A177523 -id:A177523 - cp's OEIS frontend

A230233 Number of permutations of [n] avoiding adjacent step pattern {up}^10.

1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916799, 479001577, 6227020358, 87178283010, 1307674215120, 20922786961440, 355687370176320, 6402372516146880, 121645075013280000, 2432901444395385600, 51090929159028595200, 1124000415686590747031

Offset: 0

Alois P. Heinz, Oct 12 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A049774, A117158, A177523, A177533, A177553, A230051, A230231, A230232.

Column k=1023 of A242784.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t<9, 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);

nn=20;r=10;a=Apply[Plus,Table[Normal[Series[y x^(r+1)/(1-Sum[y x^i,{i,1,r}]),{x,0,nn}]][[n]]/(n+r)!,{n,1,nn-r}]]/.y->-1;Range[0,nn]! CoefficientList[Series[1/(1-x-a),{x,0,nn}],x] (* Geoffrey Critzer, Feb 25 2014 *)
CoefficientList[Series[1/(HypergeometricPFQ[{}, {1/11, 2/11, 3/11, 4/11, 5/11, 6/11, 7/11, 8/11, 9/11, 10/11}, x^11/285311670611] - x*HypergeometricPFQ[{}, {2/11, 3/11, 4/11, 5/11, 6/11, 7/11, 8/11, 9/11, 10/11, 12/11}, x^11/285311670611]), {x, 0, 25}], x] * Range[0, 25]! (* Vaclav Kotesovec, Jan 17 2015 *)

E.g.f.: 1 / Sum_{n>=0} (11*n+1-x)*x^(11*n)/(11*n+1)!.
a(n)/n! ~ 1.000000227556759905306252970186381144189779110025896440589711080508... * (1/r)^n, where r = 1.000000022964438439732421879840792836238519233492197325926442472620564... is the root of the equation Sum_{n>=0} (r^(11*n)/(11*n)! - r^(11*n+1)/(11*n+1)!) = 0. - Vaclav Kotesovec, Jan 17 2015
E.g.f.: -11/(2*((-cos(x*cos(Pi/22)))* cosh(x*sin(Pi/22)) - cos(x*cos(3*Pi/22))* cosh(x*sin(3*Pi/22)) - cos(x*cos(5*Pi/22))* cosh(x*sin(5*Pi/22)) - cos(x*cos(Pi/22))* cosh(x*sin(Pi/22))*sin(Pi/22) + cos(x*cos(3*Pi/22))* cosh(x*sin(3*Pi/22))* sin(3*Pi/22) - cos(x*cos(5*Pi/22))* cosh(x*sin(5*Pi/22))* sin(5*Pi/22) + cos(Pi/22)* cosh(x*sin(Pi/22))* sin(x*cos(Pi/22)) + cos(3*Pi/22)*cosh( x*sin(3*Pi/22))* sin(x*cos(3*Pi/22)) + cos(5*Pi/22)*cosh( x*sin(5*Pi/22))* sin(x*cos(5*Pi/22)) - cosh(x*cos(Pi/11))* ((1 + cos(Pi/11))* cos(x*sin(Pi/11)) - sin(Pi/11)*sin(x*sin(Pi/11))) + cosh(x*cos(2*Pi/11))* ((-1 + cos(2*Pi/11))* cos(x*sin(2*Pi/11)) + sin(2*Pi/11)* sin(x*sin(2*Pi/11))) + cos(x*sin(Pi/11))* sinh(x*cos(Pi/11)) + cos(Pi/11)*cos(x*sin(Pi/11))* sinh(x*cos(Pi/11)) - sin(Pi/11)*sin(x*sin(Pi/11))* sinh(x*cos(Pi/11)) - cos(x*sin(2*Pi/11))* sinh(x*cos(2*Pi/11)) + cos(2*Pi/11)* cos(x*sin(2*Pi/11))* sinh(x*cos(2*Pi/11)) + sin(2*Pi/11)* sin(x*sin(2*Pi/11))* sinh(x*cos(2*Pi/11)) + cos(x*cos(Pi/22))* sinh(x*sin(Pi/22)) + cos(x*cos(Pi/22))*sin(Pi/22)* sinh(x*sin(Pi/22)) - cos(Pi/22)*sin(x*cos(Pi/22))* sinh(x*sin(Pi/22)) - cos(x*cos(3*Pi/22))* sinh(x*sin(3*Pi/22)) + cos(x*cos(3*Pi/22))* sin(3*Pi/22)* sinh(x*sin(3*Pi/22)) + cos(3*Pi/22)* sin(x*cos(3*Pi/22))* sinh(x*sin(3*Pi/22)) + cos(x*cos(5*Pi/22))* sinh(x*sin(5*Pi/22)) + cos(x*cos(5*Pi/22))* sin(5*Pi/22)* sinh(x*sin(5*Pi/22)) - cos(5*Pi/22)* sin(x*cos(5*Pi/22))* sinh(x*sin(5*Pi/22)))). - Vaclav Kotesovec, Jan 31 2015

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

A202214 Number of permutations avoiding the consecutive pattern 53421.

Original entry on oeis.org

1, 2, 6, 24, 119, 708, 4914, 38977, 347779, 3447902, 37600925, 447332398, 5765327905, 80020698913, 1189991560707, 18876181370841, 318136382176708, 5677225148401090, 106939954137362482, 2120412050923152077, 44145844897844974946, 962859209130749461268

Offset: 1

Ray Chandler, Dec 14 2011

nonn

a(n) is the number of permutations on [n] that avoid the consecutive pattern 53421. It is the same as the number of permutations which avoid 12435, 13245 or 54231.

Ray Chandler, Table of n, a(n) for n = 1..40
A. Baxter, B. Nakamura, and D. Zeilberger. Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes

Cf. A177523, A202213-A202236.

A202235 Number of permutations avoiding the consecutive pattern 32415.

Original entry on oeis.org

1, 2, 6, 24, 119, 708, 4915, 38992, 347990, 3450772, 37640794, 447908832, 5774077131, 80160465380, 1192342710241, 18917802834912, 318910800658728, 5692346715440768, 107249322475436984, 2127032690422975168, 44293813329642772714, 966307529386769800328

Offset: 1

Ray Chandler, Dec 15 2011

nonn

a(n) is the number of permutations on [n] that avoid the consecutive pattern 32415. It is the same as the number of permutations which avoid 15243, 34251 or 51423.

Ray Chandler, Table of n, a(n) for n = 1..40
A. Baxter, B. Nakamura, and D. Zeilberger. Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes

Cf. A177523, A202213-A202236.

A254523 Number of permutations of [n] avoiding adjacent step pattern {up}^11.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001599, 6227020775, 87178290682, 1307674357710, 20922789683040, 355687423926240, 6402373618334400, 121645098513933120, 2432901965590252800, 51090941178938707200, 1124000703770606323200

Offset: 0

Vaclav Kotesovec, Jan 31 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A049774, A117158, A177523, A177533, A177553, A230051, A230231, A230232, A230233.

Column k=2047 of A242784.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t<10, 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); # after Alois P. Heinz

CoefficientList[Series[6 / (Exp[-x] + Cos[x] + 2*Cos[x/2] * Cosh[Sqrt[3]*x/2] - Cosh[Sqrt[3]*x/2]*Sin[x/2] - Sin[x] + Cosh[x/2] * (2*Cos[Sqrt[3]*x/2] - Sqrt[3]*Sin[Sqrt[3]*x/2]) - Cos[Sqrt[3]*x/2]*Sinh[x/2] - Sqrt[3]*Cos[x/2]*Sinh[Sqrt[3]*x/2]), {x, 0, 25}], x] * Range[0, 25]!

E.g.f.: 1 / Sum_{n>=0} (12*n+1-x)*x^(12*n)/(12*n+1)!.
E.g.f.: 6 / (exp(-x) + cos(x) + 2*cos(x/2)*cosh(sqrt(3)*x/2) - cosh(sqrt(3)*x/2)*sin(x/2) - sin(x) + cosh(x/2)*(2*cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2)) - cos(sqrt(3)*x/2)*sinh(x/2) - sqrt(3)*cos(x/2)*sinh(sqrt(3)*x/2)).
a(n)/n! ~ c * (1/r)^n, where r = 1.0000000019270853046730165249753673978954992128247736041276... is the root of the equation Sum_{n>=0} (r^(12*n)/(12*n)! - r^(12*n+1)/(12*n+1)!) = 0, equivalently root of the equation exp(-r) + cos(r) + 2*cos(r/2)*cosh(sqrt(3)*r/2) - cosh(sqrt(3)*r/2)*sin(r/2) - sin(r) + cosh(r/2)*(2*cos(sqrt(3)*r/2) - sqrt(3)*sin(sqrt(3)*r/2)) - cos(sqrt(3)*r/2)*sinh(r/2) - sqrt(3)*cos(r/2)*sinh(sqrt(3)*r/2) = 0, c = 3/(r*sqrt((cosh(sqrt(3)*r/2) * sin(r/2) + sin(r))^2 + 2*sqrt(3)*cosh(r/2) * (cosh(sqrt(3)*r/2) * sin(r/2) + sin(r)) * sin(sqrt(3)*r/2) + 3*cosh(r/2)^2 * sin((sqrt(3)*r)/2)^2)) = 1.0000000210373483515818712802156496756788404534079689145773611990529818919... .

A202215 Number of permutations avoiding the consecutive pattern 43251.

Original entry on oeis.org

1, 2, 6, 24, 119, 708, 4914, 38977, 347783, 3447982, 37602245, 447353497, 5765670515, 80026435959, 1190091483826, 18877998176267, 318170909582504, 5677911026433716, 106954187252974515, 2120720309936327198, 44152804913927293967, 963022839743497051427

Offset: 1

Ray Chandler, Dec 14 2011

nonn

a(n) is the number of permutations on [n] that avoid the consecutive pattern 43251. It is the same as the number of permutations which avoid 15234, 23415 or 51432.

Ray Chandler, Table of n, a(n) for n = 1..40
A. Baxter, B. Nakamura, and D. Zeilberger. Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes

Cf. A177523, A202213-A202236.

A202216 Number of permutations avoiding the consecutive pattern 34521.

Original entry on oeis.org

1, 2, 6, 24, 119, 708, 4914, 38976, 347775, 3447912, 37601586, 447346944, 5765604885, 80025786960, 1190085736014, 18877969022976, 318171464604945, 5677939605422880, 106955077375109430, 2120744934976978944, 44153462828412744975, 963040357207773472320

Offset: 1

Ray Chandler, Dec 14 2011

nonn

a(n) is the number of permutations on [n] that avoid the consecutive pattern 34521. It is the same as the number of permutations which avoid 12453, 31245, 35421, 54213, 12543, 32145 or 54123.

Ray Chandler, Table of n, a(n) for n = 1..40
A. Baxter, B. Nakamura, and D. Zeilberger. Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes

Cf. A177523, A202213-A202236.

A202217 Number of permutations avoiding the consecutive pattern 45132.

Original entry on oeis.org

1, 2, 6, 24, 119, 708, 4914, 38976, 347776, 3447932, 37601916, 447352224, 5765690544, 80027220448, 1190110692204, 18878422622976, 318180082610048, 5678110761876224, 106958628519952128, 2120821833862388544, 44155198873515034560, 963081167548894179456

Offset: 1

Ray Chandler, Dec 14 2011

nonn

a(n) is the number of permutations on [n] that avoid the consecutive pattern 45132. It is the same as the number of permutations which avoid 21534, 23154 or 43512.

Ray Chandler, Table of n, a(n) for n = 1..40
A. Baxter, B. Nakamura, and D. Zeilberger. Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes

Cf. A177523, A202213-A202236.

A202218 Number of permutations avoiding the consecutive pattern 32451.

Original entry on oeis.org

1, 2, 6, 24, 119, 708, 4914, 38977, 347793, 3448182, 37605545, 447406214, 5766525727, 80040747361, 1190340631290, 18882526564227, 318256944873984, 5679619717431613, 106989639176572571, 2121488012937187849, 44170136425049025574, 963430265253564585315

Offset: 1

Ray Chandler, Dec 15 2011

nonn

a(n) is the number of permutations on [n] that avoid the consecutive pattern 32451. It is the same as the number of permutations which avoid 15423, 34215 or 51243.

Ray Chandler, Table of n, a(n) for n = 1..40
A. Baxter, B. Nakamura, and D. Zeilberger. Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes

Cf. A177523, A202213-A202236.

A202219 Number of permutations avoiding the consecutive pattern 45312.

Original entry on oeis.org

1, 2, 6, 24, 119, 708, 4914, 38977, 347794, 3448202, 37605875, 447411544, 5766612686, 80042208149, 1190366133003, 18882991058982, 318265784348321, 5679795502722184, 106993290279989932, 2121567147336853153, 44171924298572910714, 963472321485580749448

Offset: 1

Ray Chandler, Dec 15 2011

nonn

a(n) is the number of permutations on [n] that avoid the consecutive pattern 45312. It is the same as the number of permutations which avoid 21354.

Ray Chandler, Table of n, a(n) for n = 1..40
A. Baxter, B. Nakamura, and D. Zeilberger. Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes

Cf. A177523, A202213-A202236.

cp's OEIS Frontend

A230233 Number of permutations of [n] avoiding adjacent step pattern {up}^10.

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Maple

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

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Maple

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A202214 Number of permutations avoiding the consecutive pattern 53421.

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A202235 Number of permutations avoiding the consecutive pattern 32415.

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A254523 Number of permutations of [n] avoiding adjacent step pattern {up}^11.

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Maple

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A202215 Number of permutations avoiding the consecutive pattern 43251.

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A202216 Number of permutations avoiding the consecutive pattern 34521.

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A202217 Number of permutations avoiding the consecutive pattern 45132.

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A202218 Number of permutations avoiding the consecutive pattern 32451.

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A202219 Number of permutations avoiding the consecutive pattern 45312.

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