<s>
In	O
mathematics	O
,	O
the	O
permutoassociahedron	B-Algorithm
is	O
an	O
-dimensional	O
polytope	O
whose	O
vertices	O
correspond	O
to	O
the	O
bracketings	O
of	O
the	O
permutations	B-Algorithm
of	O
terms	O
and	O
whose	O
edges	O
connect	O
two	O
bracketings	O
that	O
can	O
be	O
obtained	O
from	O
one	O
another	O
either	O
by	O
moving	O
a	O
pair	O
of	O
brackets	O
using	O
associativity	O
or	O
by	O
transposing	B-Algorithm
two	O
consecutive	O
terms	O
that	O
are	O
not	O
separated	O
by	O
a	O
bracket	O
.	O
</s>
<s>
The	O
permutoassociahedron	B-Algorithm
was	O
first	O
defined	O
as	O
a	O
CW	O
complex	O
by	O
Mikhail	O
Kapranov	O
who	O
noted	O
that	O
this	O
structure	O
appears	O
implicitly	O
in	O
Mac	O
Lane	O
's	O
coherence	O
theorem	O
for	O
symmetric	O
and	O
braided	O
categories	O
as	O
well	O
as	O
in	O
Vladimir	O
Drinfeld	O
's	O
work	O
on	O
the	O
Knizhnik	O
–	O
Zamolodchikov	O
equations	O
.	O
</s>
<s>
When	O
,	O
the	O
vertices	O
of	O
the	O
permutoassociahedron	B-Algorithm
can	O
be	O
represented	O
by	O
bracketing	O
all	O
the	O
permutations	B-Algorithm
of	O
three	O
terms	O
,	O
,	O
and	O
.	O
</s>
<s>
There	O
are	O
six	O
such	O
permutations	B-Algorithm
,	O
,	O
,	O
,	O
,	O
,	O
and	O
,	O
and	O
each	O
of	O
them	O
admits	O
two	O
bracketings	O
(	O
obtained	O
from	O
one	O
another	O
by	O
associativity	O
)	O
.	O
</s>
<s>
Hence	O
,	O
the	O
-dimensional	O
permutoassociahedron	B-Algorithm
is	O
the	O
dodecagon	O
with	O
vertices	O
,	O
,	O
,	O
,	O
,	O
,	O
,	O
,	O
,	O
,	O
,	O
and	O
.	O
</s>
<s>
When	O
,	O
the	O
vertex	O
is	O
adjacent	O
to	O
exactly	O
three	O
other	O
vertices	O
of	O
the	O
permutoassociahedron	B-Algorithm
:	O
,	O
,	O
and	O
.	O
</s>
<s>
The	O
first	O
two	O
vertices	O
are	O
reached	O
from	O
via	O
associativity	O
and	O
the	O
third	O
via	O
a	O
transposition	B-Algorithm
.	O
</s>
<s>
Two	O
of	O
them	O
,	O
and	O
,	O
are	O
reached	O
via	O
associativity	O
,	O
and	O
the	O
other	O
two	O
,	O
and	O
,	O
via	O
a	O
transposition	B-Algorithm
.	O
</s>
<s>
This	O
illustrates	O
that	O
,	O
in	O
dimension	O
and	O
above	O
,	O
the	O
permutoassociahedron	B-Algorithm
is	O
not	O
a	O
simple	O
polytope	O
.	O
</s>
<s>
This	O
is	O
the	O
product	O
between	O
the	O
number	O
of	O
permutations	B-Algorithm
of	O
terms	O
and	O
the	O
number	O
of	O
all	O
possible	O
bracketings	O
of	O
any	O
such	O
permutation	B-Algorithm
.	O
</s>
<s>
By	O
its	O
description	O
in	O
terms	O
of	O
bracketed	O
permutations	B-Algorithm
,	O
the	O
1-skeleton	O
of	O
the	O
permutoassociahedron	B-Algorithm
is	O
a	O
flip	O
graph	O
with	O
two	O
different	O
kinds	O
of	O
flips	O
(	O
associativity	O
and	O
transpositions	B-Algorithm
)	O
.	O
</s>
