<s>
In	O
mathematics	O
,	O
the	O
Golomb	B-Algorithm
–	I-Algorithm
Dickman	I-Algorithm
constant	I-Algorithm
arises	O
in	O
the	O
theory	O
of	O
random	B-Algorithm
permutations	I-Algorithm
and	O
in	O
number	O
theory	O
.	O
</s>
<s>
Let	O
an	O
be	O
the	O
average	O
taken	O
over	O
all	O
permutations	B-Algorithm
of	O
a	O
set	O
of	O
size	O
n	O
of	O
the	O
length	O
of	O
the	O
longest	O
cycle	B-Algorithm
in	O
each	O
permutation	B-Algorithm
.	O
</s>
<s>
In	O
the	O
language	O
of	O
probability	O
theory	O
,	O
is	O
asymptotically	O
the	O
expected	O
length	O
of	O
the	O
longest	O
cycle	B-Algorithm
in	O
a	O
uniformly	O
distributed	O
random	B-Algorithm
permutation	I-Algorithm
of	O
a	O
set	O
of	O
size	O
n	O
.	O
</s>
<s>
In	O
number	O
theory	O
,	O
the	O
Golomb	B-Algorithm
–	I-Algorithm
Dickman	I-Algorithm
constant	I-Algorithm
appears	O
in	O
connection	O
with	O
the	O
average	O
size	O
of	O
the	O
largest	O
prime	O
factor	O
of	O
an	O
integer	O
.	O
</s>
<s>
The	O
Golomb	B-Algorithm
–	I-Algorithm
Dickman	I-Algorithm
constant	I-Algorithm
appears	O
in	O
number	O
theory	O
in	O
a	O
different	O
way	O
.	O
</s>
<s>
The	O
Golomb-Dickman	B-Algorithm
constant	I-Algorithm
also	O
arises	O
when	O
we	O
consider	O
the	O
average	O
length	O
of	O
the	O
largest	O
cycle	B-Algorithm
of	O
any	O
function	O
from	O
a	O
finite	O
set	O
to	O
itself	O
.	O
</s>
<s>
If	O
X	O
is	O
a	O
finite	O
set	O
,	O
if	O
we	O
repeatedly	O
apply	O
a	O
function	O
f	O
:	O
X	O
X	O
to	O
any	O
element	O
x	O
of	O
this	O
set	O
,	O
it	O
eventually	O
enters	O
a	O
cycle	B-Algorithm
,	O
meaning	O
that	O
for	O
some	O
k	O
we	O
have	O
for	O
sufficiently	O
large	O
n	O
;	O
the	O
smallest	O
k	O
with	O
this	O
property	O
is	O
the	O
length	O
of	O
the	O
cycle	B-Algorithm
.	O
</s>
<s>
Let	O
bn	O
be	O
the	O
average	O
,	O
taken	O
over	O
all	O
functions	O
from	O
a	O
set	O
of	O
size	O
n	O
to	O
itself	O
,	O
of	O
the	O
length	O
of	O
the	O
largest	O
cycle	B-Algorithm
.	O
</s>
