<s>
In	O
mathematics	O
,	O
the	O
permutohedron	B-Algorithm
of	O
order	O
n	O
is	O
an	O
(	O
n1	O
)	O
-dimensional	O
polytope	O
embedded	O
in	O
an	O
n-dimensional	O
space	O
.	O
</s>
<s>
Its	O
vertex	O
coordinates	O
(	O
labels	O
)	O
are	O
the	O
permutations	B-Algorithm
of	O
the	O
first	O
n	O
natural	O
numbers	O
.	O
</s>
<s>
The	O
edges	O
identify	O
the	O
shortest	O
possible	O
paths	O
(	O
sets	O
of	O
transpositions	B-Algorithm
)	O
that	O
connect	O
two	O
vertices	O
(	O
permutations	B-Algorithm
)	O
.	O
</s>
<s>
Two	O
permutations	B-Algorithm
connected	O
by	O
an	O
edge	O
differ	O
in	O
only	O
two	O
places	O
(	O
one	O
transposition	B-Algorithm
)	O
,	O
and	O
the	O
numbers	O
on	O
these	O
places	O
are	O
neighbors	O
(	O
differ	O
in	O
value	O
by	O
1	O
)	O
.	O
</s>
<s>
The	O
image	O
on	O
the	O
right	O
shows	O
the	O
permutohedron	B-Algorithm
of	O
order	O
4	O
,	O
which	O
is	O
the	O
truncated	O
octahedron	O
.	O
</s>
<s>
Its	O
vertices	O
are	O
the	O
24	O
permutations	B-Algorithm
of	O
(	O
1	O
,	O
2	O
,	O
3	O
,	O
4	O
)	O
.	O
</s>
<s>
The	O
6	O
edge	O
colors	O
correspond	O
to	O
the	O
6	O
possible	O
transpositions	B-Algorithm
of	O
4	O
elements	O
,	O
i.e.	O
</s>
<s>
they	O
indicate	O
in	O
which	O
two	O
places	O
the	O
connected	O
permutations	B-Algorithm
differ	O
.	O
</s>
<s>
red	O
edges	O
connect	O
permutations	B-Algorithm
that	O
differ	O
in	O
the	O
last	O
two	O
places	O
.	O
)	O
</s>
<s>
According	O
to	O
,	O
permutohedra	B-Algorithm
were	O
first	O
studied	O
by	O
.	O
</s>
<s>
The	O
alternative	O
spelling	O
permutahedron	B-Algorithm
is	O
sometimes	O
also	O
used	O
.	O
</s>
<s>
Permutohedra	B-Algorithm
are	O
sometimes	O
called	O
permutation	B-Algorithm
polytopes	I-Algorithm
,	O
but	O
this	O
terminology	O
is	O
also	O
used	O
for	O
the	O
related	O
Birkhoff	B-Algorithm
polytope	I-Algorithm
,	O
defined	O
as	O
the	O
convex	O
hull	O
of	O
permutation	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
More	O
generally	O
,	O
uses	O
that	O
term	O
for	O
any	O
polytope	O
whose	O
vertices	O
have	O
a	O
bijection	B-Algorithm
with	O
the	O
permutations	B-Algorithm
of	O
some	O
set	O
.	O
</s>
<s>
The	O
permutohedron	B-Algorithm
of	O
order	O
has	O
vertices	O
,	O
each	O
of	O
which	O
is	O
adjacent	O
to	O
others	O
.	O
</s>
<s>
More	O
generally	O
,	O
the	O
faces	O
of	O
dimensions	O
0	O
(	O
vertices	O
)	O
to	O
(	O
the	O
permutohedron	B-Algorithm
itself	O
)	O
correspond	O
to	O
the	O
strict	O
weak	O
orderings	O
of	O
the	O
set	O
.	O
</s>
<s>
The	O
images	O
above	O
show	O
the	O
face	O
lattices	O
of	O
the	O
permutohedra	B-Algorithm
of	O
order	O
3	O
and	O
4	O
(	O
without	O
the	O
empty	O
faces	O
)	O
.	O
</s>
<s>
The	O
orderings	O
are	O
partially	O
ordered	O
by	O
refinement	B-Data_Structure
,	O
with	O
the	O
finer	O
ones	O
being	O
on	O
the	O
outside	O
.	O
</s>
<s>
The	O
vertex	O
labels	O
interpreted	O
as	O
permutations	B-Algorithm
are	O
those	O
forming	O
the	O
Cayley	O
graph	O
.	O
</s>
<s>
The	O
images	O
below	O
show	O
how	O
the	O
facets	O
of	O
the	O
-permutohedron	O
correspond	O
to	O
the	O
simplical	O
projection	O
of	O
the	O
-cube	O
.	O
</s>
<s>
Nor	O
does	O
its	O
opposite	O
vertex	O
in	O
the	O
-cube	O
,	O
which	O
is	O
not	O
part	O
of	O
the	O
projection	O
.	O
)	O
</s>
<s>
The	O
number	O
of	O
faces	O
of	O
dimension	O
in	O
the	O
permutohedron	B-Algorithm
of	O
order	O
is	O
given	O
by	O
the	O
triangle	O
:	O
</s>
<s>
The	O
permutohedron	B-Algorithm
is	O
vertex-transitive	O
:	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
Sn	O
acts	O
on	O
the	O
permutohedron	B-Algorithm
by	O
permutation	B-Algorithm
of	O
coordinates	O
.	O
</s>
<s>
The	O
permutohedron	B-Algorithm
is	O
a	O
zonotope	O
;	O
a	O
translated	B-Algorithm
copy	O
of	O
the	O
permutohedron	B-Algorithm
can	O
be	O
generated	O
as	O
the	O
Minkowski	B-Algorithm
sum	I-Algorithm
of	O
the	O
n( n	O
−	O
1	O
)	O
/2	O
line	O
segments	O
that	O
connect	O
the	O
pairs	O
of	O
the	O
standard	O
basis	O
vectors	O
.	O
</s>
<s>
The	O
vertices	O
and	O
edges	O
of	O
the	O
permutohedron	B-Algorithm
are	O
isomorphic	O
to	O
one	O
of	O
the	O
Cayley	O
graphs	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
,	O
namely	O
the	O
one	O
generated	O
by	O
the	O
transpositions	B-Algorithm
that	O
swap	O
consecutive	O
elements	O
.	O
</s>
<s>
The	O
vertices	O
of	O
the	O
Cayley	O
graph	O
are	O
the	O
inverse	O
permutations	B-Algorithm
of	O
those	O
in	O
the	O
permutohedron	B-Algorithm
.	O
</s>
<s>
Its	O
edge	O
colors	O
represent	O
the	O
3	O
generating	O
transpositions	B-Algorithm
:	O
,	O
,	O
</s>
<s>
This	O
Cayley	O
graph	O
is	O
Hamiltonian	O
;	O
a	O
Hamiltonian	O
cycle	O
may	O
be	O
found	O
by	O
the	O
Steinhaus	B-Algorithm
–	I-Algorithm
Johnson	I-Algorithm
–	I-Algorithm
Trotter	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Moreover	O
,	O
this	O
hyperplane	O
can	O
be	O
tiled	B-Algorithm
by	O
infinitely	O
many	O
translated	B-Algorithm
copies	O
of	O
the	O
permutohedron	B-Algorithm
.	O
</s>
<s>
Each	O
of	O
them	O
differs	O
from	O
the	O
basic	O
permutohedron	B-Algorithm
by	O
an	O
element	O
of	O
a	O
certain	O
(	O
n1	O
)	O
-dimensional	O
lattice	O
,	O
which	O
consists	O
of	O
the	O
n-tuples	O
of	O
integers	O
that	O
sum	O
to	O
zero	O
and	O
whose	O
residues	O
(	O
modulo	O
n	O
)	O
are	O
all	O
equal	O
:	O
</s>
<s>
In	O
other	O
words	O
,	O
the	O
permutohedron	B-Algorithm
is	O
the	O
Voronoi	B-Architecture
cell	I-Architecture
for	O
.	O
</s>
<s>
Thus	O
,	O
the	O
permutohedron	B-Algorithm
of	O
order	O
4	O
shown	O
above	O
tiles	O
the	O
3-dimensional	O
space	O
by	O
translation	O
.	O
</s>
<s>
The	O
tessellations	B-Algorithm
formed	O
in	O
this	O
way	O
from	O
the	O
order-2	O
,	O
order-3	O
,	O
and	O
order-4	O
permutohedra	B-Algorithm
,	O
respectively	O
,	O
are	O
the	O
apeirogon	O
,	O
the	O
regular	O
hexagonal	O
tiling	O
,	O
and	O
the	O
bitruncated	O
cubic	O
honeycomb	O
.	O
</s>
<s>
The	O
dual	O
tessellations	B-Algorithm
contain	O
all	O
simplex	O
facets	O
,	O
although	O
they	O
are	O
not	O
regular	O
polytopes	O
beyond	O
order-3	O
.	O
</s>
