A177523 -id:A177523 - cp's OEIS frontend

A049774 Number of permutations of n elements not containing the consecutive pattern 123.

1, 1, 2, 5, 17, 70, 349, 2017, 13358, 99377, 822041, 7477162, 74207209, 797771521, 9236662346, 114579019469, 1516103040833, 21314681315998, 317288088082405, 4985505271920097, 82459612672301846, 1432064398910663705, 26054771465540507273, 495583804405888997218

Offset: 0

Tuwani A. Tshifhumulo (tat(AT)caddy.univen.ac.za)

Permutations on n letters without double falls. A permutation w has a double fall at k if w(k) > w(k+1) > w(k+2) and has an initial fall if w(1) > w(2).
Hankel transform is A055209. - Paul Barry, Jan 12 2009
Increasing colored 1-2 trees of order n with choice of two colors for the right branches of the vertices of outdegree 2 except those vertices on the path from the root to the leftmost leaf. - Wenjin Woan, May 21 2011

			Permutations without double increase and without pattern 123:
a(3) = 5: 132, 213, 231, 312, 321.
a(4) = 17: 1324, 1423, 1432, 2143, 2314, 2413, 2431, 3142, 3214, 3241, 3412, 3421, 4132, 4213, 4231, 4312, 4321.

F. N. David and D. E. Barton, Combinatorial Chance, Hafner, New York, 1962, pp. 156-157.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (5.2.17).

Alois P. Heinz, Table of n, a(n) for n = 0..464 (first 201 terms from Ray Chandler)
Martin Aigner, Catalan and other numbers: a recurrent theme, in Algebraic Combinatorics and Computer Science, a Tribute to Gian-Carlo Rota, pp.347-390, Springer, 2001.
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 1.
Paul Barry, Constructing Exponential Riordan Arrays from Their A and Z Sequences, Journal of Integer Sequences, 17 (2014), #14.2.6.
Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.
Nicolas Basset, Counting and generating permutations using timed languages, HAL Id: hal-00820373, 2013.
A. Baxter, B. Nakamura, and D. Zeilberger. Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes; Local copy [Pdf file only, no active links].
S. Elizalde, Asymptotic enumeration of permutations avoiding generalized patterns arXiv:math/0505254 [math.CO], 2015.
S. Elizalde and M. Noy, Consecutive patterns in permutations, Adv. Appl. Math. 30 (2003), 110-123.
Steven Finch, Pattern-Avoiding Permutations [Archived version]
Steven Finch, Pattern-Avoiding Permutations [Cached copy, with permission]
Ira M. Gessel and Yan Zhuang, Counting permutations by alternating descents , arXiv:1408.1886 [math.CO], 2014. See Eq. (3). - _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 and Doron Zeilberger, Increasing Consecutive Patterns in Words, arXiv:1805.06077 [math.CO], 2018.
Christopher Zhu, Enumerating Permutations and Rim Hooks Characterized by Double Descent Sets, arXiv:1910.12818 [math.CO], 2019.

Cf. A065429, A080635, A111004, A117158, A177523, A177533.
Column k=0 of A162975.
Column k=3 of A242784.
Equals 1 + A000303. - Greg Dresden, Feb 22 2020

b:= proc(u, o, t) option remember;
     `if`(u+o=0, 1, add(b(u-j, o+j-1, 0), j=1..u)+
     `if`(t=1, 0,   add(b(u+j-1, o-j, 1), j=1..o)))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Nov 04 2021

Table[Simplify[ n! SeriesCoefficient[ Series[ Sqrt[3] Exp[x/2]/(Sqrt[3] Cos[Sqrt[3]/2 x] - Sin[Sqrt[3]/2 x]), {x, 0, n}], n] ], {n, 0, 40}]
(* Second program: *)
b[u_, o_, t_, k_] := b[u, o, t, k] = If[t == k, (u + o)!, If[Max[t, u] + o < k, 0, Sum[b[u + j - 1, o - j, t + 1, k], {j, 1, o}] + Sum[b[u - j, o + j - 1, 1, k], {j, 1, u}]]];
a[n_] := b[0, n, 0, 2] - b[0, n, 0, 3] + 1;
a /@ Range[0, 40] (* Jean-François Alcover, Nov 09 2020, after Alois P. Heinz in A000303 *)

E.g.f.: 1/Sum_{i>=0} (x^(3*i)/(3*i)! - x^(3*i+1)/(3*i+1)!). [Corrected g.f. --> e.g.f. by Vaclav Kotesovec, Feb 15 2015]
Equivalently, e.g.f.: exp(x/2) * r / sin(r*x + (2/3)*Pi) where r = sqrt(3)/2. This has simple poles at (3*m+1)*x0 where x0 = Pi/sqrt(6.75) = 1.2092 approximately and m is an arbitrary integer. This yields the asymptotic expansion a(n)/n! ~ x0^(-n-1) * Sum((-1)^m * E^(3*m+1) / (3*m+1)^(n+1)) where E = exp(x0/2) = 1.8305+ and m ranges over all integers. - Noam D. Elkies, Nov 15 2001
E.g.f.: sqrt(3)*exp(x/2)/(sqrt(3)*cos(x*sqrt(3)/2) - sin(x*sqrt(3)/2) ); a(n+1) = Sum_{k=0..n} binomial(n, k)*a(k)*b(n-k) where b(n) = number of n-permutations without double falls and without initial falls. - Emanuele Munarini, Feb 28 2003
O.g.f.: A(x) = 1/(1 - x - x^2/(1 - 2*x - 4*x^2/(1 - 3*x - 9*x^2/(1 - ... - n*x - n^2*x^2/(1 - ...))))) (continued fraction). - Paul D. Hanna, Jan 17 2006
a(n) = leftmost column term of M^n*V, where M = an infinite tridiagonal matrix with (1,2,3,...) in the super, sub, and main diagonals and the rest zeros. V = the vector [1,0,0,0,...]. - Gary W. Adamson, Jun 16 2011
E.g.f.: A(x)=1/Q(0); Q(k)=1-x/((3*k+1)-(x^2)*(3*k+1)/((x^2)-3*(3*k+2)*(k+1)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 25 2011
a(n) ~ n! * exp(Pi/(3*sqrt(3))) * (3*sqrt(3)/(2*Pi))^(n+1). - Vaclav Kotesovec, Jul 28 2013
E.g.f.: T(0)/(1-x), where T(k) = 1 - x^2*(k+1)^2/( x^2*(k+1)^2 - (1-x-x*k)*(1-2*x-x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 17 2013

Corrected and extended by Vladeta Jovovic, Apr 14 2001

A242784 Number A(n,k) of permutations of [n] avoiding the consecutive step pattern given by the binary expansion of k, where 1=up and 0=down; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 2, 5, 8, 1, 1, 1, 1, 2, 6, 17, 16, 1, 1, 1, 1, 2, 6, 21, 70, 32, 1, 1, 1, 1, 2, 6, 19, 90, 349, 64, 1, 1, 1, 1, 2, 6, 21, 70, 450, 2017, 128, 1, 1, 1, 1, 2, 6, 23, 90, 331, 2619, 13358, 256, 1, 1

Offset: 0

Alois P. Heinz, May 22 2014

			A(4,5) = 19 because there are 4! = 24 permutations of {1,2,3,4} and only 5 of them do not avoid the consecutive step pattern up, down, up given by the binary expansion of 5 = 101_2: (1,3,2,4), (1,4,2,3), (2,3,1,4), (2,4,1,3), (3,4,1,2).
Square array A(n,k) begins:
  1, 1,   1,     1,     1,     1,     1,     1,     1, ...
  1, 1,   1,     1,     1,     1,     1,     1,     1, ...
  1, 1,   2,     2,     2,     2,     2,     2,     2, ...
  1, 1,   4,     5,     6,     6,     6,     6,     6, ...
  1, 1,   8,    17,    21,    19,    21,    23,    24, ...
  1, 1,  16,    70,    90,    70,    90,   111,   116, ...
  1, 1,  32,   349,   450,   331,   450,   642,   672, ...
  1, 1,  64,  2017,  2619,  1863,  2619,  4326,  4536, ...
  1, 1, 128, 13358, 17334, 11637, 17334, 33333, 34944, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns give: 0, 1: A000012, 2: A011782, 3: A049774, 4, 6: A177479, 5: A177477, 7: A117158, 8, 14: A177518, 9: A177519, 10: A177520, 11, 13: A177521, 12: A177522, 15: A177523, 16, 30: A177524, 17: A177525, 18, 22: A177526, 19, 25: A177527, 20, 26: A177528, 21: A177529, 23, 29: A177530, 24, 28: A177531, 27: A177532, 31: A177533, 32, 62: A177534, 33: A177535, 34, 46: A177536, 35, 49: A177537, 36, 54: A177538, 37, 41: A177539, 38: A177540, 39, 57: A177541, 40, 58: A177542, 42: A177543, 43, 53: A177544, 44, 50: A177545, 45: A177546, 47, 61: A177547, 48, 60: A177548, 51: A177549, 52: A177550, 55, 59: A177551, 56: A177552, 63: A177553, 127: A230051, 255: A230231, 511: A230232, 1023: A230233, 2047: A254523.
Main diagonal gives A242785.
Cf. A242783, A335308.

A:= proc(n, k) option remember; local b, m, r, h;
      if k<2 then return 1 fi;
      m:= iquo(k, 2, 'r'); h:= 2^ilog2(k);
      b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t=m and r=0, 0, add(b(u-j, o+j-1, irem(2*t, h)), j=1..u))+
      `if`(t=m and r=1, 0, add(b(u+j-1, o-j, irem(2*t+1, h)), j=1..o)))
      end; forget(b);
      b(n, 0, 0)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..15);

Clear[A]; A[n_, k_] := A[n, k] = Module[{b, m, r, h}, If[k < 2, Return[1]]; {m, r} = QuotientRemainder[k, 2]; h = 2^Floor[Log[2, k]]; b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, If[t == m && r == 0, 0, Sum[b[u - j, o + j - 1, Mod[2*t, h]], {j, 1, u}]] + If[t == m && r == 1, 0, Sum[b[u + j - 1, o - j, Mod[2*t + 1, h]], {j, 1, o}]]]; b[n, 0, 0]]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 15}] // Flatten (* Jean-François Alcover, Sep 22 2014, translated from Maple *)

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

Original entry on oeis.org

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)

A202236 Number of permutations avoiding the consecutive pattern 35241.

Original entry on oeis.org

1, 2, 6, 24, 119, 708, 4916, 39008, 348202, 3453572, 37679044, 448455648, 5782307134, 80291122976, 1194530208634, 18956382485568, 319626461800751, 5706286536509228, 107533930423507736, 2133112981295131824, 44429507242283663790, 969465861672912148324

Offset: 1

Ray Chandler, Dec 15 2011

nonn

a(n) is the number of permutations on [n] that avoid the consecutive pattern 35241. It is the same as the number of permutations which avoid 14253, 31425 or 52413.

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-A202235.

A117158 Number of permutations avoiding the consecutive pattern 1234.

Original entry on oeis.org

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

A177533 Number of permutations of 1..n avoiding adjacent step pattern up, up, up, up, up.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 719, 5027, 40168, 361080, 3606480, 39623760, 474915803, 6166512899, 86227808578, 1291868401830, 20645144452320, 350547210173280, 6302294420371031, 119600213982762899, 2389140113204434900, 50111866901959213980, 1101140993932295832120

Offset: 0

R. H. Hardin, May 10 2010

nonn

a(n) is the number of permutations of length n that avoid the consecutive pattern 123456 (or equivalently 654321).

Alois P. Heinz, Table of n, a(n) for n = 0..450 (terms n = 1..30 from Ray Chandler)
R. E. L. Aldred, M. D. Atkinson, D. J. McCaughan, Avoiding consecutive patterns in permutations Adv. in Appl. Math., 45(3), 449-461, 2010.
A. Baxter, B. Nakamura, and D. Zeilberger, Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes
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=6. - _N. J. A. Sloane_, Aug 11 2014
Mingjia Yang, Doron Zeilberger, Increasing Consecutive Patterns in Words, arXiv:1805.06077 [math.CO], 2018.

Cf. A049774, A117158, A177523.

Column k=31 of A242784.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t<4, 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

Table[n!*SeriesCoefficient[1/(Sum[x^(6*k)/(6*k)!-x^(6*k+1)/(6*k+1)!,{k,0,n}]),{x,0,n}],{n,1,20}] (* Vaclav Kotesovec, Dec 11 2013 *)
Rest[CoefficientList[Series[3/(E^(x/2) * Cos[x*Sqrt[3]/2+Pi/3] + Sqrt[3] * E^(-x/2) * Cos[x*Sqrt[3]/2+Pi/6] + E^(-x)),{x,0,20}],x] * Range[0,20]!] (* Vaclav Kotesovec, Aug 23 2014 *)

a(n)/n! ~ 1.005827831279392186... * (1/r)^n, where r = 1.0011988273240623031887... is the root of the equation Sum_{n>=0} (r^(6*n)/(6*n)! - r^(6*n+1)/(6*n+1)!) = 0. - Vaclav Kotesovec, Dec 11 2013
Equivalently, a(n)/n! ~ c * (1/r)^n, where r = 1.00119882732406230318870210972855430833421618931012450844128... is the root of the equation 2 + exp(r/2) * (3 + exp(r)) * cos(sqrt(3)*r/2) = 2 * sqrt(3) * exp(r) * cosh(r/2) * sin(sqrt(3)*r/2), c = sqrt(3) / (2 * r * cosh(r/2) * sin(sqrt(3)*r/2)) = 1.0058278312793921866941324506580803251270892126827302878865925027445... . - Vaclav Kotesovec, Aug 23 2014
E.g.f. (Aldred, Atkinson, McCaughan, 2010): 3/(exp(x/2) * cos(x*sqrt(3)/2+Pi/3) + sqrt(3) * exp(-x/2) * cos(x*sqrt(3)/2+Pi/6) + exp(-x)). - Vaclav Kotesovec, Aug 23 2014

More terms from Ray Chandler, Dec 06 2011
Minor edits by Vaclav Kotesovec, Aug 29 2014
a(0)=1 prepended by Alois P. Heinz, Aug 08 2018

A230051 Number of permutations of [n] avoiding adjacent step pattern {up}^7.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5040, 40319, 362863, 3628550, 39913170, 478947480, 6226179960, 87164597520, 1307440134000, 20918580896069, 355608034188517, 6400803479701178, 121612584595293870, 2432198062707745560, 51075033128533094520, 1123625953230764250960

Offset: 0

Alois P. Heinz, Oct 07 2013

nonn

			a(8) = 40319 = 8!-1: only permutation 12345678 does not avoid {up}^7.

R. E. L. Aldred, M. D. Atkinson, D. J. McCaughan, Avoiding consecutive patterns in permutations. Adv. in Appl. Math., 45(3), 449-461, 2010.

Alois P. Heinz, Table of n, a(n) for n = 0..450
Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, Jean-Marie Maillard, Stieltjes moment sequences for pattern-avoiding permutations, arXiv:2001.00393 [math.CO], 2020.
Mingjia Yang, Doron Zeilberger, Increasing Consecutive Patterns in Words, arXiv:1805.06077 [math.CO], 2018.

Cf. A049774, A117158, A177523, A177533, A177553.

Column k=127 of A242784.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t<6, 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=7;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[4/(E^(-x) + Cos[x] - Sin[x] + 2*Cos[x/Sqrt[2]] * Cosh[x/Sqrt[2]] - Sqrt[2] * Cos[x/Sqrt[2]] * Sinh[x/Sqrt[2]] - Sqrt[2] * Cosh[x/Sqrt[2]] * Sin[x/Sqrt[2]]),{x,0,20}],x] * Range[0,20]! (* Vaclav Kotesovec, Aug 23 2014 *)

E.g.f.: 1 / Sum_{n>=0} (8*n+1-x)*x^(8*n)/(8*n+1)!.
E.g.f. (Aldred, Atkinson, McCaughan, 2010): 4/(exp(-x) + cos(x) - sin(x) + 2*cos(x/sqrt(2))*cosh(x/sqrt(2)) - sqrt(2)*cos(x/sqrt(2))*sinh(x/sqrt(2)) - sqrt(2)*cosh(x/sqrt(2))*sin(x/sqrt(2))). - Vaclav Kotesovec, Aug 23 2014
a(n)/n! ~ c / r^n, where r = 1.0000220496837836995332841475679738951237308817759821845322... is the root of the equation exp(-r) + cos(r) - sin(r) + 2*cos(r/sqrt(2)) * cosh(r/sqrt(2)) - sqrt(2)*cos(r/sqrt(2)) * sinh(r/sqrt(2)) - sqrt(2) * cosh(r/sqrt(2)) * sin(r/sqrt(2)) = 0, c = 2*sqrt(2) / (r*sqrt(2 + cosh(sqrt(2)*r) - cos(2*r) + 2*cosh(r/sqrt(2)) * (2*sqrt(2)*sin(r) * sin(r/sqrt(2)) - cos(sqrt(2)*r) * cosh(r/sqrt(2))))) = 1.0001516144914746839400607922657094772985420791612537... . - Vaclav Kotesovec, Aug 23 2014, updated Feb 01 2015

A177553 Number of permutations of 1..n avoiding adjacent step pattern up, up, up, up, up, up.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5039, 40305, 362682, 3626190, 39881160, 478490760, 6219298800, 87055051511, 1305598835941, 20885951018102, 354999461960226, 6388879812001704, 121367620532150280, 2426930566055020080, 50956684690331669759, 1120852238721212726609

Offset: 0

R. H. Hardin, May 10 2010

nonn

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

Column k=63 of A242784.

Cf. A080635, A049774, A117158, A177533, A177523.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t<5, 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

nn=20;r=6;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 *)
Table[n!*SeriesCoefficient[1/(Sum[x^(7*k)/(7*k)!-x^(7*k+1)/(7*k+1)!,{k,0,n}]),{x,0,n}],{n,1,20}] (* Vaclav Kotesovec, Aug 29 2014 *)

a(n)/n! ~ c * (1/r)^n, where r = 1.0001738181531504504518260962714687775785823593018886... is the root of the equation Sum_{n>=0} (r^(7*n)/(7*n)! - r^(7*n+1)/(7*n+1)!) = 0, c = 1.0010191104259450282450770594076722424772755532278.... - Vaclav Kotesovec, Aug 29 2014
E.g.f.: -(7/(2*((-cos(x*cos(3*Pi/14)))*cosh(x*sin(3*Pi/14)) + cos(x*cos(3*Pi/14))*cosh(x*sin(3*Pi/14))* sin(3*Pi/14) - cosh(x*sin(Pi/14))* (cos(x*cos(Pi/14))*(1 + sin(Pi/14)) - cos(Pi/14)*sin(x*cos(Pi/14))) + cos(3*Pi/14)*cosh(x*sin(3*Pi/14))* sin(x*cos(3*Pi/14)) - cosh(x*cos(Pi/7))* ((1 + cos(Pi/7))*cos(x*sin(Pi/7)) - sin(Pi/7)*sin(x*sin(Pi/7))) + cos(x*sin(Pi/7))* sinh(x*cos(Pi/7)) + cos(Pi/7)*cos(x*sin(Pi/7))* sinh(x*cos(Pi/7)) - sin(Pi/7)*sin(x*sin(Pi/7))* sinh(x*cos(Pi/7)) + cos(x*cos(Pi/14))* sinh(x*sin(Pi/14)) + cos(x*cos(Pi/14))*sin(Pi/14)* sinh(x*sin(Pi/14)) - cos(Pi/14)*sin(x*cos(Pi/14))* sinh(x*sin(Pi/14)) - cos(x*cos(3*Pi/14))* sinh(x*sin(3*Pi/14)) + cos(x*cos(3*Pi/14))* sin(3*Pi/14)*sinh(x*sin(3*Pi/14)) + cos(3*Pi/14)*sin(x*cos(3*Pi/14))* sinh(x*sin(3*Pi/14))))). - Vaclav Kotesovec, Jan 31 2015
In closed form, c = 7 / (r * (2*cos(r*sin(Pi/7))*cosh(r*cos(Pi/7)) + cos(Pi/7 - r*sin(Pi/7)) * cosh(r*cos(Pi/7)) + cos(Pi/7 - r*sin(Pi/7)) * cosh(r*cos(Pi/7)) + 2*cos(r*cos(Pi/14)) * cosh(r*sin(Pi/14)) + 2*cos(r*cos(3*Pi/14)) * cosh(r*sin(3*Pi/14)) + 2*cosh(r*sin(Pi/14)) * sin(Pi/14 + r*cos(Pi/14)) - 2*cosh(r*sin(3*Pi/14)) * sin(3*Pi/14 - r*cos(3*Pi/14)) - 2*cos(r*sin(Pi/7)) * sinh(r*cos(Pi/7)) - cos(Pi/7 - r*sin(Pi/7)) * sinh(r*cos(Pi/7)) - cos(Pi/7 - r*sin(Pi/7)) * sinh(r*cos(Pi/7)) - 2*cos(r*cos(Pi/14)) * sinh(r*sin(Pi/14)) - 2*sin(Pi/14 + r*cos(Pi/14))*sinh(r*sin(Pi/14)) + 2*cos(r*cos(3*Pi/14)) * sinh(r*sin(3*Pi/14)) - 2*sin((3*Pi)/14 - r*cos(3*Pi/14)) * sinh(r*sin(3*Pi/14)))). - Vaclav Kotesovec, Feb 01 2015

a(18)-a(22) from Alois P. Heinz, Oct 07 2013

a(0)=1 prepended by Alois P. Heinz, Aug 08 2018

A230231 Number of permutations of [n] avoiding adjacent step pattern {up}^8.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362879, 3628781, 39916492, 478996716, 6226941864, 87176969880, 1307651304960, 20922368987520, 355679390626560, 6402213152423659, 121641748198554547, 2432828930036156696, 51089280818439941448, 1123961390341566969192

Offset: 0

Alois P. Heinz, Oct 12 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450
Mingjia Yang, Doron Zeilberger, Increasing Consecutive Patterns in Words, arXiv:1805.06077 [math.CO], 2018.

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

Column k=255 of A242784.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t<7, 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=8;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[{}, {1/9, 2/9, 1/3, 4/9, 5/9, 2/3, 7/9, 8/9}, x^9/387420489] - x*HypergeometricPFQ[{}, {2/9, 1/3, 4/9, 5/9, 2/3, 7/9, 8/9, 10/9}, x^9/387420489]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Feb 01 2015 *)

E.g.f.: 1 / Sum_{n>=0} (9*n+1-x)*x^(9*n)/(9*n+1)!.
a(n)/n! ~ 1.0000195665100891649606434859189953881417919885320660432331680939719... * (1/r)^n, where r = 1.0000024802134092668222044475851121972165291678378389183730077680957571... is the root of the equation Sum_{n>=0} (r^(9*n)/(9*n)! - r^(9*n+1)/(9*n+1)!) = 0. - Vaclav Kotesovec, Jan 17 2015
E.g.f.: 1/(1/3 * cos((sqrt(3)*x)/2) * cosh(x/2) + 2/9 * cos(x * sin(Pi/9)) * cosh(x * cos(Pi/9)) + 2/9 * cos(Pi/9) * cos(x * sin(Pi/9)) * cosh(x * cos(Pi/9)) + 2/9 * cos(x * cos(Pi/18)) * cosh(x * sin(Pi/18)) + 2/9 * cos(x/2*(sqrt(3) * cos(Pi/9) - sin(Pi/9)))* cosh(x/2*(cos(Pi/9) + sqrt(3) * sin(Pi/9))) - 1/9 * cos(Pi/9) * cos(x/2*(sqrt(3) * cos(Pi/9) - sin(Pi/9))) * cosh(x/2*(cos(Pi/9) + sqrt(3) * sin(Pi/9))) - 2/9 * cos(x * cos(Pi/18))* cosh(x * sin(Pi/18)) * sin(Pi/18) - (cos(x/2*(sqrt(3) * cos(Pi/9) - sin(Pi/9)))* cosh(x/2*(cos(Pi/9) + sqrt(3) * sin(Pi/9)))* sin(Pi/9))/(3 * sqrt(3)) - (cosh(x/2) * sin((sqrt(3)*x)/2))/(3 * sqrt(3)) - 2/9 * cos(Pi/18) * cosh(x * sin(Pi/18)) * sin(x * cos(Pi/18)) - (cos(Pi/9) * cosh(x/2*(cos(Pi/9) + sqrt(3) * sin(Pi/9)))* sin(x/2*(sqrt(3) * cos(Pi/9) - sin(Pi/9))))/ (3 * sqrt(3)) + 1/9 * cosh(x/2*(cos(Pi/9) + sqrt(3) * sin(Pi/9)))* sin(Pi/9) * sin(x/2*(sqrt(3) * cos(Pi/9) - sin(Pi/9))) - 2/9 * cosh(x * cos(Pi/9)) * sin(Pi/9)* sin(x * sin(Pi/9)) - 1/3 * cos((sqrt(3)*x)/2)* sinh(x/2) + (sin((sqrt(3)*x)/2) * sinh(x/2))/ (3 * sqrt(3)) - 2/9 * cos(x * sin(Pi/9))* sinh(x * cos(Pi/9)) - 2/9 * cos(Pi/9) * cos(x * sin(Pi/9))* sinh(x * cos(Pi/9)) + 2/9 * sin(Pi/9) * sin(x * sin(Pi/9))* sinh(x * cos(Pi/9)) + 2/9 * cos(x * cos(Pi/18))* sinh(x * sin(Pi/18)) - 2/9 * cos(x * cos(Pi/18))* sin(Pi/18) * sinh(x * sin(Pi/18)) - 2/9 * cos(Pi/18)* sin(x * cos(Pi/18)) * sinh(x * sin(Pi/18)) + 2/9 * cos(x/2*(sqrt(3) * cos(Pi/9) - sin(Pi/9)))* sinh(x/2*(cos(Pi/9) + sqrt(3) * sin(Pi/9))) - 1/9 * cos(Pi/9) * cos(x/2*(sqrt(3) * cos(Pi/9) - sin(Pi/9))) * sinh(x/2*(cos(Pi/9) + sqrt(3) * sin(Pi/9))) - (cos(x/2*(sqrt(3) * cos(Pi/9) - sin(Pi/9))) * sin(Pi/9)* sinh(x/2*(cos(Pi/9) + sqrt(3) * sin(Pi/9))))/ (3 * sqrt(3)) - (cos(Pi/9) * sin(x/2*(sqrt(3) * cos(Pi/9) - sin(Pi/9)))* sinh(x/2*(cos(Pi/9) + sqrt(3) * sin(Pi/9))))/ (3 * sqrt(3)) + 1/9 * sin(Pi/9)* sin(x/2*(sqrt(3) * cos(Pi/9) - sin(Pi/9)))* sinh(x/2*(cos(Pi/9) + sqrt(3) * sin(Pi/9)))). - Vaclav Kotesovec, Feb 01 2015

A230232 Number of permutations of [n] avoiding adjacent step pattern {up}^9.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628799, 39916779, 479001228, 6227014404, 87178179816, 1307672369640, 20922752672640, 355686706327680, 6402359109968640, 121644792614741760, 2432895242801771955, 51090787299486057355, 1123997039003038423610

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, A230233.

Column k=511 of A242784.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t<8, 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=9;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 *)
FullSimplify[CoefficientList[Series[10/(2/E^x - Sqrt[2*(5 - Sqrt[5])]* Cosh[(1/4)*(1 + Sqrt[5])*x]* Sin[Sqrt[(1/8)*(5 - Sqrt[5])]*x] - Sqrt[2*(5 + Sqrt[5])]*Cosh[(1/4)*(Sqrt[5] - 1)* x]*Sin[Sqrt[(1/8)*(5 + Sqrt[5])]*x] + Cos[Sqrt[(1/8)*(5 + Sqrt[5])]*x]* (4*Cosh[(1/4)*(Sqrt[5] - 1)*x] - (Sqrt[5] - 1)*Sinh[(1/4)*(Sqrt[5] - 1)*x]) - Cos[Sqrt[(1/8)*(5 - Sqrt[5])]*x]* ((1 + Sqrt[5])*Sinh[(1/4)*(1 + Sqrt[5])*x] - 4*Cosh[(1/4)*(1 + Sqrt[5])*x])), {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 31 2015 *)

E.g.f.: 1 / Sum_{n>=0} (10*n+1-x)*x^(10*n)/(10*n+1)!.
a(n)/n! ~ c * (1/r)^n, where r = 1.0000002505217051890946793081039639693008257169189079028339632923816... is the root of the equation Sum_{n>=0} (r^(10*n)/(10*n)! - r^(10*n+1)/(10*n+1)!) = 0, c = 1.000002229648140602899529055054469878816530201510267349345270187155... . - Vaclav Kotesovec, Jan 17 2015
E.g.f.: 10 / (2/exp(x) - sqrt(2*(5 - sqrt(5))) * cosh((1/4)*(1 + sqrt(5))*x) * sin(sqrt((1/8)*(5 - sqrt(5)))*x) - sqrt(2*(5 + sqrt(5))) * cosh((1/4)*(sqrt(5) - 1)*x) * sin(sqrt((1/8)*(5 + sqrt(5)))*x) + cos(sqrt((1/8)*(5 + sqrt(5)))*x) * (4*cosh((1/4)*(sqrt(5) - 1)*x) - (sqrt(5) - 1)*sinh((1/4)*(sqrt(5) - 1)*x)) - cos(sqrt((1/8)*(5 - sqrt(5)))*x) * ((1 + sqrt(5))*sinh((1/4)*(1 + sqrt(5))*x) - 4*cosh((1/4)*(1 + sqrt(5))*x))). - Vaclav Kotesovec, Jan 31 2015
In closed form, c = 5 / (r * (sqrt(10 - 2*sqrt(5)) * cosh((sqrt(5)+1)*r/4) * sin(sqrt((5 - sqrt(5))/2)*r/2) + sqrt(2*(5 + sqrt(5))) * cosh((sqrt(5)-1)*r/4) * sin(sqrt((5 + sqrt(5))/2)*r/2))). - Vaclav Kotesovec, Feb 01 2015

cp's OEIS Frontend

A049774 Number of permutations of n elements not containing the consecutive pattern 123.

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A242784 Number A(n,k) of permutations of [n] avoiding the consecutive step pattern given by the binary expansion of k, where 1=up and 0=down; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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

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A202236 Number of permutations avoiding the consecutive pattern 35241.

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A117158 Number of permutations avoiding the consecutive pattern 1234.

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A177533 Number of permutations of 1..n avoiding adjacent step pattern up, up, up, up, up.

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A230051 Number of permutations of [n] avoiding adjacent step pattern {up}^7.

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A177553 Number of permutations of 1..n avoiding adjacent step pattern up, up, up, up, up, up.