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Otto Baxter
Artikel: Baxter permutation GudangMovies21 Rebahinxxi
In combinatorial mathematics, a Baxter permutation is a permutation
σ
∈
S
n
{\displaystyle \sigma \in S_{n}}
which satisfies the following generalized pattern avoidance property:
There are no indices
i
<
j
<
k
{\displaystyle i
such that
σ
(
j
+
1
)
<
σ
(
i
)
<
σ
(
k
)
<
σ
(
j
)
{\displaystyle \sigma (j+1)<\sigma (i)<\sigma (k)<\sigma (j)}
or
σ
(
j
)
<
σ
(
k
)
<
σ
(
i
)
<
σ
(
j
+
1
)
{\displaystyle \sigma (j)<\sigma (k)<\sigma (i)<\sigma (j+1)}
.
Equivalently, using the notation for vincular patterns, a Baxter permutation is one that avoids the two dashed patterns
2
−
41
−
3
{\displaystyle 2-41-3}
and
3
−
14
−
2
{\displaystyle 3-14-2}
.
For example, the permutation
σ
=
2413
{\displaystyle \sigma =2413}
in
S
4
{\displaystyle S_{4}}
(written in one-line notation) is not a Baxter permutation because, taking
i
=
1
{\displaystyle i=1}
,
j
=
2
{\displaystyle j=2}
and
k
=
4
{\displaystyle k=4}
, this permutation violates the first condition.
These permutations were introduced by Glen E. Baxter in the context of mathematical analysis.
Enumeration
For
n
=
1
,
2
,
3
,
…
{\displaystyle n=1,2,3,\ldots }
, the number
a
n
{\displaystyle a_{n}}
of Baxter permutations of length
n
{\displaystyle n}
is
1, 2, 6, 22, 92, 422, 2074, 10754, 58202, 326240, 1882960, 11140560, 67329992, 414499438, 2593341586, 16458756586,...
This is sequence OEIS: A001181 in the OEIS. In general,
a
n
{\displaystyle a_{n}}
has the following formula:
a
n
=
∑
k
=
1
n
(
n
+
1
k
−
1
)
(
n
+
1
k
)
(
n
+
1
k
+
1
)
(
n
+
1
1
)
(
n
+
1
2
)
.
{\displaystyle a_{n}\,=\,\sum _{k=1}^{n}{\frac {{\binom {n+1}{k-1}}{\binom {n+1}{k}}{\binom {n+1}{k+1}}}{{\binom {n+1}{1}}{\binom {n+1}{2}}}}.}
In fact, this formula is graded by the number of descents in the permutations, i.e., there are
(
n
+
1
k
−
1
)
(
n
+
1
k
)
(
n
+
1
k
+
1
)
(
n
+
1
1
)
(
n
+
1
2
)
{\displaystyle {\frac {{\binom {n+1}{k-1}}{\binom {n+1}{k}}{\binom {n+1}{k+1}}}{{\binom {n+1}{1}}{\binom {n+1}{2}}}}}
Baxter permutations in
S
n
{\displaystyle S_{n}}
with
k
−
1
{\displaystyle k-1}
descents.
Other properties
The number of alternating Baxter permutations of length
2
n
{\displaystyle 2n}
is
(
C
n
)
2
{\displaystyle (C_{n})^{2}}
, the square of a Catalan number, and of length
2
n
+
1
{\displaystyle 2n+1}
is
C
n
C
n
+
1
{\displaystyle C_{n}C_{n+1}}
.
The number of doubly alternating Baxter permutations of length
2
n
{\displaystyle 2n}
and
2
n
+
1
{\displaystyle 2n+1}
(i.e., those for which both
σ
{\displaystyle \sigma }
and its inverse
σ
−
1
{\displaystyle \sigma ^{-1}}
are alternating) is the Catalan number
C
n
{\displaystyle C_{n}}
.
Baxter permutations are related to Hopf algebras, planar graphs, and tilings.
Motivation: commuting functions
Baxter introduced Baxter permutations while studying the fixed points of commuting continuous functions. In particular, if
f
{\displaystyle f}
and
g
{\displaystyle g}
are continuous functions from the interval
[
0
,
1
]
{\displaystyle [0,1]}
to itself such that
f
(
g
(
x
)
)
=
g
(
f
(
x
)
)
{\displaystyle f(g(x))=g(f(x))}
for all
x
{\displaystyle x}
, and
f
(
g
(
x
)
)
=
x
{\displaystyle f(g(x))=x}
for finitely many
x
{\displaystyle x}
in
[
0
,
1
]
{\displaystyle [0,1]}
, then:
the number of these fixed points is odd;
if the fixed points are
x
1
<
x
2
<
…
<
x
2
k
+
1
{\displaystyle x_{1}
then
f
{\displaystyle f}
and
g
{\displaystyle g}
act as mutually-inverse permutations on
{
x
1
,
x
3
,
…
,
x
2
k
+
1
}
{\displaystyle \{x_{1},x_{3},\ldots ,x_{2k+1}\}}
and
{
x
2
,
x
4
,
…
,
x
2
k
}
{\displaystyle \{x_{2},x_{4},\ldots ,x_{2k}\}}
;
the permutation induced by
f
{\displaystyle f}
on
{
x
1
,
x
3
,
…
,
x
2
k
+
1
}
{\displaystyle \{x_{1},x_{3},\ldots ,x_{2k+1}\}}
uniquely determines the permutation induced by
f
{\displaystyle f}
on
{
x
2
<
,
x
4
,
…
,
x
2
k
}
{\displaystyle \{x_{2}<,x_{4},\ldots ,x_{2k}\}}
;
under the natural relabeling
x
1
→
1
{\displaystyle x_{1}\to 1}
,
x
3
→
2
{\displaystyle x_{3}\to 2}
, etc., the permutation induced on
{
1
,
2
,
…
,
k
+
1
}
{\displaystyle \{1,2,\ldots ,k+1\}}
is a Baxter permutation.
See also
Enumerations of specific permutation classes
References
External links
OEIS sequence A001181 (Number of Baxter permutations of length n)