The smallest size of components in random decomposable combinatorial structures is studied in a general framework. The results apply to several combinatorial structures in both the labelled and the unlabelled case. Typical examples are the cycle decomposition of permutations and the factorization of polynomials over finite fields into irreducible factors.
(L): F(z) 



(U): F(z) 



(1) 
I(z)~ 

a log 

+c_{0}+O((1z/r) 

), (2) 
F(z)~ e^{I(z)}~ c_{1}(1z)^{a}, c_{1}=e 

. (3) 

~ (ae 

G(a)) 

. 
uw(u)=1 (1£ u £ 2),  (  uw(u)  )  '=w(u1) (u>2). 
Pr  {L_{n}£ m}=[z^{n}]P 

(z), Pr{S_{n}> m}=[z^{n}]P^{[> m]}(z). 
log 

=s_{m}(z)+r_{m}(z), s_{m}(z):= 


, r_{m}(z):= 


. 
E[S_{n}]º 

P_{n}^{[>m]} = e 

log n(1+o(1)), 
Pr{S_{n}>m}= 

w 
æ ç ç è 

ö ÷ ÷ ø 
+O 
æ ç ç è 

+ 

ö ÷ ÷ ø 
, 
E(v):=  ó õ 

e^{w} 

. 
r_{m}(e^{h})=E(mh)+O 
æ ç ç è 

ö ÷ ÷ ø 
. 
r_{m}(e^{h})= 

ó õ 

e^{s} 

ds, 

= 
æ ç ç è 

 

ö ÷ ÷ ø 
+ 

, 

(9) 
This document was translated from L^{A}T_{E}X by H^{E}V^{E}A.