<s>
In	O
the	O
mathematics	O
of	O
permutations	B-Algorithm
,	O
a	O
layered	B-Algorithm
permutation	I-Algorithm
is	O
a	O
permutation	B-Algorithm
that	O
reverses	O
contiguous	O
blocks	O
of	O
elements	O
.	O
</s>
<s>
Equivalently	O
,	O
it	O
is	O
the	O
direct	B-Algorithm
sum	I-Algorithm
of	O
decreasing	O
permutations	B-Algorithm
.	O
</s>
<s>
One	O
of	O
the	O
earlier	O
works	O
establishing	O
the	O
significance	O
of	O
layered	B-Algorithm
permutations	I-Algorithm
was	O
,	O
which	O
established	O
the	O
Stanley	O
–	O
Wilf	O
conjecture	O
for	O
classes	O
of	O
permutations	B-Algorithm
forbidding	O
a	O
layered	B-Algorithm
permutation	I-Algorithm
,	O
before	O
the	O
conjecture	O
was	O
proven	O
more	O
generally	O
.	O
</s>
<s>
The	O
layered	B-Algorithm
permutations	I-Algorithm
can	O
also	O
be	O
equivalently	O
described	O
as	O
the	O
permutations	B-Algorithm
that	O
do	O
not	O
contain	O
the	O
permutation	B-Algorithm
patterns	I-Algorithm
231	O
or	O
312	O
.	O
</s>
<s>
That	O
is	O
,	O
no	O
three	O
elements	O
in	O
the	O
permutation	B-Algorithm
(	O
regardless	O
of	O
whether	O
they	O
are	O
consecutive	O
)	O
have	O
the	O
same	O
ordering	O
as	O
either	O
of	O
these	O
forbidden	O
triples	O
.	O
</s>
<s>
A	O
layered	B-Algorithm
permutation	I-Algorithm
on	O
the	O
numbers	O
from	O
to	O
can	O
be	O
uniquely	O
described	O
by	O
the	O
subset	O
of	O
the	O
numbers	O
from	O
to	O
that	O
are	O
the	O
first	O
element	O
in	O
a	O
reversed	O
block	O
.	O
</s>
<s>
Because	O
there	O
are	O
subsets	O
of	O
the	O
numbers	O
from	O
to	O
,	O
there	O
are	O
also	O
layered	B-Algorithm
permutation	I-Algorithm
of	O
length	O
.	O
</s>
<s>
The	O
layered	B-Algorithm
permutations	I-Algorithm
are	O
Wilf	O
equivalent	O
to	O
other	O
permutation	B-Algorithm
classes	O
,	O
meaning	O
that	O
the	O
numbers	O
of	O
permutations	B-Algorithm
of	O
each	O
length	O
are	O
the	O
same	O
.	O
</s>
<s>
For	O
instance	O
,	O
the	O
Gilbreath	O
permutations	B-Algorithm
are	O
counted	O
by	O
the	O
same	O
function	O
.	O
</s>
<s>
The	O
shortest	O
superpattern	B-Algorithm
of	O
the	O
layered	B-Algorithm
permutations	I-Algorithm
of	O
length	O
is	O
itself	O
a	O
layered	B-Algorithm
permutation	I-Algorithm
.	O
</s>
<s>
Its	O
length	O
is	O
a	O
sorting	O
number	O
,	O
the	O
number	O
of	O
comparisons	O
needed	O
for	O
binary	B-Algorithm
insertion	I-Algorithm
sort	I-Algorithm
to	O
sort	O
elements	O
.	O
</s>
<s>
Every	O
layered	B-Algorithm
permutation	I-Algorithm
is	O
an	O
involution	B-Algorithm
.	O
</s>
<s>
They	O
are	O
exactly	O
the	O
231-avoiding	O
involutions	B-Algorithm
,	O
and	O
they	O
are	O
also	O
exactly	O
the	O
312-avoiding	O
involutions	B-Algorithm
.	O
</s>
<s>
The	O
layered	B-Algorithm
permutations	I-Algorithm
are	O
a	O
subset	O
of	O
the	O
stack-sortable	B-Algorithm
permutations	I-Algorithm
,	O
which	O
forbid	O
the	O
pattern	O
231	O
but	O
not	O
the	O
pattern	O
312	O
.	O
</s>
<s>
Like	O
the	O
stack-sortable	B-Algorithm
permutations	I-Algorithm
,	O
they	O
are	O
also	O
a	O
subset	O
of	O
the	O
separable	B-Algorithm
permutations	I-Algorithm
,	O
the	O
permutations	B-Algorithm
formed	O
by	O
recursive	O
combinations	O
of	O
direct	O
and	O
skew	O
sums	O
.	O
</s>
