultaneously consecutive patterns 1243, 1342, and 1324.

Original entry on oeis.org

1, 1, 2, 6, 21, 91, 467, 2755, 18523, 139740, 1169616, 10763807, 108028386, 1174391384, 13748315494, 172439034531, 2306986699190, 32792999417180, 493559520202535, 7841127918788283, 131127477517244419, 2302491655047553206, 42355105188617740229

Offset: 0

Alois P. Heinz, Nov 04 2013

nonn

			a(4) = 24 - 3 = 21 because 1243, 1342, 1324 are avoided.

Alois P. Heinz, Table of n, a(n) for n = 0..70
A. Baxter, B. Nakamura, and D. Zeilberger, Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes, 2011.
S. Kitaev and T. Mansour, On multi-avoidance of generalized patterns, arXiv:math/0209340 [math.CO], 2002.

Cf. A113228, A117156, A117226.

b:= proc(u, o, s, t) option remember; `if`(u+o=0, 1,
       add(b(u-j, o+j-1, `if`(t>0, t, 0), `if`(t>0, -j, 0)),
           j=`if`(s>0 and t>0,s+t-1,1)..u)+
       add(b(u+j-1, o-j, `if`(t>0, t, 0), +j),
           j=1..`if`(s>0 and t<0 and -t b(n, 0$3):
seq(a(n), n=0..25);

b[u_, o_, s_, t_] := b[u, o, s, t] = If[u + o == 0, 1, Sum[b[u - j, o + j - 1, If[t > 0, t, 0], If[t > 0, -j, 0]], {j, If[s > 0 && t > 0, s + t - 1, 1], u}] + Sum[b[u + j - 1, o - j, If[t > 0, t, 0], +j], {j, 1, If[s > 0 && t < 0 && -t < s, -t - 1, o]}]];
a[n_] := b[n, 0, 0, 0];
a /@ Range[0, 25] (* Jean-François Alcover, Feb 27 2020, after Alois P. Heinz *)

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A177475 Number of permutations of order n avoiding the consecutive pattern 131'2.

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A177476 Number of partitions of order n avoiding the consecutive pattern 231'1.

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A177480 Number of permutations of order n avoiding the consecutive pattern egfh with e

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A177481 Number of permutations of order n avoiding the consecutive pattern efgh with e

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A201692 Number of permutations that avoid the consecutive pattern 1423.

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A201693 Number of permutations that avoid the consecutive pattern 2413.

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A231166 Number of permutations of [n] avoiding simultaneously consecutive patterns 1243, 1342, and 1324.

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