<s>
is	O
a	O
bijection	B-Algorithm
between	O
two	O
specified	O
subsets	O
of	O
S	O
.	O
That	O
is	O
,	O
it	O
is	O
defined	O
by	O
two	O
subsets	O
U	O
and	O
V	O
of	O
equal	O
size	O
,	O
and	O
a	O
one-to-one	O
mapping	O
from	O
U	O
to	O
V	O
.	O
Equivalently	O
,	O
it	O
is	O
a	O
partial	B-Algorithm
function	I-Algorithm
on	O
S	O
that	O
can	O
be	O
extended	O
to	O
a	O
permutation	B-Algorithm
.	O
</s>
<s>
In	O
this	O
case	O
,	O
a	O
partial	B-Algorithm
permutation	I-Algorithm
may	O
be	O
represented	O
by	O
a	O
string	O
of	O
n	O
symbols	O
,	O
some	O
of	O
which	O
are	O
distinct	O
numbers	O
in	O
the	O
range	O
from	O
1	O
to	O
and	O
the	O
remaining	O
ones	O
of	O
which	O
are	O
a	O
special	O
"	O
hole	O
"	O
symbol	O
◊	O
.	O
</s>
<s>
In	O
this	O
formulation	O
,	O
the	O
domain	B-Algorithm
U	O
of	O
the	O
partial	B-Algorithm
permutation	I-Algorithm
consists	O
of	O
the	O
positions	O
in	O
the	O
string	O
that	O
do	O
not	O
contain	O
a	O
hole	O
,	O
and	O
each	O
such	O
position	O
is	O
mapped	O
to	O
the	O
number	O
in	O
that	O
position	O
.	O
</s>
<s>
For	O
instance	O
,	O
the	O
string	O
"	O
1	O
◊	O
2	O
"	O
would	O
represent	O
the	O
partial	B-Algorithm
permutation	I-Algorithm
that	O
maps	O
1	O
to	O
itself	O
and	O
maps	O
3	O
to	O
2	O
.	O
</s>
<s>
in	O
which	O
the	O
ith	O
term	O
counts	O
the	O
number	O
of	O
partial	B-Algorithm
permutations	I-Algorithm
with	O
support	O
of	O
size	O
i	O
,	O
that	O
is	O
,	O
the	O
number	O
of	O
partial	B-Algorithm
permutations	I-Algorithm
with	O
i	O
non-hole	O
entries	O
.	O
</s>
<s>
partial	B-Algorithm
permutations	I-Algorithm
where	O
the	O
final	O
elements	O
of	O
each	O
set	O
are	O
omitted	O
:	O
</s>
<s>
partial	B-Algorithm
permutations	I-Algorithm
where	O
the	O
final	O
elements	O
of	O
each	O
set	O
map	O
to	O
each	O
other	O
.	O
</s>
<s>
,	O
the	O
partial	B-Algorithm
permutations	I-Algorithm
included	O
in	O
both	O
counts	O
3	O
and	O
4	O
,	O
those	O
permutations	B-Algorithm
where	O
the	O
final	O
elements	O
of	O
both	O
sets	O
are	O
included	O
,	O
but	O
do	O
not	O
map	O
to	O
each	O
other	O
.	O
</s>
<s>
or	O
the	O
range	O
of	O
the	O
bijection	B-Algorithm
is	O
forced	O
to	O
consist	O
of	O
the	O
first	O
k	O
items	O
in	O
the	O
set	O
of	O
n	O
items	O
being	O
permuted	O
,	O
for	O
some	O
k	O
.	O
In	O
the	O
former	O
case	O
,	O
a	O
partial	B-Algorithm
permutation	I-Algorithm
of	O
length	O
k	O
from	O
an	O
n-set	O
is	O
just	O
a	O
sequence	O
of	O
k	O
terms	O
from	O
the	O
n-set	O
without	O
repetition	O
.	O
</s>
<s>
(	O
In	O
elementary	O
combinatorics	O
,	O
these	O
objects	O
are	O
sometimes	O
confusingly	O
called	O
"	O
k-permutations	O
"	O
of	O
the	O
n-set	O
.	O
)	O
</s>
