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

Author

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

Keywords

• nonn
• nice
• easy

Comments

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

Examples

			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.
		

References

• 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).

Links

• 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.

Crossrefs

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

Programs

Maple

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

Mathematica

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 *)

Formula

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

Extensions

Corrected and extended by Vladeta Jovovic, Apr 14 2001