<s>
The	O
size	O
of	O
the	O
orbit	O
is	O
called	O
the	O
length	O
of	O
the	O
corresponding	O
cycle	O
;	O
when	O
,	O
the	O
single	O
element	O
in	O
the	O
orbit	O
is	O
called	O
a	O
fixed	O
point	O
of	O
the	O
permutation	B-Algorithm
.	O
</s>
<s>
A	O
permutation	B-Algorithm
is	O
determined	O
by	O
giving	O
an	O
expression	O
for	O
each	O
of	O
its	O
cycles	B-Algorithm
,	O
and	O
one	O
notation	O
for	O
permutations	B-Algorithm
consist	O
of	O
writing	O
such	O
expressions	O
one	O
after	O
another	O
in	O
some	O
order	O
.	O
</s>
<s>
be	O
a	O
permutation	B-Algorithm
that	O
maps	O
1	O
to	O
2	O
,	O
6	O
to	O
8	O
,	O
etc	O
.	O
</s>
<s>
It	O
is	O
typical	O
,	O
but	O
not	O
necessary	O
,	O
to	O
not	O
write	O
the	O
cycles	B-Algorithm
of	O
length	O
one	O
in	O
such	O
an	O
expression	O
.	O
</s>
<s>
Thus	O
,	O
π	O
=	O
(	O
1	O
2	O
4	O
3	O
)	O
(	O
6	O
8	O
)	O
,	O
would	O
be	O
an	O
appropriate	O
way	O
to	O
express	O
this	O
permutation	B-Algorithm
.	O
</s>
<s>
There	O
are	O
different	O
ways	O
to	O
write	O
a	O
permutation	B-Algorithm
as	O
a	O
list	O
of	O
its	O
cycles	B-Algorithm
,	O
but	O
the	O
number	O
of	O
cycles	B-Algorithm
and	O
their	O
contents	O
are	O
given	O
by	O
the	O
partition	O
of	O
S	O
into	O
orbits	O
,	O
and	O
these	O
are	O
therefore	O
the	O
same	O
for	O
all	O
such	O
expressions	O
.	O
</s>
<s>
The	O
unsigned	O
Stirling	O
number	O
of	O
the	O
first	O
kind	O
,	O
s(k, j )	O
counts	O
the	O
number	O
of	O
permutations	B-Algorithm
of	O
k	O
elements	O
with	O
exactly	O
j	O
disjoint	O
cycles	B-Algorithm
.	O
</s>
<s>
(	O
1	O
)	O
There	O
is	O
only	O
one	O
way	O
to	O
construct	O
a	O
permutation	B-Algorithm
of	O
k	O
elements	O
with	O
k	O
cycles	B-Algorithm
:	O
Every	O
cycle	O
must	O
have	O
length	O
1	O
so	O
every	O
element	O
must	O
be	O
a	O
fixed	O
point	O
.	O
</s>
<s>
(	O
2.a	O
)	O
Every	O
cycle	O
of	O
length	O
k	O
may	O
be	O
written	O
as	O
permutation	B-Algorithm
of	O
the	O
number	O
1	O
to	O
k	O
;	O
there	O
are	O
k	O
!	O
</s>
<s>
of	O
these	O
permutations	B-Algorithm
.	O
</s>
<s>
(	O
3	O
)	O
There	O
are	O
two	O
different	O
ways	O
to	O
construct	O
a	O
permutation	B-Algorithm
of	O
k	O
elements	O
with	O
j	O
cycles	B-Algorithm
:	O
</s>
<s>
(	O
3.a	O
)	O
If	O
we	O
want	O
element	O
k	O
to	O
be	O
a	O
fixed	O
point	O
we	O
may	O
choose	O
one	O
of	O
the	O
permutations	B-Algorithm
with	O
elements	O
and	O
cycles	B-Algorithm
and	O
add	O
element	O
k	O
as	O
a	O
new	O
cycle	O
of	O
length	O
1	O
.	O
</s>
<s>
(	O
3.b	O
)	O
If	O
we	O
want	O
element	O
k	O
not	O
to	O
be	O
a	O
fixed	O
point	O
we	O
may	O
choose	O
one	O
of	O
the	O
permutations	B-Algorithm
with	O
elements	O
and	O
j	O
cycles	B-Algorithm
and	O
insert	O
element	O
k	O
in	O
an	O
existing	O
cycle	O
in	O
front	O
of	O
one	O
of	O
the	O
elements	O
.	O
</s>
<s>
The	O
value	O
counts	O
the	O
number	O
of	O
permutations	B-Algorithm
of	O
k	O
elements	O
with	O
exactly	O
j	O
fixed	O
points	O
.	O
</s>
<s>
(	O
3	O
)	O
There	O
are	O
three	O
different	O
methods	O
to	O
construct	O
a	O
permutation	B-Algorithm
of	O
k	O
elements	O
with	O
j	O
fixed	O
points	O
:	O
</s>
<s>
(	O
3.a	O
)	O
We	O
may	O
choose	O
one	O
of	O
the	O
permutations	B-Algorithm
with	O
elements	O
and	O
fixed	O
points	O
and	O
add	O
element	O
k	O
as	O
a	O
new	O
fixed	O
point	O
.	O
</s>
<s>
(	O
3.b	O
)	O
We	O
may	O
choose	O
one	O
of	O
the	O
permutations	B-Algorithm
with	O
elements	O
and	O
j	O
fixed	O
points	O
and	O
insert	O
element	O
k	O
in	O
an	O
existing	O
cycle	O
of	O
length	O
>	O
1	O
in	O
front	O
of	O
one	O
of	O
the	O
elements	O
.	O
</s>
<s>
(	O
3.c	O
)	O
We	O
may	O
choose	O
one	O
of	O
the	O
permutations	B-Algorithm
with	O
elements	O
and	O
fixed	O
points	O
and	O
join	O
element	O
k	O
with	O
one	O
of	O
the	O
fixed	O
points	O
to	O
a	O
cycle	O
of	O
length	O
2	O
.	O
</s>
