<s>
In	O
the	O
theory	O
of	O
permutation	B-Algorithm
patterns	I-Algorithm
,	O
a	O
skew-merged	B-Algorithm
permutation	I-Algorithm
is	O
a	O
permutation	B-Algorithm
that	O
can	O
be	O
partitioned	O
into	O
an	O
increasing	O
sequence	O
and	O
a	O
decreasing	O
sequence	O
.	O
</s>
<s>
The	O
two	O
smallest	O
permutations	B-Algorithm
that	O
cannot	O
be	O
partitioned	O
into	O
an	O
increasing	O
and	O
a	O
decreasing	O
sequence	O
are	O
3412	O
and	O
2143	O
.	O
was	O
the	O
first	O
to	O
establish	O
that	O
a	O
skew-merged	B-Algorithm
permutation	I-Algorithm
can	O
also	O
be	O
equivalently	O
defined	O
as	O
a	O
permutation	B-Algorithm
that	O
avoids	O
the	O
two	O
patterns	O
3412	O
and	O
2143	O
.	O
</s>
<s>
A	O
permutation	B-Algorithm
is	O
skew-merged	O
if	O
and	O
only	O
if	O
its	O
associated	O
permutation	B-Algorithm
graph	O
is	O
a	O
split	O
graph	O
,	O
a	O
graph	O
that	O
can	O
be	O
partitioned	O
into	O
a	O
clique	O
(	O
corresponding	O
to	O
the	O
descending	O
subsequence	O
)	O
and	O
an	O
independent	O
set	O
(	O
corresponding	O
to	O
the	O
ascending	O
subsequence	O
)	O
.	O
</s>
<s>
The	O
two	O
forbidden	O
patterns	O
for	O
skew-merged	B-Algorithm
permutations	I-Algorithm
,	O
3412	O
and	O
2143	O
,	O
correspond	O
to	O
two	O
of	O
the	O
three	O
forbidden	O
induced	O
subgraphs	O
for	O
split	O
graphs	O
,	O
a	O
four-vertex	O
cycle	O
and	O
a	O
graph	O
with	O
two	O
disjoint	O
edges	O
,	O
respectively	O
.	O
</s>
<s>
The	O
third	O
forbidden	O
induced	O
subgraph	O
,	O
a	O
five-vertex	O
cycle	O
,	O
cannot	O
exist	O
in	O
a	O
permutation	B-Algorithm
graph	O
(	O
see	O
)	O
.	O
</s>
<s>
Another	O
derivation	O
of	O
the	O
generating	O
function	O
for	O
skew-merged	B-Algorithm
permutations	I-Algorithm
was	O
given	O
by	O
.	O
</s>
<s>
Testing	O
whether	O
one	O
permutation	B-Algorithm
is	O
a	O
pattern	O
in	O
another	O
can	O
be	O
solved	O
efficiently	O
when	O
the	O
larger	O
of	O
the	O
two	O
permutations	B-Algorithm
is	O
skew-merged	O
,	O
as	O
shown	O
by	O
.	O
</s>
