The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A038205 Number of derangements of n where minimal cycle size is at least 3. 110
 1, 0, 0, 2, 6, 24, 160, 1140, 8988, 80864, 809856, 8907480, 106877320, 1389428832, 19452141696, 291781655984, 4668504894480, 79364592318720, 1428562679845888, 27142690734936864, 542853814536802656, 11399930109077490560, 250798462399300784640 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Permutations with no cycles of length 1 or 2. Related to (and bounded by) "derangements" (A000166). Minimal cycle size 3 is interesting because of its physical analog. Consider a fully-connected network of n nodes where the objects stored at the nodes must derange but can't do so in such a way that any two objects would collide along the connecting "pipe" between their nodes. REFERENCES G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. H. S. Wilf, Generatingfunctionology, Academic Press, NY, 1990, p. 147, Eq. 5.2.9 (q=2). LINKS Alois P. Heinz, Table of n, a(n) for n = 0..200 (first 51 terms from Arie Bos) Joerg Arndt, Generating Random Permutations, PhD thesis, Australian National University, Canberra, Australia, (2010). Poly H. da Silva, Arash Jamshidpey, and Simon Tavaré, Random derangements and the Ewens Sampling Formula, arXiv:2006.04840 [math.PR], 2020. J. East and R. D. Gray, Idempotent generators in finite partition monoids and related semigroups, arXiv preprint arXiv:1404.2359, 2014. See Table 4. Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010. G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission] H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, p. 176, Eq. 5.2.9 (q=2). FORMULA a(n) = Sum_{i=3..n} binomial(n-1,i-1) * (i-1)! * a(n-i). E.g.f.: exp(-x-x^2/2)/(1-x) = exp( Sum_{k>2} x^k / k ). a(n) = n! * Sum_{m=1..n} ((Sum_{k=0..m} k!*(-1)^(m-k) *binomial(m,k) *Sum_{i=0..n-m} abs(stirling1(i+k,k)) *binomial(m-k,n-m-i) *2^(-n+m+i) /(i+k)!))/m!; a(0)=1. - Vladimir Kruchinin, Feb 01 2011 a(n) = A000166(n) - A158243(n). - Paul Weisenhorn, May 29 2010 a(n) = (n-1)*a(n-1) + (n-1)*(n-2)*a(n-3). - Peter Bala, Apr 18 2012 a(n) ~ n! * exp(-3/2). - Vaclav Kotesovec, Jul 30 2013 EXAMPLE a(5) = 24 because, with a minimum cycle size of 3, the only way to derange all 5 elements is to have them move around in one large 5-cycle. The number of possible moves is (5-1)! = 4! = 24. MAPLE with(combstruct): ZL2:=[S, {S=Set(Cycle(Z, card>2))}, labeled] :seq(count(ZL2, size=n), n=0..21); # Zerinvary Lajos, Sep 26 2007 with(combstruct):a:=proc(m) [ZZ, {ZZ=Set(Cycle(Z, card>m))}, labeled]; end: A038205:=a(2):seq(count(A038205, size=n), n=0..21); # Zerinvary Lajos, Oct 02 2007 G:= exp(-x-x^2/2)/(1-x): Gser:=series(G, x, 26): a:= n-> n!*coeff(Gser, x, n): seq(a(n), n=0..25); # Paul Weisenhorn, May 29 2010 MATHEMATICA max = 21; f[x_] := Exp[-x - x^2/2]/(1 - x); CoefficientList[ Series[ f[x], {x, 0, max}], x]*Range[0, max]! (* Jean-François Alcover, Dec 07 2011, after Vladimir Kruchinin *) PROG (PARI) x='x+O('x^30); Vec(serlaplace(exp(-x-x^2/2)/(1-x))) \\ G. C. Greubel, Jun 25 2018 (Magma) m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(-x-x^2/2)/(1-x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jun 25 2018 CROSSREFS Cf. A000166, A047865, A158243. Sequence in context: A275955 A009608 A012715 * A012361 A308111 A121773 Adjacent sequences: A038202 A038203 A038204 * A038206 A038207 A038208 KEYWORD nonn,easy,nice AUTHOR Charles G. Moore, N. J. A. Sloane EXTENSIONS Definition corrected by Brendan McKay, Jun 02 2007 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified February 28 04:16 EST 2024. Contains 370379 sequences. (Running on oeis4.)