A262970 Total cycle length of all iteration trajectories of all elements of random mappings from [n] to [n].

1, 10, 117, 1648, 27425, 528336, 11581885, 284878336, 7772592897, 233010784000, 7614411069221, 269412832512000, 10261487793254113, 418636033893726208, 18213563455467238125, 841799936112774086656, 41189866031118283907585, 2127207204243268173103104

Offset: 1

Author

Marko Riedel, Oct 05 2015

Keywords

• nonn

Comments

An iteration trajectory is the directed graph obtained by iterating the mapping starting from one of the n elements until a cycle appears and consists of a tail attached to a cycle.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..380
• P. Flajolet and A. M. Odlyzko, Random Mapping Statistics, INRIA RR 1114, 1989.
• Math StackExchange, Generating functions for tail length and rho-length

Crossrefs

Cf. A036360.

Programs

Maple

proc(n) 1/2*n!*add(n^q*(n + 1 - q)*(n - q)/q!, q = 0 .. n - 1) end proc

Mathematica

Table[n!/2 Sum[n^q (n + 1 - q) (n - q)/q!, {q, 0, n - 1}], {n, 21}] (* Michael De Vlieger, Oct 06 2015 *)
a[n_] := E^n n Gamma[n + 1, n] / 2;
Table[a[n], {n, 1, 19}]  (* Peter Luschny, Jul 20 2024 *)

PARI

a(n) = n! * sum(q=0, n-1, n^q*(n+1-q)*(n-q)/q!)/2;

Formula

E.g.f.: T/(1-T)^4, where T is the labeled tree function, average over all mappings and values asymptotic to sqrt(Pi*n/8).

a(n) = e^n * n * Gamma(n + 1, n) / 2. - Peter Luschny, Jul 20 2024