A117156 -id:A117156 - cp's OEIS frontend

A117158 Number of permutations avoiding the consecutive pattern 1234.

1, 1, 2, 6, 23, 111, 642, 4326, 33333, 288901, 2782082, 29471046, 340568843, 4263603891, 57482264322, 830335952166, 12793889924553, 209449977967081, 3630626729775362, 66429958806679686, 1279448352687538463, 25874432578888440471, 548178875969847203202

Offset: 0

Steven Finch, Apr 26 2006

nonn

a(n) is the number of permutations on [n] that avoid the consecutive pattern 1234. It is the same as the number of permutations which avoid 4321.

F. N. David and D. E. Barton, Combinatorial Chance, Hafner, New York, 1962, pages 156-157.

Alois P. Heinz, Table of n, a(n) for n = 0..450 (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.
Sergi Elizalde, Asymptotic enumeration of permutations avoiding generalized patterns, Adv. Appl. Math. 36 (2006), 138-155.
Sergi Elizalde and Marc Noy, Consecutive patterns in permutations, Adv. Appl. Math. 30 (2003), 110-125.
Steven Finch, Pattern-Avoiding Permutations. [Archived version]
Steven Finch, Pattern-Avoiding Permutations. [Cached copy, with permission]
Ira M. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory A 53 (1990), 257-285.
Ira M. Gessel, Yan Zhuang, Counting permutations by alternating descents, arXiv:1408.1886 [math.CO], 2014. See displayed equation before Eq. (3) and set m=4. - _N. J. A. Sloane_, Aug 11 2014
Kaarel Hänni, Asymptotics of descent functions, arXiv:2011.14360 [math.CO], Nov 29 2020, p. 14.
Mingjia Yang, Doron Zeilberger, Increasing Consecutive Patterns in Words, arXiv:1805.06077 [math.CO], 2018.
Mingjia Yang, An experimental walk in patterns, partitions, and words, Ph. D. Dissertation, Rutgers University (2020).

Cf. A022558, A049774, A111004, A113228, A113229, A117156, A117226, A177523, A177533, A201692, A201693, A230051, A324132.
Column k=0 of A220183.
Column k=7 of A242784.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t<2, add(b(u+j-1, o-j, t+1), j=1..o), 0)+
      add(b(u-j, o+j-1, 0), j=1..u))
    end:
a:= n-> b(n, 0, 0):
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 07 2013

a[n_]:=Coefficient[Series[2/(Cos[x]-Sin[x]+Exp[ -x]),{x,0,30}],x^n]*n!
(* second program: *)
b[u_, o_, t_] := b[u, o, t] = If[u+o==0, 1, If[t<2, Sum[b[u+j-1, o-j, t+1], {j, 1, o}], 0] + Sum[b[u-j, o+j-1, 0], {j, 1, u}]]; a[n_] := b[n, 0, 0]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 23 2015, after Alois P. Heinz *)

From Sergei N. Gladkovskii, Nov 30 2011: (Start)
E.g.f.: 2/(exp(-x) + cos(x) - sin(x)) = 1/W(0) with continued fraction
W(k) = 1 + (x^(2*k))/(f + f*x/(4*k + 1 - x - (4*k + 1)*b/R)), where R := x^(2*k) + b -(x^(4*k+1))/(c + (x^(2*k+1)) + x*c/T); T := 4*k + 3 - x - (4*k + 3)*d/(d +(x^(2*k+1))/W(k+1)), and
f := (4*k)!/(2*k)!; b := (4*k + 1)!/(2*k + 1)!; c := (4*k + 2)!/(2*k + 1)!; and d :=(4*k + 3)!/(2*k + 2)!. (End)
a(n) ~ n! / (sin(r)*r^(n+1)), where r = 1.0384156372665563... is the root of the equation exp(-r) + cos(r) = sin(r). - Vaclav Kotesovec, Dec 11 2013

A022558 Number of permutations of length n avoiding the pattern 1342.

Original entry on oeis.org

1, 1, 2, 6, 23, 103, 512, 2740, 15485, 91245, 555662, 3475090, 22214707, 144640291, 956560748, 6411521056, 43478151737, 297864793993, 2059159989914, 14350039389022, 100726680316559, 711630547589023, 5057282786190872, 36132861123763276, 259423620328055093

Offset: 0

Miklos Bona

Differs from A117156 which counts permutations avoiding the *consecutive* pattern 1342. - Ray Chandler, Dec 06 2011
Also, number of permutation of length n avoiding the pattern 3142 (see Stankova (1994) below). - Alexander Burstein, Aug 09 2013
Conjecture: a(n) is the number of permutations of length n simultaneously avoiding patterns 2143 and 415263. - Alexander Burstein, Mar 21 2019

			a(4) = 23 because obviously all permutations of length 4 with the exception of 1342 avoid 1342.

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 768, Th. 12.1.14.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.48.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, A family of Bell transformations, arXiv:1803.07727 [math.CO], 2018.
Miklos Bona, Exact enumeration of 1342-avoiding permutations: A close link with labeled trees and planar maps, arXiv:math/9702223 [math.CO], 1997.
Miklos Bona, Exact enumeration of 1342-avoiding permutations; A close link with labeled trees and planar maps, J. Combinatorial Theory, A80 (1997), 257-272.
Alexander Burstein and Jay Pantone, Two examples of unbalanced Wilf-equivalence, J. Combin. 6 (2015), no. 1-2, 55-67.
Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, and Jean-Marie Maillard, Stieltjes moment sequences for pattern-avoiding permutations, arXiv:2001.00393 [math.CO], 2020.
A. R. Conway and A. J. Guttmann, On 1324-avoiding permutations, Adv. Appl. Math. 64 (2015), 50-69.
A. L. L. Gao, S. Kitaev, and P. B. Zhang. On pattern avoiding indecomposable permutations, arXiv:1605.05490 [math.CO], 2016.
Elizabeth Hartung, Hung Phuc Hoang, Torsten Mütze, and Aaron Williams, Combinatorial generation via permutation languages. I. Fundamentals, arXiv:1906.06069 [cs.DM], 2019.
C. Homberger, Patterns in Permutations and Involutions: A Structural and Enumerative Approach, arXiv preprint arXiv:1410.2657 [math.CO], 2014.
Hsien-Kuei Hwang, Mihyun Kang, and Guan-Huei Duh, Asymptotic Expansions for Sub-Critical Lagrangean Forms, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2018.
W. Mlotkowski, K. A. Penson, A Fuss-type family of positive definite sequences, arXiv:1507.07312 (2015), eq. (36).
Z. E. Stankova, Forbidden subsequences, Discrete Math., 132 (1994), no. 1-3, 291-316.
Zvezdelina Stankova-Frenkel and Julian West, A new class of Wilf-equivalent permutations, arXiv:math/0103152 [math.CO], 2001. See Fig. 11.

Essentially the same as A004040.
Cf. A117158.
A005802, A022558, A061552 are representatives for the three Wilf classes for length-four avoiding permutations (cf. A099952).

a := proc (n) options operator, arrow: (1/2)*(-1)^(n-1)*(7*n^2-3*n-2)+3*(sum((-1)^(n-i)*2^(i+1)*factorial(2*i-4)*binomial(n-i+2, 2)/(factorial(i)*factorial(i-2)), i = 2 .. n)) end proc: seq(a(n), n = 0 .. 30); # Emeric Deutsch, Oct 15 2014

Table[SeriesCoefficient[32*x/(1+20*x-8*x^2-(1-8*x)^(3/2)),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 07 2012 *)
Table[1/2*(-1)^(n-1) * (-2-3*n+7*n^2) + 1/4*(-1)^n * (1+n) * (-2-13*n+(n+2) * Hypergeometric2F1[-3/2,-n,-2-n,-8]),{n,0,20}] (* Vaclav Kotesovec, Aug 24 2014 *)

x='x+O('x^66); Vec( 32*x/(1+20*x-8*x^2-(1-8*x)^(3/2)) ) \\ Joerg Arndt, May 04 2013

a(n) = (7*n^2-3*n-2)/2 * (-1)^(n-1) + 3*Sum_{i=2..n} 2^(i+1) * (2*i-4)!/(i!*(i-2)!) * binomial(n-i+2, 2) * (-1)^(n-i).
G.f.: 32*x/(1 + 20*x - 8*x^2 - (1 - 8*x)^(3/2)). - Emeric Deutsch, Mar 13 2004
Recurrence: n*a(n) = (7*n-22)*a(n-1) + 4*(2*n-1)*a(n-2). - Vaclav Kotesovec, Oct 07 2012
a(n) ~ 2^(3*n+6)/(243*sqrt(Pi)*n^(5/2)). - Vaclav Kotesovec, Oct 07 2012

Minor edits by Vaclav Kotesovec, Aug 24 2014

A117226 Number of permutations of [n] avoiding the consecutive pattern 1243.

Original entry on oeis.org

1, 1, 2, 6, 23, 110, 630, 4204, 32054, 274914, 2619692, 27459344, 313990182, 3889585408, 51888955808, 741668212080, 11307669002720, 183174676857608, 3141820432768752, 56882461258572976, 1084056190235653304, 21692744773505849952, 454758269790599361968

Offset: 0

Steven Finch, Apr 26 2006

nonn

a(n) is the number of permutations on [n] that avoid the consecutive pattern 1243. It is the same as the number of permutations which avoid 3421, 4312 or 2134.

Alois P. Heinz, Table of n, a(n) for n = 0..200 (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, 2011.
Sergi Elizalde and Marc Noy, Consecutive patterns in permutations, Adv. Appl. Math. 30 (2003), 110-125; see p. 120.
Sergi Elizalde, Asymptotic enumeration of permutations avoiding generalized patterns, arXiv:math/0505254 [math.CO], 2005.
Sergi Elizalde, Asymptotic enumeration of permutations avoiding generalized patterns, Adv. Appl. Math. 36 (2006), 138-155.
Steven Finch, Pattern-Avoiding Permutations. [Archived copy]
Steven Finch, Pattern-Avoiding Permutations. [Cached copy, with permission]
Eric Weisstein's World of Mathematics, Pochhammer Symbol.
Wikipedia, Falling and rising factorials.

Cf. A113228, A113229, A117156, A117158, A176730, A201692, A201693, A231166.

Row m = 1 of A327722.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      add(b(u-j, o+j-1, 0), j=`if`(t<0, -t, 1)..u)+
      add(b(u+j-1, o-j, `if`(t=0, j, -j)), j=1..o))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Nov 07 2013

A[x_]:=Integrate[AiryAi[ -t],{t,0,x}]; B[x_]:=Integrate[AiryBi[ -t],{t,0,x}];
c=-3^(2/3)*Gamma[2/3]/2; d=-3^(1/6)*Gamma[2/3]/2;
a[n_]:=SeriesCoefficient[1/(c*A[x]+d*B[x]+1),{x,0,n}]*n!; Table[a[n],{n,0,10}] (* fixed by Vaclav Kotesovec, Aug 23 2014 *)
(* constant d: *) 1/x/.FindRoot[3^(2/3)*Gamma[2/3]/2 * Integrate[AiryAi[-t],{t,0,x}] + 3^(1/6)*Gamma[2/3]/2 * Integrate[AiryBi[-t],{t,0,x}]==1,{x,1},WorkingPrecision->50] (* Vaclav Kotesovec, Aug 23 2014 *)

a(n) ~ c * d^n * n!, where d = 0.952891423325053197208702817349165942637814..., c = 1.169657787464830219717093446929792145316... . - Vaclav Kotesovec, Aug 23 2014
From Petros Hadjicostas, Nov 01 2019: (Start)
E.g.f.: 1/W(z), where W(z) := 1 + Sum_{n >= 0} (-1)^(n+1)* z^(3*n+1)/(b(n)*(3*n+1)) with b(n) = A176730(n) = (3*n)!/(3^n*(1/3)_n). (Here (x)_n = x*(x + 1)*...*(x + n - 1) is the Pochhammer symbol, or rising factorial, which is denoted by (x)^n in some papers and books.) The function W(z) satisfies the o.d.e. W'''(z) + z*W'(z) = 0 with W(0) = 1, W'(0) = -1, and W''(0) = 0. [See Theorem 4.3 (Case 1243 with u = 0) in Elizalde and Noy (2003).]
a(n) = Sum_{m = 0..floor((n-1)/3)} (-3)^m * (1/3)_m * binomial(n, 3*m+1) * a(n-3*m-1) for n >= 1 with a(0) = 1. (End)

A113228 a(n) is the number of permutations of [1..n] that avoid the consecutive pattern 1324 (equally, the permutations that avoid 4231).

Original entry on oeis.org

1, 1, 2, 6, 23, 110, 632, 4229, 32337, 278204, 2659223, 27959880, 320706444, 3985116699, 53328433923, 764610089967, 11693644958690, 190015358010114, 3269272324528547, 59373764638615449, 1135048629795612125, 22783668363316052016, 479111084084119883217

Offset: 0

David Callan, Oct 19 2005

nonn

			In 24135, the entries 2435 are in relative order 1324 but they do not occur consecutively and 24135 avoids the consecutive 1324 pattern.

Alois P. Heinz, Table of n, a(n) for n = 0..200 (terms n = 0..60 from Ray Chandler)
Andrew Baxter, Brian Nakamura, and Doron Zeilberger, Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes
Colin Defant, Noah Kravitz, and Nathan Williams, The Ungar Games, arXiv:2302.06552 [math.CO], 2023.
Sergi Elizalde, Asymptotic enumeration of permutations avoiding generalized patterns, arXiv:math/0505254 [math.CO], 2005.
Sergi Elizalde, Asymptotic enumeration of permutations avoiding generalized patterns, Adv. in Appl. Math. 36 (2006), no. 2, 138-155.
Sergi Elizalde and Marc Noy, Consecutive patterns in permutations, Adv. Appl. Math. 30 (2003), 110-125.
Steven Finch, Pattern-Avoiding Permutations [Broken link?]
Steven Finch, Pattern-Avoiding Permutations [Cached copy, with permission]

Cf. A113229, A117156, A117158, A117226, A201692, A201693, A231166.

Column k=0 of A264173.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
       add(b(u-j, o+j-1, `if`(t>0 and j b(n, 0, 0):
seq(a(n), n=0..25);  # Alois P. Heinz, Nov 07 2013

Clear[u, v, w]; w[0]=1; w[1]=1;w[2]=2; w[n_]/;n>=3 := w[n] = Sum[w[n, a], {a, n}]; w[1, 1] = w[2, 1] = w[2, 2] = 1; w[n_, a_]/;n>=3 && 1<=a<=n := Sum[u[n, a, b], {b, a+1, n}] + v[n, a]; v[1, 1]=1; v[n_, a_]/;n>=2 && a==1 := 0; v[n_, a_]/;n>=2 && 2<=a<=n := wCumulative[n-1, a-1]; wCumulative[n_, k_]/;Not[1<=k<=n] := 0; wCumulative[n_, k_]/;1<=k<=n := wCumulative[n, k] = Sum[w[n, a], {a, k}]; u[n_, a_, b_]/;Not[1<=a=4 && 1<=a0 && j < t, -j, 0]], {j, 1, u}] + Sum[b[u+j-1, o-j, j], {j, 1, If[t<0, Min[-t-1, o], o]}]]; a[n_] := b[n, 0, 0]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jan 19 2017, after Alois P. Heinz *)

In the recurrence coded in Mathematica below, w[n, a] = #1324-avoiding permutations on [n] with first entry a; u[n, a, b] is the number that start with an ascent a=2). The main sum for u[n, a, b] counts by length k of the longest initial increasing subsequence. The cases k=2, k=3, k>=4 are considered separately.

a(n) ~ c * d^n * n!, where d = 0.9558503134742499886507376383060906722796..., c = 1.15104449887019137479444895134035262624... . - Vaclav Kotesovec, Aug 23 2014

A177478 Permutations avoiding the consecutive patterns 4312 and 4213.

Original entry on oeis.org

1, 1, 2, 6, 22, 100, 540, 3388, 24248, 195048, 1742860, 17127880, 183617280, 2132433940, 26669752928, 357375269160, 5108084756320, 77574769941760, 1247401873186560, 21172559509803520, 378282904982091200, 7096584257305845120, 139471475802695196160

Offset: 0

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

nonn

a(n) gives the number of permutations of [n] which avoid both the pattern 4312 and 4213 consecutively. Also the number avoiding the pairs {2134, 3124}, {1243, 1342}, or {3421, 2431} (by symmetry).

This can also be considered avoiding a partially ordered pattern: Suppose p

The Baxter-Nakamura-Zeilberger paper has an associated Maple package. See Links.

Alois P. Heinz, Table of n, a(n) for n = 0..120 (terms n = 1..40 from Ray Chandler)
A. Baxter, B. Nakamura, and D. Zeilberger. Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes
S. Kitaev, Introduction to partially ordered patterns, Discrete Applied Mathematics 155 (2007), 929-944.

Cf. A117226, A117156.

b:= proc(u, o, s, t) option remember; `if`(u+o=0, 1,
       add(b(u-j, o+j-1, t, j), j=1..u)+
       add(b(u+j-1, o-j, 0, 0), j=`if`(s>0, s+t-1, 1)..o))
    end:
a:= n-> b(0, n, 0, 0):
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 25 2013

b[u_, o_, s_, t_] := b[u, o, s, t] = If[u+o == 0, 1, Sum[b[u-j, o+j-1, t, j], {j, 1, u}] + Sum[b[u+j-1, o-j, 0, 0], {j, If[s > 0, s+t-1, 1], o}]];
a[n_] := b[0, n, 0, 0];
a /@ Range[0, 25] (* Jean-François Alcover, Nov 03 2020, after Alois P. Heinz *)

a(n) ~ c * d^n * n!, where d = 0.89333294588184091624317413051..., c = 1.4839698712287023868073431417... . - Vaclav Kotesovec, Aug 24 2014

More terms, succinct title, additional comments, new references from Andrew Baxter, Jan 21 2011

A113229 Number of permutations avoiding the consecutive pattern 3412.

Original entry on oeis.org

1, 1, 2, 6, 23, 110, 631, 4223, 32301, 277962, 2657797, 27954521, 320752991, 3987045780, 53372351265, 765499019221, 11711207065229, 190365226548070, 3276401870322033, 59523410471007913, 1138295039078030599, 22856576346825690128, 480807130959249565541

Offset: 0

David Callan, Oct 19 2005

nonn

a(n) is the number of permutations on [n] that avoid the consecutive pattern 3412 (also number that avoid 2143).

			The 5! - a(5) = 10 permutations on [5] not counted by a(5) are 14523, 24513, 34125, 34512, 35124, 43512, 45123, 45132, 45231, 53412.

Alois P. Heinz, Table of n, a(n) for n = 0..200 (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 [math.CO], 2011.
Sergi Elizalde, Asymptotic enumeration of permutations avoiding generalized patterns, arXiv:math/0505254 [math.CO], 2005.
Sergi Elizalde, Asymptotic enumeration of permutations avoiding generalized patterns, Adv. in Appl. Math. 36 (2006), no. 2, 138-155.
S. Elizalde and M. Noy, Consecutive patterns in permutations, Adv. Appl. Math. 30 (2003), 110-125.
Steven Finch, Pattern-Avoiding Permutations [Broken link?]
Steven Finch, Pattern-Avoiding Permutations [Cached copy, with permission]

Cf. A113228, A117156, A117158, A117226, A201692, A201693.

Column k=0 of A264319.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      add(b(u-j, o+j-1, `if`(t>0 and j>t, t-j, 0)), j=1..u)+
      add(b(u+j-1, o-j, j), j=`if`(t<0,1-t,1)..o))
    end:
a:= n-> 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[t>0 && j>t, t-j, 0]], {j, 1, u}] + Sum[b[u+j-1, o-j, j], {j, Range[If[t<0, 1-t, 1], o]}]]; a[n_] := b[n, 0, 0]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 13 2015, after Alois P. Heinz *)

The Dotsenko et al. 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.

a(n) ~ c * d^n * n!, where d = 0.9561742431150784273897350385923872770208469..., c = 1.1465405299007850875068632404058971045769... . - Vaclav Kotesovec, Aug 23 2014

A177470 Number of permutations of order n avoiding the consecutive pattern 11'22'.

Original entry on oeis.org

1, 1, 2, 6, 18, 70, 300, 1435, 7910, 47376, 316008, 2314158, 18331236, 158024724, 1462752720, 14497475850, 153488070450, 1724035906450, 20515906356660, 257720354712106, 3406481187714176, 47293781230517640, 687760952277462336, 10456003715906638162, 165890170459303164420

Offset: 0

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

nonn

To avoid 11'22' means not to have four consecutive letters such that the first letter is less than the third one and the second letter is less than the fourth one.

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

Cf. A117156, A177471, A177472, A177473, A177475, A177476, A177477, A177478, A177479, A177480, A177481, A177482, A177483, A177484, A376694.

C
```
// See Links section.
```

a(0) = 1 and a(10)-a(16) from Rémy Sigrist, Mar 13 2023

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

A376694 Number of permutations of order n avoiding consecutive pattern 121'3.

Original entry on oeis.org

1, 1, 2, 6, 22, 100, 548, 3482, 25256, 206298, 1871704, 18676354, 203323724, 2397969518, 30455963576, 414446765490, 6015821216380, 92778775395190, 1515047281161392, 26114701159308242, 473827422862284740, 9027024454944900390, 180165845677134636856, 3759286756628732868754, 81850596163629861103004

Offset: 0

Max Alekseyev, Oct 01 2024

nonn

To avoid 121'3 means not to have four consecutive letters such that the first letter and the third one is less than the second, and the second one is less than and the fourth one.

S. Kitaev, Introduction to partially ordered patterns, Discrete Applied Mathematics 155 (2007), 929-944. [WARNING: The sequence is erroneously given as 1, 2, 6, 20, 83, 411, ...]

Cf. A117156, A177470, A177471, A177472, A177473, A177475, A177476, A177477, A177478, A177479, A177480, A177481, A177482, A177483, A177484.

A177472 Number of permutations of order n avoiding the consecutive pattern 11'2'2.

Original entry on oeis.org

1, 1, 2, 6, 18, 71, 322, 1665, 9789, 64327, 468914, 3748920, 32699022, 308710917, 3138821688, 34186918427, 397173376849, 4902547569617, 64073734206528, 883940288032392, 12836250526983672, 195724485831901029, 3126468139092090818, 52211816154041174670, 909843384081141844312

Offset: 0

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

nonn

To avoid 11'2'2 means not to have four consecutive letters such that the first letter is less than the last one, and the second letter is less than the third one.

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

Cf. A117156, A177470, A177471, A177473, A177475, 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

A177473 Number of permutations of order n avoiding the consecutive pattern 12'1'2.

Original entry on oeis.org

1, 1, 2, 6, 18, 61, 272, 1410, 8048, 51550, 372995, 2976679, 25686748, 239687103, 2419267562, 26194183096, 301838412516, 3692782460824, 47891164866100, 655887513203263, 9449915113835659, 142923476094740969, 2265214890150056647, 37539217881003574022, 649054317768293760078

Offset: 0

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

nonn

To avoid 12'1'2 means not to have four consecutive letters such that the first one is less than the fourth letter and the second letter is larger than the third one.

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

Cf. A117156, A177470, A177471, A177472, A177475, A177476, A177477, A177478, A177479, A177480, A177481, A177482, A177483, A177484, A376694.

Edited and terms a(0), a(10)-a(24) added by Max Alekseyev, Oct 01 2024

Showing 1-10 of 17 results. Next

cp's OEIS Frontend

A117158 Number of permutations avoiding the consecutive pattern 1234.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A022558 Number of permutations of length n avoiding the pattern 1342.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A117226 Number of permutations of [n] avoiding the consecutive pattern 1243.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A113228 a(n) is the number of permutations of [1..n] that avoid the consecutive pattern 1324 (equally, the permutations that avoid 4231).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A177478 Permutations avoiding the consecutive patterns 4312 and 4213.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A113229 Number of permutations avoiding the consecutive pattern 3412.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A177470 Number of permutations of order n avoiding the consecutive pattern 11'22'.

Views

Author

Keywords

Comments

Links