A262973 Total tail length of all iteration trajectories of all elements of random mappings from [n] to [n].

0, 2, 36, 624, 11800, 248400, 5817084, 150660608, 4285808496, 133010784000, 4475982692500, 162419627132928, 6324111407554824, 263067938335913984, 11645155099754347500, 546652030933421260800, 27126781579050558916576, 1418971858887930496745472

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

Programs

Maple

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

Mathematica

Table[n!/2 Sum[n^q (n - q) (n - 1 - q)/q!, {q, 0, n - 2}], {n, 21}] (* Michael De Vlieger, Oct 06 2015 *)

Formula

E.g.f.: T^2/(1-T)^4 where T is the labeled tree function, average over all mappings and values is asymptotic to sqrt(Pi*n/8).