OFFSET
0,3
COMMENTS
a(n) is the number of permutations of length n that avoid the consecutive pattern 12345 (or equivalently 54321).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..400 (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
Ira M. Gessel, Yan Zhuang, Counting permutations by alternating descents , 2014. See displayed equation before Eq. (3), and set m=5. - 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).
FORMULA
E.g.f.: 1/( Sum_{n>=0} x^(5*n)/(5*n)! - x^(5*n+1)/(5*n+1)! ).
a(n)/n! ~ c * (1/r)^n, where r = 1.007187547786015395418998654... is the root of the equation Sum_{n>=0} (r^(5*n)/(5*n)! - r^(5*n+1)/(5*n+1)!) = 0, c = 1.02806793756750152.... - Vaclav Kotesovec, Dec 11 2013
Equivalently, r = 1.00718754778601539541899865400272701484... is the root of the equation (5+sqrt(5)) * cos(sqrt((5-sqrt(5))/2)*r/2) + (5-sqrt(5)) * exp(sqrt(5)*r/2) * cos(sqrt((5+sqrt(5))/2)*r/2) - sqrt(2*(5-sqrt(5))) * sin(sqrt((5-sqrt(5))/2)*r/2) - sqrt(2*(5+sqrt(5))) * exp(sqrt(5)*r/2) * sin(sqrt((5+sqrt(5))/2)*r/2) = 0. - Vaclav Kotesovec, Aug 29 2014
E.g.f.: 10*exp((1+sqrt(5))*x/4) / ((5+sqrt(5)) * cos(sqrt((5-sqrt(5))/2)*x/2) + (5-sqrt(5)) * exp(sqrt(5)*x/2) * cos(sqrt((5+sqrt(5))/2)*x/2) - sqrt(2*(5-sqrt(5))) * sin(sqrt((5-sqrt(5))/2)*x/2) - sqrt(2*(5+sqrt(5))) * exp(sqrt(5)*x/2) * sin(sqrt((5+sqrt(5))/2)*x/2)). - Vaclav Kotesovec, Aug 29 2014
In closed form, c = 5*exp((1+sqrt(5))*r/4) / (r*((5 + sqrt(5)) * cos(sqrt((5 - sqrt(5))/2)*r/2) + (5 - sqrt(5)) * exp(sqrt(5)*r/2) * cos(sqrt((5 + sqrt(5))/2)*r/2))) = 1.0280679375675015201596831656779442465978511664638... . Vaclav Kotesovec, Feb 01 2015
EXAMPLE
E.g.f.: A(x) = 1 + x + 2*x^2/2! + 6*x^3/3! + 24*x^4/4! + 119*x^5/5! + 709*x^6/6! +...
where A(x) = 1/(1 - x + x^5/5! - x^6/6! + x^10/10! - x^11/11! + x^15/15! - x^16/16! + x^20/20! +...).
MATHEMATICA
Table[n!*SeriesCoefficient[1/(Sum[x^(5*k)/(5*k)!-x^(5*k+1)/(5*k+1)!, {k, 0, n}]), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 11 2013 *)
FullSimplify[CoefficientList[Series[10*E^((1+Sqrt[5])*x/4) / ((5+Sqrt[5]) * Cos[Sqrt[(5-Sqrt[5])/2]*x/2] + (5-Sqrt[5]) * E^(Sqrt[5]*x/2) * Cos[Sqrt[(5+Sqrt[5])/2]*x/2] - Sqrt[2*(5-Sqrt[5])] * Sin[Sqrt[(5-Sqrt[5])/2]*x/2] - Sqrt[2*(5+Sqrt[5])] * E^(Sqrt[5]*x/2) * Sin[Sqrt[(5+Sqrt[5])/2]*x/2]), {x, 0, 20}], x]*Range[0, 20]!] (* Vaclav Kotesovec, Aug 29 2014 *)
PROG
(PARI) {a(n)=n!*polcoeff(1/sum(m=0, n\5+1, x^(5*m)/(5*m)!-x^(5*m+1)/(5*m+1)!+x^2*O(x^n)), n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
R. H. Hardin, May 10 2010
EXTENSIONS
More terms from Ray Chandler, Dec 06 2011
a(0)=1 prepended by Alois P. Heinz, Jan 13 2015
STATUS
approved