A202236 -id:A202236 - cp's OEIS frontend

A202213 Number of permutations of [n] avoiding the consecutive pattern 45321.

1, 1, 2, 6, 24, 119, 708, 4914, 38976, 347765, 3447712, 37598286, 447294144, 5764747515, 80011430240, 1189835682714, 18873422539776, 318085061976105, 5676223254661760, 106919460527212950, 2119973556022047744, 44136046410218669055, 962630898723772565760

Offset: 0

Ray Chandler, Dec 14 2011

nonn

a(n) is the number of permutations on [n] that avoid the consecutive pattern 45321. It is the same as the number of permutations which avoid 12354, 21345 or 54312.

Alois P. Heinz, Table of n, a(n) for n = 0..200 (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, 2011.
Sergi Elizalde and Marc Noy, Consecutive patterns in permutations, Adv. Appl. Math. 30 (2003), 110-125; see Theorem 3.2 (p. 116) with m = a = 3 and u = 0.
Eric Weisstein's World of Mathematics, Pochhammer Symbol.
Wikipedia, Falling and rising factorials.

Cf. A177523, A202214-A202236, A329070.

Column k = 0 of A264781 and row m = 2 of A327722.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      add(b(u+j-1, o-j, `if`(u+j-10, -1, `if`(t=-1, -2, 0)))), j=1..u)))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Nov 19 2013

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, Sum[b[u+j-1, o-j, If[u+j-1 < j, 0, j]], {j, 1, o}] + If[t == -2, 0, Sum[b[u-j, o+j-1, If[j0, -1, If[t == -1, -2, 0]]]], {j, 1, u}]]]; a[n_] := b[n, 0, 0]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 12 2015, after Alois P. Heinz *)

From Petros Hadjicostas, Nov 02 2019: (Start)
E.g.f.: 1/W(z), where W(z) = 1 + Sum_{n >= 0} (-1)^(n+1)* z^(4*n+1)/(b(n)*(4*n+1)) with b(n) = A329070(n,4) = (4*n)!/(4^n*(1/4)_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^(4)(z) + z*W'(z) = 0 with W(0) = 1, W'(0) = -1, and W^(k)(0) = 0 for k = 2..3. [See Theorem 3.2 (with m = a = 3 and u = 0) in Elizalde and Noy (2003).]
a(n) = Sum_{m = 0..floor((n-1)/4)} (-4)^m * (1/4)_m * binomial(n, 4*m+1) * a(n-4*m-1) for n >= 1 with a(0) = 1. (End)

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.

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.

A202220 Number of permutations avoiding the consecutive pattern 53241.

Original entry on oeis.org

1, 2, 6, 24, 119, 708, 4914, 38979, 347815, 3448442, 37608834, 447450930, 5767170785, 80050654711, 1190502148797, 18885313699983, 318307710883720, 5680593325882836, 107009253612595865, 2121902210596869336, 44179286193550935095, 963641311893918317652

Offset: 1

Ray Chandler, Dec 15 2011

nonn

a(n) is the number of permutations on [n] that avoid the consecutive pattern 53241. It is the same as the number of permutations which avoid 13425, 14235 or 53241.

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.

A202221 Number of permutations avoiding the consecutive pattern 41523.

Original entry on oeis.org

1, 2, 6, 24, 119, 708, 4914, 38976, 347790, 3448212, 37606536, 447426144, 5766888120, 80047243080, 1190459039364, 18884750846976, 318300264871440, 5680496863401840, 107008122578788800, 2121893387875882944, 44179385761361239200, 963649662245743556160

Offset: 1

Ray Chandler, Dec 15 2011

nonn

a(n) is the number of permutations on [n] that avoid the consecutive pattern 41523. It is the same as the number of permutations which avoid 24153, 31524, 35142, 42513, 25143, 32514 or 34152.

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

A202213 Number of permutations of [n] avoiding the consecutive pattern 45321.

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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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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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A202220 Number of permutations avoiding the consecutive pattern 53241.

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A202221 Number of permutations avoiding the consecutive pattern 41523.

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