5.
S
EPARABILITY
The direct sum operation introduced in the Lecture 3 has an obvious sym
metry. The
skew sum
of the permutation
π
of length
m
and the permutation
σ
of length
n
is the permutation
π
σ
defined by
π
σ
i
π
i
n
for
1
i
m
,
σ
i
m
for
m
1
i
m
n
.
Figure 5.1 shows an example.
A permutation is called
separable
if it can be built from the permutation
1
with sums and skew sums. For example, the permutation
576984132
from
Figure 5.1 is separable:
576984132
13254
4132
1
2143
1
132
1
21
21
1
1
21
1
1
1
1
1
1
1
1
1
.
The separable permutations form a permutation class, as the reader is asked
to verify in Exercise 1.
5.1.
B
ASIS AND
E
NUMERATION
We have specified the separable permutations by describing their structure,
so a natural first goal is to find the basis of this class.
Theorem 5.1.
The basis of the class of separable permutations consists of
2413
and
3142
.
Proof.
It is easy to check that
2413
and
3142
are not separable, so since
the separable permutations form a class (Exercise 1), they are contained in
Av
2413
,
3142
.
Figure 5.1:
An example of a skew sum:
13254
4132
57698 4132
.
23
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L
ECTURE
5
S
EPARABILITY
π
i
1
n
π
i
1
n
π
j
Figure 5.2:
Two situations which arise in the proof of Proposition 5.1.
The gray regions represent areas where entries cannot lie. There is a
gray region above
π
i
because we chose
π
i
as the largest entry to
the left of the
1
; similarly, there is a gray region to the right of
π
j
because of the conditions on our choice of
π
j
.
Now we must show that every element of
Av
2413
,
3142
is separa
ble.
We prove this by induction on the length of the permutation.
For
n
3
the claim is trivial, as all such permutations are separable, so take
π
Av
2413
,
3142
of length
n
4
and suppose that the claim is true for all
shorter permutations. Clearly
π
is separable if and only if its reverse is sep
arable, so without loss of generality, we may assume that the entry
1
occurs
before (to the left of) the entry
n
in
π
. Let
π
i
denote the greatest entry to
the left of the entry
1
. There cannot be an entry to the right of
n
which lies
below
π
i
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 Spring '11
 Vetter
 Combinatorics, Permutations, Generating function, Combinatorial principles, Binomial type, separable permutations

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