OFFSET
0,3
COMMENTS
a(n) is the number of permutations of length n that avoid the consecutive pattern 123456 (or equivalently 654321).
LINKS
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.
FORMULA
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
MAPLE
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
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
R. H. Hardin, May 10 2010
EXTENSIONS
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
STATUS
approved