A177260 Number of derangements of {1,2,...,n} having no adjacent 4-cycles (an adjacent 4-cycle is a cycle of the form (i,i+1,i+2,i+3)).

1, 0, 1, 2, 8, 44, 262, 1846, 14789, 133232, 1333112, 14669758, 176081478, 2289458896, 32056423888, 480890367598, 7694774125983, 130818028518432, 2354820682603399, 44743035640567412, 894883797133726171, 18792952193893804872, 413452012727711517437

Offset: 0

Author

Emeric Deutsch, May 08 2010

Keywords

• nonn

Examples

			a(6)=262 because among the 265 (= A000166(6)) derangements of {1,2,3,4,5,6} only (1234)(56), (16)(2345), and (12)(3456) have adjacent 4-cycles.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..445
• R. A. Brualdi and Emeric Deutsch, Adjacent q-cycles in permutations, arXiv:1005.0781 [math.CO], 2010.

Crossrefs

Cf. A000166, A177258, A177259.

Programs

Magma

F:=Factorial;
A177258:= func< n | (&+[(&+[(-1)^(j+k)*F(n-3*k)/(F(j)*F(k)): k in [0..Floor((n-j)/4)]]): j in [0..n]]) >;
[A177258(n): n in [0..40]]; // G. C. Greubel, May 13 2024

Maple

a := proc (n) local ct, t, s: ct := 0: for s from 0 to n do for t from 0 to (1/4)*n do if s+4*t <= n then ct := ct+(-1)^(s+t)*factorial(n-3*t)/(factorial(s)*factorial(t)) else end if end do end do: ct end proc; seq(a(n), n = 0 .. 22);

Mathematica

a[n_] := Module[{ct = 0, t, s}, For[s = 0, s <= n, s++, For[t = 0, t <= n/3, t++, If[s + 4*t <= n, ct = ct + (-1)^(s + t)*Factorial[n - 3*t] / (Factorial[s]*Factorial[t])]]]; ct];
Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Nov 24 2017, translated from Maple *)

SageMath

f=factorial;
def A177260(n): return sum(sum((-1)^(j+k)*f(n-3*k)/(f(j)*f(k)) for k in range(1+(n-j)//4)) for j in range(n+1))
[A177260(n) for n in range(41)] # G. C. Greubel, May 13 2024

Formula

a(n) = Sum_{s=0..n} Sum_{t=0..floor((n-s)/4)} (-1)^(s+t)*(n-3*t)!/(s!*t!).

a(n) ~ exp(-1) * n!. - Vaclav Kotesovec, Dec 10 2021

Conjecture: D-finite with recurrence a(n) = (n-1)*a(n-1) + (n-1)*a(n-2) + 2*a(n-4) + (n-1)*a(n-5) + 3*a(n-8). - R. J. Mathar, Jul 26 2022

G.f.: Sum_{k>=0} k! * x^k / (1+x+x^4)^(k+1). - Seiichi Manyama, Feb 22 2024