Discrete Mathematics and Theoretical Computer Science 3, 1999, 151–154
Permutations Containing and Avoiding
and
Patterns
Aaron Robertson Department of Mathematics, Colgate University, Hamilton, NY 13346
received April 5, 1999, revised May 12, 1999, accepted May 19, 1999.
We prove that the number of permutations which avoid
-patterns and have exactly one
-pattern, equals
, for
. We then give a bijection onto the set of permutations which avoid
-patterns and have exactly
one
-pattern. Finally, we show that the number of permutations which contain exactly one
-pattern and exactly
one
-pattern is
, for
.
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Keywords: Patterns, Words
1 Introduction
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In 1990, Herb Wilf asked the following: How many permutations of length avoid a given pattern, ? By
pattern-avoiding we mean the following: Let be a permutation of length and let
be a permutation of length
(we will call this a pattern of length ). Let be a set of integers, and
let
. Define
to be if is the smallest element in , if it is the second smallest, ...,
and if it is the largest. The permutation avoids the pattern if and only if there does not exist a set of
indices
, such that
.
In two beautiful papers ([B1] and [N]), the number of subsequences containing exactly one
-pattern
and exactly one
-pattern are enumerated. Noonan shows in [N] that the number of permutations
containing exactly one
-pattern is the simple formula
. Bóna proves that the even simpler
formula
enumerates the number of permutations containing exactly one
-pattern. Bóna’s result
proved a conjecture first made by Noonan and Zeilberger in [NZ].
Noonan and Zeilberger considered in [NZ] the number of permutations of length which contain exactly p-patterns, for
. Bóna, in [B2], made further progress concerning the number of permutations
with exactly
-patterns. In this article we work towards the following generalization: How many per, and contain
-patterns, for
,
? We will
mutations of length avoid patterns , for
first consider the permutations of length which avoid
-patterns, but contain exactly one
-pattern.
We then define a natural bijection between these permutations and the permutations of length which
avoid
-patterns, but contain exactly one
-pattern. Finally, we will calculate the number of permutations which contain one
-pattern and one
-pattern. These results address questions first raised in
[NZ].
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email: arobertson@mail.colgate.edu
1365–8050 c 1999 Maison de l’Informatique et des Mathématiques Discrètes (MIMD), Paris, France
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152
Aaron Robertson
2 Known Results
For completeness, two results which are already known are given below.
Lemma 1: The number of permutations of length with one -pattern is
.
Proof: Induct on . The base case is trivial. A permutation, , of length with one -pattern must have
or
. If
, by induction we get
permutation. If
, then we must
have
(or we would have more than one -pattern). The rest of the entries of must be
decreasing. Hence we get more permutation from this second case, for a total of
.
Lemma 2: The number of permutations which avoid both the pattern
and
is
.
Proof: Let
denote the number of permutations we are interested in. Then
with
. To see this, let be a permutation of length
. Insert the element into the
position of
. Let be this new permutation of length . To assure that avoids the
-pattern, we must have all
entries preceding in be larger than the entries following . To assure that avoids the
-pattern,
the entries preceding must be in decreasing order. This argument gives the sum in the recursion. The
recursion holds by noting that if
, there is one permutation which avoids both patterns. To complete
the proof note that
.
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3 One 123-pattern, but no 132-pattern
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Theorem 1: The number of permutations of length which have exactly one
-pattern, and avoid the
-pattern is
.
Proof: Call a permutation good if it has exactly one
-pattern and avoids the
-pattern, and let
denote the number of good permutations of length . Let be a permutation of length
. Insert the
element into the
position of . Call this newly constructed permutation of length , . To assure
that avoids the
pattern, we must have all elements preceding in be larger than the elements
following in . For to be a good permutation, we must consider two disjoint cases.
Case I: The pattern
appears in the elements following in . This forces the elements preceding
to be in decreasing order. Summing over , this case accounts for
permutations.
Case II: The pattern
appears in the elements preceding and including in . This forces the in
the pattern to be . Hence the elements preceding must contain exactly one -pattern. (Further there
must be at least elements. Hence must be at least ). From Lemma 1, this number is
. We are also
forced to avoid both patterns in the elements following . Lemma 2 implies that there are
such
permutations. Summing over , this case accounts for
permutations.
We have established that the recurrence relation
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eF} [
which holds for 'ctY († ~ *,g$0W†U.Œ*Lgˆ0W†72K*,g ), enumerates the permutations of length ' which avoid
the pattern HYZI and contain exactly one HIY -pattern.
One easy way to proceed would be to find the generating function of † \ . However, in this article
we would like to employ a different, and in many circumstances more powerful, tool. We will use
the Maple procedure findrec in Doron Zeilberger’s Maple package EKHAD  . (The Maple shareware
Ž Available for download at www.math.temple.edu/˜zeilberg/
Permutations Containing and Avoiding
HI7Y
and
HYZI
Patterns
153
package gfun could have also been used.) Instructions for its use are available online. To use find)
. We
rec we compute the first few terms of . These are (for
type findrec([4,12,32,80,192,448,1024],0,2,n,N) and are given the recurrence
for
. Define
and
, and it is routine to verify that
for
. Another routine calculation shows us that
for
, thereby
proving the statement of the theorem.
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4 One 132-pattern, but no 123-pattern
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Theorem 2: The number of permutations of length which have exactly one
-pattern, and avoid the
-pattern is
.
Proof: We prove this by exhibiting a (natural) bijection from the permutations counted in Theorem 1 to
the permutations counted in this theorem. Define
avoids
-pattern and contains one
-pattern and
avoids
-pattern and contains one
-pattern . We will show that
, by using the following bijection:
Let
. Let
, and let
be the
-pattern in . Then acts on the elements of as
follows:
if
,
, and
. In other words, all elements keep their positions
except and switch places. An easy examination of several cases shows that this is a bijection, thereby
proving the theorem.
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HIY
Theorem 3: The number of permutations of length which have exactly one
-pattern and one
pattern is
.
Proof: We use the same insertion technique as in the proof of Theorem 1. Call a permutation good if it has
exactly one
-pattern and exactly one
-pattern and let denote the number of good permutations of
length . Let be a permutation of length
. Insert the element into the
position of . Let be
-pattern cannot consist of elements
this newly constructed permutation of length . We note that the
only preceding . If this were the case, we would have two
-patterns ending with . For to be a
good permutation, we must consider the following disjoint cases.
Case I: The
-pattern consists of elements following . In this case all elements preceding must be
larger than the elements following .
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-pattern consists of elements following . Summing over we get
Subcase A: The
permutations in this subcase.
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good
Subcase B: The elements preceding have exactly one -pattern. This gives a
-pattern where the
3 in the pattern is . We must also avoid the
-pattern in the elements following . Summing over
and using Lemma 1 and Theorem 1, we get
good permutations in this
subcase.
Case II: The
-pattern has the first element preceding , the last element following , and as the
middle element. The elements preceding must be
, where
immediately precedes in . See [B1] for a more detailed argument as to why this must be true.
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Subcase A: The elements preceding have exactly one
element of the pattern is . We must also avoid both the
'
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M
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HI -pattern. This gives a HIY -pattern where the last
HIY and the HYUI pattern in the elements following
154
Aaron Robertson
' . Summing over M and using Lemma 1 and Lemma 2 we have { \DeF}ba ² . N+ M±qYZ6PI \Da e a . good permutations
in this subcase.
Subcase B: The HIY -pattern consists of elements following ' . We must have the elements preceding '
in ) be decreasing to avoid another HI7Y -pattern. Further, the elements following
' must not contain a
HYZI -pattern. Using Theorem 1 and summing over M , we get a total of { \DeF}³a 2 [ +N'qhMfqI76PI \Da e a [ good
permutations in this subcase.
In total, we find that the following recurrence enumerates the permutations of length ' which contain
exactly one HIY -pattern and one HYZI -pattern.
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for 'sc¬´ and † . *¬† 2 *† *¬† ² *…g .
[
Using findrec again by typing findrec([2,12,48,160,480,1344,3584],1,1,n,N)
2µ \7\ _ 2P¶ z \ ,
(where the list is the first few terms of our recurrence for 'sc´ ) we get the recurrence z \_ . *
with z7. *?I . After reindexing, another routine calculation shows that z \ *  \ . Solving z \ for an explicit
\a3° .
answer, we find that † \ *L+N'qYZ6+1'xqsD6OI
We conjecture that the generating function for the number of permutations with exactly zero or exactly
_
one HYUI -pattern and exactly <KHI7Y -patterns is ·k+1¸D6P¹$+WH q®I¸D6Wº . , where ·k+1¸D6 is a polynomial. For more
evidence, and further extensions see [RWZ].
Acknowledgements
The author would like to thank Robin Chapman for his useful comments.
References
HYZI
[B1] M. Bóna, Permutations with one or two
-subsequences., Discrete Mathematics, 181, 1998, 267274.
-subsequences is -recursive in the Size!.,
[B2] M. Bóna, The Number of Permutations with Exactly
Advances in Applied Mathematics, 18, 1997, 510-522.
[N] J. Noonan, The Number of Permutations Containing Exactly One Increasing Subsequence of Length
, Discrete Mathematics, 152, 1996, 307-313.
[NZ] J. Noonan and D. Zeilberger, The Enumeration of Permutations with a Prescribed Number of “Forbidden” Patterns, Advances in Applied Mathematics, 17, 1996, 381-407.
[RWZ] A. Robertson, H. Wilf, and D. Zeilberger, Patterns and Fractions, submitted.
See www.math.temple.edu/˜[aaron,zeilberg]/ for preprint.
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