<s>
In	O
combinatorics	O
,	O
the	O
skew	O
sum	O
and	O
direct	B-Algorithm
sum	I-Algorithm
of	I-Algorithm
permutations	I-Algorithm
are	O
two	O
operations	O
to	O
combine	O
shorter	O
permutations	B-Algorithm
into	O
longer	O
ones	O
.	O
</s>
<s>
The	O
skew	O
sum	O
of	O
the	O
permutations	B-Algorithm
π	O
=	O
2413	O
and	O
σ	O
=	O
35142	O
is	O
796835142	O
(	O
the	O
last	O
five	O
entries	O
are	O
equal	O
to	O
σ	O
,	O
while	O
the	O
first	O
four	O
entries	O
come	O
from	O
shifting	O
the	O
entries	O
of	O
π	O
)	O
while	O
their	O
direct	O
sum	O
is	O
241379586	O
(	O
the	O
first	O
four	O
entries	O
are	O
equal	O
to	O
π	O
,	O
while	O
the	O
last	O
five	O
come	O
from	O
shifting	O
the	O
entries	O
of	O
σ	O
)	O
.	O
</s>
<s>
Skew	B-Algorithm
and	I-Algorithm
direct	I-Algorithm
sums	I-Algorithm
of	I-Algorithm
permutations	I-Algorithm
appear	O
(	O
among	O
other	O
places	O
)	O
in	O
the	O
study	O
of	O
pattern	B-Algorithm
avoidance	I-Algorithm
in	O
permutations	B-Algorithm
.	O
</s>
<s>
Breaking	O
permutations	B-Algorithm
down	O
as	O
skew	O
and/or	O
direct	O
sums	O
of	O
a	O
maximal	O
number	O
of	O
parts	O
(	O
that	O
is	O
,	O
decomposing	O
into	O
indecomposable	O
parts	O
)	O
is	O
one	O
of	O
several	O
possible	O
techniques	O
used	O
to	O
study	O
the	O
structure	O
of	O
,	O
and	O
so	O
to	O
enumerate	O
,	O
pattern	O
classes	O
.	O
</s>
<s>
Permutations	B-Algorithm
whose	O
decomposition	O
by	O
skew	O
and	O
direct	O
sums	O
into	O
a	O
maximal	O
number	O
of	O
parts	O
,	O
that	O
is	O
,	O
can	O
be	O
built	O
up	O
from	O
the	O
permutations	B-Algorithm
(	O
1	O
)	O
,	O
are	O
called	O
separable	B-Algorithm
permutations	I-Algorithm
;	O
they	O
arise	O
in	O
the	O
study	O
of	O
sortability	O
theory	O
,	O
and	O
can	O
also	O
be	O
characterized	O
as	O
permutations	B-Algorithm
avoiding	O
the	O
permutation	B-Algorithm
patterns	I-Algorithm
2413	O
and	O
3142	O
.	O
</s>
<s>
The	O
skew	O
and	O
direct	O
sums	O
are	O
associative	O
but	O
not	O
commutative	O
,	O
and	O
they	O
do	O
not	O
associate	O
with	O
each	O
other	O
(	O
i.e.	O
,	O
for	O
permutations	B-Algorithm
π	O
,	O
σ	O
and	O
τ	O
we	O
typically	O
have	O
)	O
.	O
</s>
<s>
Given	O
a	O
permutation	B-Algorithm
ω	O
,	O
define	O
its	O
reverse	O
rev(ω )	O
to	O
be	O
the	O
permutation	B-Algorithm
whose	O
entries	O
appear	O
in	O
the	O
opposite	O
order	O
of	O
those	O
of	O
ω	O
when	O
written	O
in	O
one-line	O
notation	O
;	O
for	O
example	O
,	O
the	O
reverse	O
of	O
25143	O
is	O
34152	O
.	O
</s>
<s>
(	O
As	O
permutation	B-Algorithm
matrices	I-Algorithm
,	O
this	O
operation	O
is	O
reflection	O
across	O
a	O
horizontal	O
axis	O
.	O
)	O
</s>
