<s>
In	O
combinatorial	O
mathematics	O
,	O
a	O
separable	B-Algorithm
permutation	I-Algorithm
is	O
a	O
permutation	B-Algorithm
that	O
can	O
be	O
obtained	O
from	O
the	O
trivial	O
permutation	B-Algorithm
1	O
by	O
direct	B-Algorithm
sums	I-Algorithm
and	O
skew	B-Algorithm
sums	I-Algorithm
.	O
</s>
<s>
Separable	B-Algorithm
permutations	I-Algorithm
may	O
be	O
characterized	O
by	O
the	O
forbidden	O
permutation	B-Algorithm
patterns	I-Algorithm
2413	O
and	O
3142	O
;	O
they	O
are	O
also	O
the	O
permutations	B-Algorithm
whose	O
permutation	B-Algorithm
graphs	O
are	O
cographs	O
and	O
the	O
permutations	B-Algorithm
that	O
realize	O
the	O
series-parallel	O
partial	O
orders	O
.	O
</s>
<s>
It	O
is	O
possible	O
to	O
test	O
in	O
polynomial	O
time	O
whether	O
a	O
given	O
separable	B-Algorithm
permutation	I-Algorithm
is	O
a	O
pattern	O
in	O
a	O
larger	O
permutation	B-Algorithm
,	O
or	O
to	O
find	O
the	O
longest	O
common	O
subpattern	O
of	O
two	O
separable	B-Algorithm
permutations	I-Algorithm
.	O
</s>
<s>
define	O
a	O
separable	B-Algorithm
permutation	I-Algorithm
to	O
be	O
a	O
permutation	B-Algorithm
that	O
has	O
a	O
separating	O
tree	O
:	O
a	O
rooted	O
binary	O
tree	O
in	O
which	O
the	O
elements	O
of	O
the	O
permutation	B-Algorithm
appear	O
(	O
in	O
permutation	B-Algorithm
order	O
)	O
at	O
the	O
leaves	O
of	O
the	O
tree	O
,	O
and	O
in	O
which	O
the	O
descendants	O
of	O
each	O
tree	O
node	O
form	O
a	O
contiguous	O
subset	O
of	O
these	O
elements	O
.	O
</s>
<s>
There	O
may	O
be	O
more	O
than	O
one	O
tree	O
for	O
a	O
given	O
permutation	B-Algorithm
:	O
if	O
two	O
nodes	O
that	O
are	O
adjacent	O
in	O
the	O
same	O
tree	O
have	O
the	O
same	O
sign	O
,	O
then	O
they	O
may	O
be	O
replaced	O
by	O
a	O
different	O
pair	O
of	O
nodes	O
using	O
a	O
tree	B-Data_Structure
rotation	I-Data_Structure
operation	O
.	O
</s>
<s>
Each	O
subtree	O
of	O
a	O
separating	O
tree	O
may	O
be	O
interpreted	O
as	O
itself	O
representing	O
a	O
smaller	O
separable	B-Algorithm
permutation	I-Algorithm
,	O
whose	O
element	O
values	O
are	O
determined	O
by	O
the	O
shape	O
and	O
sign	O
pattern	O
of	O
the	O
subtree	O
.	O
</s>
<s>
A	O
one-node	O
tree	O
represents	O
the	O
trivial	O
permutation	B-Algorithm
,	O
a	O
tree	O
whose	O
root	O
node	O
is	O
positive	O
represents	O
the	O
direct	B-Algorithm
sum	I-Algorithm
of	I-Algorithm
permutations	I-Algorithm
given	O
by	O
its	O
two	O
child	O
subtrees	O
,	O
and	O
a	O
tree	O
whose	O
root	O
node	O
is	O
negative	O
represents	O
the	O
skew	B-Algorithm
sum	I-Algorithm
of	O
the	O
permutations	B-Algorithm
given	O
by	O
its	O
two	O
child	O
subtrees	O
.	O
</s>
<s>
In	O
this	O
way	O
,	O
a	O
separating	O
tree	O
is	O
equivalent	O
to	O
a	O
construction	O
of	O
the	O
permutation	B-Algorithm
by	O
direct	O
and	O
skew	B-Algorithm
sums	I-Algorithm
,	O
starting	O
from	O
the	O
trivial	O
permutation	B-Algorithm
.	O
</s>
<s>
As	O
prove	O
,	O
separable	B-Algorithm
permutations	I-Algorithm
may	O
also	O
be	O
characterized	O
in	O
terms	O
of	O
permutation	B-Algorithm
patterns	I-Algorithm
:	O
a	O
permutation	B-Algorithm
is	O
separable	O
if	O
and	O
only	O
if	O
it	O
contains	O
neither	O
2413	O
nor	O
3142	O
as	O
a	O
pattern	O
.	O
</s>
<s>
The	O
separable	B-Algorithm
permutations	I-Algorithm
also	O
have	O
a	O
characterization	O
from	O
algebraic	O
geometry	O
:	O
if	O
a	O
collection	O
of	O
distinct	O
real	O
polynomials	O
all	O
have	O
equal	O
values	O
at	O
some	O
number	O
,	O
then	O
the	O
permutation	B-Algorithm
that	O
describes	O
how	O
the	O
numerical	O
ordering	O
of	O
the	O
polynomials	O
changes	O
at	O
is	O
separable	O
,	O
and	O
every	O
separable	B-Algorithm
permutation	I-Algorithm
can	O
be	O
realized	O
in	O
this	O
way	O
.	O
</s>
<s>
The	O
separable	B-Algorithm
permutations	I-Algorithm
are	O
enumerated	O
by	O
the	O
Schröder	O
numbers	O
.	O
</s>
<s>
This	O
result	O
was	O
proven	O
for	O
a	O
class	O
of	O
permutation	B-Algorithm
matrices	I-Algorithm
equivalent	O
to	O
the	O
separable	B-Algorithm
permutations	I-Algorithm
by	O
,	O
by	O
using	O
a	O
canonical	O
form	O
of	O
the	O
separating	O
tree	O
in	O
which	O
the	O
right	O
child	O
of	O
each	O
node	O
has	O
a	O
different	O
sign	O
than	O
the	O
node	O
itself	O
and	O
then	O
applying	O
the	O
theory	O
of	O
generating	O
functions	O
to	O
these	O
trees	O
.	O
</s>
<s>
Another	O
proof	O
applying	O
more	O
directly	O
to	O
separable	B-Algorithm
permutations	I-Algorithm
themselves	O
,	O
was	O
given	O
by	O
.	O
</s>
<s>
showed	O
that	O
it	O
is	O
possible	O
to	O
determine	O
in	O
polynomial	O
time	O
whether	O
a	O
given	O
separable	B-Algorithm
permutation	I-Algorithm
is	O
a	O
pattern	O
in	O
a	O
larger	O
permutation	B-Algorithm
,	O
in	O
contrast	O
to	O
the	O
same	O
problem	O
for	O
non-separable	O
permutations	O
,	O
which	O
is	O
NP-complete	O
.	O
</s>
<s>
The	O
problem	O
of	O
finding	O
the	O
longest	O
separable	O
pattern	O
that	O
is	O
common	O
to	O
a	O
set	O
of	O
input	O
permutations	B-Algorithm
may	O
be	O
solved	O
in	O
polynomial	O
time	O
for	O
a	O
fixed	O
number	O
of	O
input	O
permutations	B-Algorithm
,	O
but	O
is	O
NP-hard	O
when	O
the	O
number	O
of	O
input	O
permutations	B-Algorithm
may	O
be	O
variable	O
,	O
and	O
remains	O
NP-hard	O
even	O
when	O
the	O
inputs	O
are	O
all	O
themselves	O
separable	O
.	O
</s>
<s>
Separable	B-Algorithm
permutations	I-Algorithm
first	O
arose	O
in	O
the	O
work	O
of	O
,	O
who	O
showed	O
that	O
they	O
are	O
precisely	O
the	O
permutations	B-Algorithm
which	O
can	O
be	O
sorted	O
by	O
an	O
arbitrary	O
number	O
of	O
pop-stacks	O
in	O
series	O
,	O
where	O
a	O
pop-stack	O
is	O
a	O
restricted	O
form	O
of	O
stack	B-Application
in	O
which	O
any	O
pop	O
operation	O
pops	O
all	O
items	O
at	O
once	O
.	O
</s>
<s>
considered	O
separable	B-Algorithm
permutations	I-Algorithm
again	O
in	O
their	O
study	O
of	O
bootstrap	O
percolation	O
,	O
a	O
process	O
in	O
which	O
an	O
initial	O
permutation	B-Algorithm
matrix	I-Algorithm
is	O
modified	O
by	O
repeatedly	O
changing	O
to	O
one	O
any	O
matrix	O
coefficient	O
that	O
has	O
two	O
or	O
more	O
orthogonal	O
neighbors	O
equal	O
to	O
one	O
.	O
</s>
<s>
As	O
they	O
show	O
,	O
the	O
class	O
of	O
permutations	B-Algorithm
that	O
are	O
transformed	O
by	O
this	O
process	O
into	O
the	O
all-one	O
matrix	O
is	O
exactly	O
the	O
class	O
of	O
separable	B-Algorithm
permutations	I-Algorithm
.	O
</s>
<s>
The	O
term	O
"	O
separable	B-Algorithm
permutation	I-Algorithm
"	O
was	O
introduced	O
later	O
by	O
,	O
who	O
considered	O
them	O
for	O
their	O
algorithmic	O
properties	O
.	O
</s>
<s>
Every	O
permutation	B-Algorithm
can	O
be	O
used	O
to	O
define	O
a	O
permutation	B-Algorithm
graph	O
,	O
a	O
graph	O
whose	O
vertices	O
are	O
the	O
elements	O
of	O
the	O
permutation	B-Algorithm
and	O
whose	O
edges	O
are	O
the	O
inversions	B-Algorithm
of	O
the	O
permutation	B-Algorithm
.	O
</s>
<s>
In	O
the	O
case	O
of	O
a	O
separable	B-Algorithm
permutation	I-Algorithm
,	O
the	O
structure	O
of	O
this	O
graph	O
can	O
be	O
read	O
off	O
from	O
the	O
separation	O
tree	O
of	O
the	O
permutation	B-Algorithm
:	O
two	O
vertices	O
of	O
the	O
graph	O
are	O
adjacent	O
if	O
and	O
only	O
if	O
their	O
lowest	O
common	O
ancestor	O
in	O
the	O
separation	O
tree	O
is	O
negative	O
.	O
</s>
<s>
Thus	O
,	O
the	O
separable	B-Algorithm
permutations	I-Algorithm
are	O
exactly	O
the	O
permutations	B-Algorithm
whose	O
permutation	B-Algorithm
graphs	O
are	O
cographs	O
.	O
</s>
<s>
The	O
forbidden	O
graph	O
characterization	O
of	O
the	O
cographs	O
(	O
they	O
are	O
the	O
graphs	O
with	O
no	O
four-vertex	O
induced	O
path	O
)	O
corresponds	O
to	O
the	O
two	O
four-element	O
forbidden	O
patterns	O
of	O
the	O
separable	B-Algorithm
permutations	I-Algorithm
.	O
</s>
<s>
Separable	B-Algorithm
permutations	I-Algorithm
are	O
also	O
closely	O
related	O
to	O
series-parallel	O
partial	O
orders	O
,	O
the	O
partially	O
ordered	O
sets	O
whose	O
comparability	O
graphs	O
are	O
the	O
cographs	O
.	O
</s>
<s>
As	O
with	O
the	O
cographs	O
and	O
separable	B-Algorithm
permutations	I-Algorithm
,	O
the	O
series-parallel	O
partial	O
orders	O
may	O
also	O
be	O
characterized	O
by	O
four-element	O
forbidden	O
suborders	O
.	O
</s>
<s>
Every	O
permutation	B-Algorithm
defines	O
a	O
partial	O
order	O
whose	O
order	O
dimension	O
is	O
two	O
,	O
in	O
which	O
the	O
elements	O
to	O
be	O
ordered	O
are	O
the	O
elements	O
of	O
the	O
permutation	B-Algorithm
,	O
and	O
in	O
which	O
x≤y	O
whenever	O
x	O
has	O
a	O
smaller	O
numerical	O
value	O
than	O
y	O
and	O
is	O
left	O
of	O
it	O
in	O
the	O
permutation	B-Algorithm
.	O
</s>
<s>
The	O
permutations	B-Algorithm
for	O
which	O
this	O
partial	O
order	O
is	O
series-parallel	O
are	O
exactly	O
the	O
separable	B-Algorithm
permutations	I-Algorithm
.	O
</s>
<s>
Separable	B-Algorithm
permutations	I-Algorithm
may	O
also	O
be	O
used	O
to	O
describe	O
hierarchical	O
partitions	O
of	O
rectangles	O
into	O
smaller	O
rectangles	O
(	O
so-called	O
"	O
slicing	O
floorplans	O
"	O
,	O
used	O
for	O
instance	O
in	O
the	O
design	O
of	O
integrated	O
circuits	O
)	O
by	O
using	O
the	O
positive	O
and	O
negative	O
signs	O
of	O
the	O
separating	O
tree	O
to	O
describe	O
horizontal	O
and	O
vertical	O
slices	O
of	O
a	O
rectangle	O
into	O
smaller	O
rectangles	O
.	O
</s>
<s>
The	O
separable	B-Algorithm
permutations	I-Algorithm
include	O
as	O
a	O
special	O
case	O
the	O
stack-sortable	B-Algorithm
permutations	I-Algorithm
,	O
which	O
avoid	O
the	O
pattern	O
231	O
.	O
</s>
