Transcript Motzkin Path - East China Normal University

Combinatorics of
Paths and Permutations
William Y. C. Chen
Center for Combinatorics, LPMC
Nankai University, Tianjin 300071, P. R. China
Email: [email protected]
Joint work with
Eva Y. P. Deng, Rosena R. X. Du, Toufik Mansour,
Sherry H. F. Yan and Laura L. M. Yang
Restricted Permutation
Permutation
Permutation
Containing
Certain Patterns
Restricted
Permutation
Restricted
Involution
Restricted
Matching
Restricted Partition
Permutation
Let
be the set of permutations on
For example
Pattern
For a permutation of
positive integers, the
pattern of is defined as a permutation on
obtained from by substituting the minimum
element by 1, the second minimum element by
2, ..., and the maximum element by .
For example
The pattern of 914 is 312.
The pattern of 37925 is 24513.
Restricted Permutation
For a permutation
and a permutation
, we say that is -avoiding if and
only if there is no subsequence
whose pattern is . We write
of -avoiding permutations of
for the set
.
For example
512673849 avoids 321 pattern.
But
512673849 contains 3412 pattern,
since
512673849;
512673849.
512673849;
For example
Stack Sorting Problem (Knuth, 1960’s)
312-avoiding
8 7 6 5 4 3 2 1
Question (Herbert Wilf, 1990’s)
How many permutations of length
do avoid a given subsequence
of
length k ?
For k=3
In 1972, Hammersley gave the first explicit
enumeration for
In 1973, Knuth first proved that
is enumerated by Catalan numbers.
For k=4
J. West (1990), Z. Stankova (1990’s) classified
the permutations with forbidden patterns of
length 4, i.e.
1234, 1243, 2143, 1432
1342, 2413
1324
For k=4
1234, 1243, 2143, 1432
In 1990, Ira M. Gessel gave the generating
function by using symmetric functions.
1342, 2413
In 1997, M. Bόna gave the exactly formula.
1324
D. Marinov & R. Radoicic (2003) gave the first
few numbers.
Open Problems
Conjecture ( Stanley and Wilf, 1990’s)
For each pattern , there is an
absolute constant
so that
holds.
M. Bόna, The solution of a conjecture of Stanley and
Wilf for all layered patterns, JCTA 85 (1999).
Richard Arratia, On the Stanley-Wilf conjecture for the
number of permutations avoiding a given pattern,
Electron. J. Combin. 6 (1999).
Noga Alon, Ehud Friedgut, On the number of
permutations avoiding a given pattern, JCTA 89
(2000).
M. Klazar, The Fueredi-Hajnal conjecture implies the
Stanley-Wilf conjecture. Formal power series and
algebraic combinatorics (Moscow, 2000), 250-255,
Springer, Berlin, 2000.
A. Marcus and G. Tardos, Excluded permutation
matrices and the Stanley-Wilf conjecture, JCTA 107
(2004) 1, 153-160.
Combinatorics for Restricted
Permutation
Since Catalan numbers have more than 60 kinds
of combinatorial descriptions, it is a question to
give restricted permutations some combinatorial
correspondings.
Dyck Path
A Dyck path of semilength n is a lattice path
in the plane from the origin (0,0) to (2n,0)
consisting of up steps (1,1) and down steps (1,-1)
that never run below the x-axis.
For example
•n=1
•n=2
•n=3
Restricted Permutation
and Dyck Path
Krattenthaler (2001) gave bijections between
and Dyck paths respectively.
Banlow and Killpatrick (2001) gave bijections
between 123 (132)-avoiding permutations and
Dyck paths.
(with Eva Y.P. Deng & Rosena R.X. Du)
Labelling Schemes for Lattice Paths
For example
For example
Schröder Path
A Schröder path of semilength n is a lattice
path in the plane from the origin (0,0) to (2n,0)
consisting of up steps (1,1), down steps (1,-1)
and double horizontal steps (2,0) that never run
below the x-axis.
Restricted Permutation
and Schröder Path
Kremer (2000) proved that for ten pairs of patterns
of length 4, permutations avoiding these patterns are
all enumerated by the Schröder numbers.
Bandlow-Egge-Killpatrick (2002) gave a bijection
between Schröder paths and
In 2003, Egge-Mansour gave a bijection
between Schröder paths and
Motzkin Path
A Motzkin path of length n is a lattice path in
the plane from the origin (0,0) to (n,0) consisting
of up steps (1,1), down steps (1,-1) and horizontal
steps (1,0) that never run below the x-axis.
Restricted Permutation
and Motzkin Path
In 1993, S. Gire discovered that
and
are enumerated by the
Motzkin numbers.
Barcucci-Del Lungo-Pergola-Pinzani (2000’s),
Guibert (1995) studied the two kinds of resricted
permutations by using generating trees.
Restricted Permutation
and Motzkin Path
(with Eva Y.P. Deng & Laura L.M. Yang)
Motzkin Paths and Reduced Decompositions for
Permutations with Forbidden Patterns, Elect. J.
Combin. 9(2) (2003), R15
Discrete Continuity
Barcucci-Del Lungo-Pergola-Pinzani (2000)
provided a ``discrete continuity" between the
Motzkin and the Catalan sequences. And they
posed a question of searching for a combinatorial
description.
(with Eva Y.P. Deng & Rosena R.X. Du &
Sherry H.F. Yan & Laura L.M. Yang) Discrete
Continuity, give combinatorial descriptions from
Motzkin to Catalan permutations and from
Catalan to Schröder permutations.
Involution
Let
only if
We say
is an involution if and
The set of involutions in
by
.
For example
I3={123, 132, 213, 321}.
is denoted
Restricted Involution
The set of involutions in
pattern
is denoted by
which avoid the
.
For example
65782134 is an involution avoiding 3214.
Question:
How many involutions length do
avoid a given subsequence
of length
k?
For k=3
Simion and Schmidt (1985) gave explicit expressions,
i.e.
They also gave the formulas for the number of
involutions avoiding several patterns of length 3.
For k=4
Guibert, Phd. Thesis, 1995.
Guibert-Pergola-Pinzani, Ann. Combin. 5
(2001).
For k=4
Guibert et. al. conjectured that
A.D. Jaggard (Elect. J. Combin. 9 (2003))
gave an affirmative answer to this conjecture by
introducing the equivalence of In(1234) and In(3214).
But it is still interesting to find a bijection between In(3214)
and the set of Motzkin paths of length n.
For k=4
(with Sherry H.F. Yan & Laura L.M. Yang)
3214-Avoiding Involutions, 321-Avoiding
Involutions and Motzkin Paths
For k≥5
A. Regev (1981) obtained an asymptotic
formula for the number of 12··· k-avoiding
involutions by using Young diagrams.
Open Problems
How about the others involutions avoiding a
pattern of length 4 ?
How about the involutions avoiding a pattern of
length greater than 4 ?
Is there others restricted involutions that can be
corresponding to lattice paths or other simpler
combinatorial objections ?
Partition
A partition of
is a collection
of
nonempty disjoint subsets of
called blocks,
whose union is
Any partition P can be expressed by its
canonical sequential form.
For Example
Question:
How many partitions of length n do
avoid a given subsequence ?
Noncrossing Partition
For example
Noncrossing Partition
Davenport-Schinzel sequence
RNA secondary structures
They can be corresponding to some special cases
of noncrossing partitions.
Noncrossing Partition
R. Simion and D. Ullman (1991)
M. Klazar (1990’s)
gave enumerations and some combinatorial
descriptions for noncrossing partitions.
Noncrossing Partition
(with Eva Y.P. Deng & Rosena R.X. Du)
Reduction of Regular Noncrossing Partitions,
European J. Combin., to appear.
Noncrossing Partition
(with Sherry H.F. Yan & Laura L.M. Yang)
Colored combinatorial objects
This paper defines and studies colored Dyck paths,
plane trees, hilly poor noncrossing partitions
and Motkzin paths, and answers two problems
posed by C. Coker (2003).
Nonnesting Partition
Open Problems
How about k-noncrossing parititions? (k≥3)
How about k-nonnesting partitions? (k≥2)
Matching
A matching is a special case of partition with each
block of cardinality two.
k-Nonnesting Matching
A. Regev (1981) gave asymptotic values
for k-nonnesting matchings.
D. Gouyou-Beauchamps (1989) gave the
enumeration for 3-nonnesting matchings
by using Young tableaux.
k-Noncrossing Matching
M. Klazar (1990’s, 2003) studied 2-noncrossing
matchings, and said that even for k=3, the
enumerations is still open.
(with Eva Y.P. Deng & Rosena R.X. Du)
Matching with forbidden substructure
• k-nonnesting matching is equivalent to
k-noncrossing mathing.
Open Problems
The exactly formula for k-noncrossing (knonnesting) matchings? (k≥4)
How about matchings avoiding the pattern
which is not noncrossing nor nonnesting.
Permutations containing a given
number of a certain patterns
Question:
How many permutations of length are
there that contain exactly r occurrences of
pattern ?
Permutations containing a given
number of a certain patterns
Noonan (1996) enumerated permutations
contain exactly one 123 pattern.
Bόna (1998) enumerated permutations contain
exactly one 132 pattern.
Fulmek (2003) enumerated permutations contain
exactly two subsequences of 132 pattern.
Conjecture (Noonan, Zeilberger, 1996)
For any given subsequence and for any
given r, then number of n-permutations
containing exactly r subsequence is a
P-recursive function of n.
Bόna (1997) proved that permutations
containing a fix number of subsequence
132 is P-recursive.
Involutions containing a given
number of a certain patterns
Question:
How many involutions of length do
contain exactly r occurrences of pattern
?
Involutions containing a given
number of a certain patterns
Guibert and Mansour (2002) gave an explicit
expression for involutions containing one 132.
Deutsch, Robertson and Saracion (2004)
enumerated involutions containing one 231.
Involutions containing a given
number of a certain patterns
(with Toufik Mansour & Sherry H.F. Yan)
P-recursiveness of the number of involutions
with a fixed number of 132-subsequences
Open Problems
How about the enumerations for the other cases
in permutations (involutions) containing a fixed
number of certain pattern ?
How about partitions or matchings ?
Thanks !