<s>
In	O
combinatorial	O
mathematics	O
,	O
a	O
Baxter	B-Algorithm
permutation	I-Algorithm
is	O
a	O
permutation	B-Algorithm
which	O
satisfies	O
the	O
following	O
generalized	O
pattern	B-Algorithm
avoidance	I-Algorithm
property	O
:	O
</s>
<s>
Equivalently	O
,	O
using	O
the	O
notation	O
for	O
vincular	O
patterns	O
,	O
a	O
Baxter	B-Algorithm
permutation	I-Algorithm
is	O
one	O
that	O
avoids	O
the	O
two	O
dashed	O
patterns	O
2-41-3	O
and	O
3-14-2	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
permutation	B-Algorithm
σ	O
=	O
2413	O
in	O
S4	O
(	O
written	O
in	O
one-line	O
notation	O
)	O
is	O
not	O
a	O
Baxter	B-Algorithm
permutation	I-Algorithm
because	O
,	O
taking	O
i	O
=	O
1	O
,	O
j	O
=	O
2	O
and	O
k	O
=	O
4	O
,	O
this	O
permutation	B-Algorithm
violates	O
the	O
first	O
condition	O
.	O
</s>
<s>
These	O
permutations	B-Algorithm
were	O
introduced	O
by	O
Glen	O
E	O
.	O
Baxter	O
in	O
the	O
context	O
of	O
mathematical	O
analysis	O
.	O
</s>
<s>
In	O
fact	O
,	O
this	O
formula	O
is	O
graded	O
by	O
the	O
number	O
of	O
descents	O
in	O
the	O
permutations	B-Algorithm
,	O
i.e.	O
,	O
there	O
are	O
Baxter	B-Algorithm
permutations	I-Algorithm
in	O
Sn	O
with	O
k	O
–	O
1	O
descents	O
.	O
</s>
<s>
The	O
number	O
of	O
alternating	B-Algorithm
Baxter	B-Algorithm
permutations	I-Algorithm
of	O
length	O
2n	O
is	O
(	O
Cn	O
)	O
2	O
,	O
the	O
square	O
of	O
a	O
Catalan	O
number	O
,	O
and	O
of	O
length	O
2n	O
+	O
1	O
is	O
CnCn+1	O
.	O
</s>
<s>
The	O
number	O
of	O
doubly	O
alternating	B-Algorithm
Baxter	B-Algorithm
permutations	I-Algorithm
of	O
length	O
2n	O
and	O
2n	O
+	O
1	O
(	O
i.e.	O
,	O
those	O
for	O
which	O
both	O
σ	O
and	O
its	O
inverse	O
σ−1	O
are	O
alternating	B-Algorithm
)	O
is	O
the	O
Catalan	O
number	O
Cn	O
.	O
</s>
<s>
Baxter	B-Algorithm
permutations	I-Algorithm
are	O
related	O
to	O
Hopf	O
algebras	O
,	O
planar	O
graphs	O
,	O
and	O
tilings	B-Algorithm
.	O
</s>
<s>
Baxter	O
introduced	O
Baxter	B-Algorithm
permutations	I-Algorithm
while	O
studying	O
the	O
fixed	O
points	O
of	O
commuting	O
continuous	O
functions	O
.	O
</s>
<s>
if	O
the	O
fixed	O
points	O
are	O
x1	O
<	O
x2	O
<	O
...	O
<	O
x2k	O
+	O
1	O
then	O
f	O
and	O
g	O
act	O
as	O
mutually-inverse	O
permutations	B-Algorithm
on	O
{	O
x1	O
,	O
x3	O
,	O
...	O
,	O
x2k	O
+	O
1}	O
and	O
{	O
x2	O
,	O
x4	O
,	O
...	O
,	O
x2k}	O
;	O
</s>
<s>
the	O
permutation	B-Algorithm
induced	O
by	O
f	O
on	O
{	O
x1	O
,	O
x3	O
,	O
...	O
,	O
x2k	O
+	O
1}	O
uniquely	O
determines	O
the	O
permutation	B-Algorithm
induced	O
by	O
f	O
on	O
{	O
x2	O
,	O
x4	O
,	O
...	O
,	O
x2k}	O
;	O
</s>
<s>
under	O
the	O
natural	O
relabeling	O
x1	O
→	O
1	O
,	O
x3	O
→	O
2	O
,	O
etc.	O
,	O
the	O
permutation	B-Algorithm
induced	O
on	O
{	O
1	O
,	O
2	O
,	O
...	O
,	O
k	O
+	O
1}	O
is	O
a	O
Baxter	B-Algorithm
permutation	I-Algorithm
.	O
</s>
