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Number of cycle-up-down permutations of {1,2,...,n} having no fixed points. A permutation is said to be cycle-up-down if it is a product of up-down cycles. A cycle (b(1), b(2), ...) is said to be up-down if, when written with its smallest element in the first position, it satisfies b(1)<b(2)>b(3)<... .
4

%I #22 Nov 07 2020 06:00:40

%S 1,0,1,1,5,15,71,341,1945,12135,84091,635281,5212085,46091955,

%T 437198711,4426839821,47657861425,543551916975,6546911178931,

%U 83039587809961,1106307936885965,15445529882517195,225502102290364751,3436240674908121701,54555087491802061705

%N Number of cycle-up-down permutations of {1,2,...,n} having no fixed points. A permutation is said to be cycle-up-down if it is a product of up-down cycles. A cycle (b(1), b(2), ...) is said to be up-down if, when written with its smallest element in the first position, it satisfies b(1)<b(2)>b(3)<... .

%C a(n) = A186363(n,0).

%C Hankel transform is A154604. Binomial transform is A000111(n+1). - _Paul Barry_, Apr 11 2011

%H Nathaniel Johnston, <a href="/A186364/b186364.txt">Table of n, a(n) for n = 0..198</a>

%H Emeric Deutsch and Sergi Elizalde, <a href="http://arxiv.org/abs/0909.5199">Cycle up-down permutations</a>, arXiv:0909.5199 [math.CO], 2009; and <a href="https://ajc.maths.uq.edu.au/pdf/50/ajc_v50_p187.pdf">also</a>, Australas. J. Combin. 50 (2011), 187-199.

%F E.g.f.: exp(-z)/(1-sin(z)).

%F G.f.: 1/(1-x^2/(1-x-3*x^2/(1-2*x-6*x^2/(1-3*x-10*x^2/(1-.../(1-n*x-((n+1)*(n+2)/2)*x^2/(1-... (continued fraction). - _Paul Barry_, Apr 11 2011

%F a(n) ~ n! * n * exp(-Pi/2) * 2^(n+3) / Pi^(n+2). - _Vaclav Kotesovec_, Oct 08 2013

%F G.f.: conjecture: T(0), where T(k) = 1 - x^2*(k+1)*(k+2)/(x^2*(k+1)*(k+2) - 2*(1-x*k)*(1-x*(k+1))/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Nov 21 2013

%e a(4) = 5 because we have (12)(34),(13)(24),(1324),(1423), and (14)(23).

%p g := exp(-z)/(1-sin(z)): gser := series(g, z = 0, 28): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 24);

%t CoefficientList[Series[E^(-x)/(1-Sin[x]), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Oct 08 2013 *)

%Y Cf. A186363.

%K nonn

%O 0,5

%A _Emeric Deutsch_, Feb 28 2011