<s>
In	O
mathematics	O
,	O
there	O
are	O
two	O
natural	O
interpretations	O
of	O
the	O
place-permutation	B-Algorithm
action	I-Algorithm
of	O
symmetric	B-Algorithm
groups	I-Algorithm
,	O
in	O
which	O
the	O
group	O
elements	O
act	O
on	O
positions	O
or	O
places	O
.	O
</s>
<s>
Each	O
may	O
be	O
regarded	O
as	O
either	O
a	O
left	O
or	O
a	O
right	O
action	O
,	O
depending	O
on	O
the	O
order	O
in	O
which	O
one	O
chooses	O
to	O
compose	O
permutations	B-Algorithm
.	O
</s>
<s>
There	O
are	O
just	O
two	O
interpretations	O
of	O
the	O
meaning	O
of	O
"	O
acting	O
by	O
a	O
permutation	B-Algorithm
"	O
but	O
these	O
lead	O
to	O
four	O
variations	O
,	O
depending	O
whether	O
maps	O
are	O
written	O
on	O
the	O
left	O
or	O
right	O
of	O
their	O
arguments	O
.	O
</s>
<s>
When	O
regarding	O
the	O
group	O
algebra	O
of	O
a	O
symmetric	B-Algorithm
group	I-Algorithm
as	O
a	O
diagram	O
algebra	O
it	O
is	O
natural	O
to	O
write	O
maps	O
on	O
the	O
right	O
so	O
as	O
to	O
compute	O
compositions	O
of	O
diagrams	O
from	O
left	O
to	O
right	O
.	O
</s>
<s>
Let	O
be	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
on	O
letters	O
,	O
with	O
compositions	O
computed	O
from	O
right	O
to	O
left	O
.	O
</s>
<s>
Now	O
what	O
does	O
it	O
mean	O
to	O
act	O
by	O
“	O
place-permutation	O
”	O
on	O
?	O
</s>
<s>
Thus	O
,	O
when	O
encountering	O
an	O
instance	O
of	O
a	O
"	O
place-permutation	O
"	O
action	O
one	O
must	O
take	O
care	O
to	O
determine	O
from	O
the	O
context	O
which	O
interpretation	O
is	O
intended	O
,	O
if	O
the	O
author	O
does	O
not	O
give	O
specific	O
formulas	O
.	O
</s>
<s>
Now	O
if	O
we	O
act	O
on	O
this	O
by	O
another	O
permutation	B-Algorithm
then	O
we	O
need	O
to	O
first	O
relabel	O
the	O
items	O
by	O
writing	O
.	O
</s>
<s>
In	O
order	O
to	O
act	O
on	O
this	O
by	O
another	O
permutation	B-Algorithm
,	O
again	O
we	O
first	O
relabel	O
the	O
items	O
by	O
writing	O
.	O
</s>
<s>
This	O
is	O
a	O
convenient	O
convention	O
to	O
adopt	O
when	O
working	O
with	O
symmetric	B-Algorithm
groups	I-Algorithm
as	O
diagram	O
algebras	O
,	O
for	O
instance	O
,	O
since	O
then	O
one	O
may	O
read	O
compositions	O
from	O
left	O
to	O
right	O
instead	O
of	O
from	O
right	O
to	O
left	O
.	O
</s>
<s>
The	O
question	O
is	O
:	O
how	O
does	O
this	O
affect	O
the	O
two	O
interpretations	O
of	O
the	O
place-permutation	B-Algorithm
action	I-Algorithm
of	O
a	O
symmetric	B-Algorithm
group	I-Algorithm
?	O
</s>
