!#"
$&%')(+*,,*.-0/2143568798:&;=<>?
@
-Species and the A -Mehler Formula
Hung Quang Ngo B
Abstract
In this paper, we present a bijective proof of the C -Mehler formula. The proof is in
the same style as Foata’s proof of the Mehler formula. Since Foata’s proof was extended
to show the Kibble-Slepian formula, a very general multilinear extension of the Mehler
formula, we hope that the proof provided in this paper helps find some multilinear extension
of the C -Mehler formula.
The basic idea to obtain this proof comes from generalizing a result by Gessel. The
generalization leads to the notion of species on permutations and the C -generating series for
these species. The bijective proof is then obtained by applying this new exponential formula
to a certain type of species on permutations and a weight preserving bijection relating this
species to the C -Mehler formula. Some by-products of the C -exponential formula shall also
be derived.
1 Introduction
The Hermite polynomials are well-studied and have applications in diverse areas of Mathematics [2, 7]. They can be defined in terms of their generating function, whose bilinear extension
is the well-known Mehler formula. The most general multilinear extension is known as the
Kibble-Slepian formula [17,20]. Foata [10] discovered a combinatorial proof of the Mehler formula, which was later extended to show the Kibble-Slepian formula combinatorially by Foata
and Garcia [12].
A D -analogue of the Hermite polynomials, called the D -Hermite polynomials, was introduced by Rogers [19], who used them to prove Rogers-Ramanujan identities. Up to rescaling,
there are other variants of the D -Hermite polynomials [1, 8, 9, 15]. The Mehler formula for
the D -Hermite is known, but no D -analogue of the Kibble-Slepian formula has been discovered
yet. Similar to the normal Hermite polynomials, one hopes that a Foata-style combinatorial
approach to the D -Mehler formula helps find a D -Kibble-Slepian. A few proofs of the D -Mehler
formula is known (see, e.g., [4, 8, 15]), of which the one in [15] is combinatorial. However, the
combinatorial objects were vector spaces over finite fields, which are difficult to be dealt with
in bijective arguments.
This paper presents a Foata-style proof of the D -Mehler formula. Along the way, we also
introduce a new kind of species and a few consequences. Throughout this paper, we use E$FHGIKJ
(avoiding confusion with ELIMJ0N ) to denote the set of integers from F to I . The following standard
O
Computer Science and Engineering Department, 201 Bell Hall, State University of New York at Buffalo,
Amherst, NY 14260, U.S.A. hungngo@cse.buffalo.edu
notations shall also be used:
E !J N
E8IKJ N
D
F
F
D
I!#"
F
D
F D D
G I F
&%+*,
$
D
F) D % F) D
&%' (%
-
G 0/
+
D
D
F D
F. D
N
!
/
I
Most often, the subscript D is dropped when there is no potential confusion. Lastly, we use
to denote the set of all matchings on I points E F=GIMJ .
1
2 Preliminaries
2.1 The Mehler formula and its extensions
There are several variations of the Hermite polynomials, which are all the same up to rescaling.
A typical definition of the Hermite polynomials and their generating function, respectively, are
435678
2
A
F
:9-;=<>
2
B
C35D
9
IFE
,
> 3
@
9 ?; <
*
;+GH?GI<
G
(1)
(2)
To interpret the Hermite
polynomials combinatorially, another variant43of
the Hermite poly5
2 C35
2
nomials, denoted by J
, was introduced. The definition of the J
, their generating
function and its bilinear extension are as follows, respectively:
2
A
@
2
B
A
@
2
B
,
J
,
#35 2
J
#35K
ISE
J
WV Q
IFE
NO
G
*
9-;+TU?T < P
C35RQ
J
C3,L?M
P
N
2
M
*
(3)
G
(4)
N
F
F
Q
*XY[Z]\
Q
3^V
*
Q
N
C
)
F
*
43
Q
*
V
*
_
(5)
Identity (5) is the celebrated Mehler’s formula, which was shown combinatorially by Foata
[10]. For a discussion of this proof and its relation to other combinatorial results on orthogonal
polynomials, the reader is referred to Stanton [21].
Carlitz [5, 6] found several multilinear extensions. Kibble [17], and later independently
Slepian [20] found an extension, known as the Kibble-Slepian formula, whose specializations
include all other extensions. Louck [18] proposed another extension which was proved combinatorially to be equivalent to the Kibble-Slepian formula by Foata [11].
To
describe the Kibble-Slepian formula, let us first introduce some notation. For each integer
N
I , define a symmetric I` I matrix a by
a
bc)ed
bfc
D
ji
if g.h ,
otherwise,
F
2
bcb c
?
where D
is an infinite sequence of indeterminates. Let 8
bc-G G i be a vector of
I indeterminates.
matrices ( F/ gG i/ I ) of order
Rbb Let be the set of all symmetric
bc
I such that
for all g/
I , and that is a non-negative integer for all g h . Also, for a
fixed , let the g th row sum of be
b b b
*
b[
R
The Kibble-Slepian formula reads
2
J
A
+ 2
J
[
bfc
D"!# $
b c
Rbc
bc
F
%
M
E
N)(
\
X=Y[Z
X'& a
F
*
,+
a
_
(6)
Foata and Garsia [12] extended Foata’s proof [10] of the Mehler formula to give a combinatorial proof of the Kibble-Slepian formula. The left hand side of (6) was interpreted as the
exponential generating function of the so-called I -involutionary graphs, while the right hand
side could be written as the exponential of the series
F
F
/ %
N.-
N
X'& a
F
A
(0
b c
bc
bc
a
b
c
+ (7)
They showed that expression (7) is the generating function for the “connected components” of
the I -involutionary graphs. Consequently, the exponential formula applies, proving (6).
2.2 The 1 -Mehler formula and its extensions
A D -analogue of the Hermite polynomials called the D -Hermite polynomials, obtained from the
so-called
Rogers-Szegő
polynomials [4, 9, 15, 19] can be defined by their generating function
3
2
G Q32 D as follows:
2
3
Q 2 D
G4
A
@
#3
2
B
2
To get the corresponding D -version
2
Q
2
D
C35
J
;
2
2
*
N3
, F
% B
D of
2
@
$
D
C3
J
#3
J
D
2
,
Q
D
J
*,=3
2
D
F3 2
J
C3
D
2
%
G 2 Q265 F
2
C3
2
(8)
D . Define
M
F
F PD 2 D
G
*
D
(9)
F6 D D
Q
* *
D
, we also normalize the
with the new three term recurrence:
2
F%
2
=43
J
#3
2
2
D
+
(10)
2 D [15]. The interpreThis recurrence43relation
yields a combinatorial interpretation for J
2
2 435
2 D as a D -analogue of the matching polynomials J
tation gives J
. Notice that each
matching 78 1
can be viewed as an involution on E F=GIKJ . Define a new statistic on 7 as
9
follows:
A
7
9 ;9 :
9
where the sum goes over all edges of 7 , and if
9
2
!
2
!
g
5
i
5
3
G
G
g
i#
G and 7
i
,g 5
!
5
, then
i<
2
-
!
9
9
Pictorially, imagine putting I points F=G
G I in this order
on a horizontal
line, then drawing all
!
9
edges of 7 on the upper
half plane. The statistic
for an edge is the number of points
i
lying between g and such that is eitheri a fixed point or an end-point of some edge 7 ,
both of whose end-points are onFthe left N of (see Figure 1). Let 2 7 2 denote the number of edges
I 2 7 2 be the number of fixed points of 7 . It follows that
in the matching 7 , and let 7
9
b
%
b
%
%
b
( :
9
2
where
C3
J
J
7
: F
7
J
3
( : -
:
c
8
, where
A
F
D
2
9
9
%
Figure 1: Illustration of c
%
( :
9
9
9
c
i
,g 5
.
G
(11)
:
(
D
i#
G
g
(12)
On the same line of reasoning as in the previous section, one would hope that (11) helps
combinatorially discover the D -analogues of the Mehler formula and its extensions. This turned
out to be not easy.
There
are several known equivalent forms of the D -Mehler formula [4, 15].
2 43
2 D , it reads
In terms of
A
@
2
B
,
C3
2
D
2
#V
Q
D
2
D
$
@
%+B
%
,
(
F Q D
C3
2 D
On the other hand,
let
%
C3
F
% B
% 3
2 D
,
, then
A
3^V
N
* *
D
Q
%
@
N3
F
*
,
C3
2
D *
NV
%
Q
D 3^V
%
Q
< <
D +
(13)
be the generating function for the number of subspaces of N :
@
B
*
Q
CWV
2
D
Q
D D
3^V
Q
@
3
Q
@
Q
WV
*
Q
@
@
43^V
Q
@
(14)
It is this form of the D -Mehler formula which has the only known combinatorial proof [15]
using the vector space interpretation. However, it does not seem to be possible to extend this
proof along Foata and Garsia’s [12] line to find a multilinear extension of the D -Mehler formula.
Firstly, we need a D -analogue of the exponential formula, which is not known in general. (A
somewhat specialized D -analogue of the exponential formula was devised by Gessel [13], but
I do not know how to use his method on linear spaces over finite fields.) Secondly, linear
4
subspaces, although very useful in enumeration arguments, are difficult to be dealt with in
bijective arguments. Hence, beside needing a D -analogue of the exponential formula, we also
need a different combinatorial proof of D -Mehler formula which uses some easier-to-describe
combinatorial objects, which is precisely the main result presented in this paper.
Several multilinear extensions have been found by Karande and Thakare [16], and Ismail
and Stanton [14]. However, their formulas all involve an infinite sum, and not as general as the
Kibble-Slepian formula. It would be interesting to have combinatorial proofs of their findings.
2.3 A 1 -exponential formula
Gessel [13] gave a partial answer to the question raised near the end of the previous section. Let
us briefly summarize here his main result which we shall make use of later. Define a D -analogue
of the derivative:
Suppose
Let
9
F
Q
is a function with
F
Q
D
F) D
Q
(15)
Q
and D -exponential form
Q
@
A
F
B
Q
(16)
IFEN
,
9
E J be the D -analogue of the function
where the coefficients
Q
Q
, namely
F
@
A
B
,
Q
G
IFEN
(17)
are defined recursively by
Then, we can write
*,
d
F
% B
, %
&%
if I
if I
% *,
G
F
(18)
as an infinite product:
Q
@
, F)
B
F D D
F
Q
D
Q
H
(19)
3 Main Results
3.1 Main theorem and approach
The main result of this paper is to provide a D -analogue of Foata’s proof of the Mehler’s formula.
This
is done by first introducing another variant of43the D -Hermite
polynomials denoted
2 C3
2
2 D . The new variant is just a rescaling of the old J
2 D , and can also be defined com
2 #3
2 D is then obtained by generalizing Gessel’s
binatorially. The D -Mehler formula for the
result summarized in Section 2.3.
Specifically, define
3
2
43
2
D
g
D
< 2
J
\
.g
M
D
2
D
_
Then, these new D -Hermite polynomials can be combinatorially defined as follows.
5
(20)
Proposition 3.1. Let be a weight function on matchings defined by
* 7
3 ( : - : (: - Then,
2
D
7
C3
A
F
D
2
: 7
2
Proof. Recall the combinatorial interpretation (11) of the
2
C3
2
D
P
D
g
D
g
*
2
J
g
: F
: D
C3
J
D , we get
2
_
:
(
-
\
M
(: 3
g
_
D
(: - 3 (: g
*
( : - : ( : D
3
A
:
F
D
2
D
(22)
3
g
M
\
: (
: A
<
A
(21)
+
D
<
< * ( : The following is the main theorem of the paper.
Theorem 3.2. The polynomials
A
@
2
B
C3
,
2
43
D satisfy the following Mehler-type identity:
2
2
D
2
#V
@ B
%+
2
D
FQ
IFEN
, F
Q
* *
D
%+*
**
Q
D%
* *
Q
@
F. D D ( F
Q
* *
D
%+*
*
3^V
Q
D
%+*,
43
*
V
*
+
(23)
Note that (23) is the same as (13) and (14), up to a change of variables.
Theorem 3.2 shall be shown in several steps. Notice that the right hand side of (23) looks
similar to the right hand side of (19). We shall find a function so that the two are identical.
The coefficients
in the D -exponential expansion of shall be interpreted as enumerating a
certain kind of new species (Section 3.3) called -species (Section 3.2). Another -species
defined from is enumerated by a sequence
which satisfes relation (18). The -species
basically gives
a
combinatorial
interpretation
of
relation
(18). The last step is to bijectively
2 CWV
2 C3
2 D
2 D (Section 3.4). The species is completely analogous to the
show that
bicolored involutionary graphs introduced by Foata in his proof of the Mehler’s formula. Thus,
our proof of Theorem 3.2 can be thought of as a D -analogue of Foata’s proof. A few simple
by-products of the new kind of species shall also be derived.
3.2 Weighted -species
In this section, we generalize Theorem 5.2 in [13] by introducing a new kind of species [3],
which then
gives a combinatorial interpretation of identity (18).
V
Let denote the symmetric group on E$FHGIKJ as usual. More generally, we use to
denote the set of all permutations on a totally ordered set of size I . Each word g
g
6
where Rg G
i.e. V G
6
could be thought of as a permutation on written in one line notation,
. The set is called the content of , and is denoted by I Q . For any
, we use to denote the number of inversions in . For any two subsets and
of , let G
denote the number of inversions created by pairs of elements in and ,
namely
i#
i
i
g
V
G
G
2
Gg G -
Let be an integral domain and E LD G Q G Q * G
J be a ring of formalpower series
Q
or
of polynomials over
on the variables D4G G
. An -weighted
set is a pair G where
WC
2
2 g g
<
is a function associating a weight
is a set and to each
element
.
D Q Q*
An -weighted set G is said to be summable
if for each monomial , the
WC
number of elements whose weight contributes a non-zero coefficient to is finite.
We are now ready to describe the new species.
Definition 3.3. An -weighted -species is a rule which
(i) to each totally ordered set
, and each permutation weighted set E G J G ,
V
, associates an -
(ii) to each increasing bijection * , and each permutation associates a weight-preserving bijection
where V
E G J
and * (
< ),
E G J G E * G * J G G
V
* are derived from in the natural way.
Moreover, these functions E G J must also satisfy the functorial properties:
> !
E G J
E#
"$ G J
>
G
E G J%" E& G J
(24)
(25)
Basically, the functorial properties say that the weighted sets E G J G depend only on
the fact that is totally ordered and on ’s cardinality. When 2 2
I we shall use ELI G J to
denote E G J , and ELIMJ to denote ' ( ELI G J .
Definition 3.4. Let be an -weighted -species with weight
function . The -generating
)
*
Q32 D
series of is the D -exponential formal power series
with coefficients in defined by
+)
D
Q32
A
,
2
ELIMJ 2
)
Q
ISEN
G
(26)
)
where the D -inventory 2 E8IKJ 2 is defined by
2
E8IKJ
2
)
A
A
( , !
WC
D.-
( (27)
Theorem 5.2 of [13] was about partitioning permutations into basic blocks with a multiplicative weight function on the blocks. We generalize this notion by defining the so-called
permutation partition.
Definition
3.5. %Given
/
G
G
such that
, a permutation partition / of is a sequence of non empty words
*
%
in one line notation, and that the largest elements of G
G
We shall write /
0 for “/ is a permutation partition of .”
7
%
form an increasing sequence.
We are now ready to define a -species whose “connected components” are structures of
another -species.
Definition
Let be a weighted -species with weight function . Define the -species
)
) ) 3.6.
with weight function as follows. For each totally ordered set and V
, define
7
%
%
E G J
J ` #` E G J G
(28)
E G
where /
G
-
G
%R
b , and 8
-
G
we associate
I
Q
G
%
, for all g
S
E G
b
F=G
J
`
G . Moreover, for each
%
%
E G
#`
-
!
-
J
%R (29)
This
is the analogue of the multiplicative property in Theorem 5.2 of [13]. The fact that
)
is a -species is easy to verify. At last, the promised generalization of Theorem 5.2
in [13] can now be stated:
. Let
Theorem
weight function
3.7. ) Let be a -species of structures with
6 @ B
, by ,
species defined as above. Define a sequence
F and
Let
@ B
2
Then,
F6
A
@
B,
Q
IFEN
%+*,
A @ FJ
Q
2
Q
D
2 !G for
9
E
, F
B
@
IFEN
)
!
E
be the -
, and
B,
namely
)
!
2
9
E8IKJ 2 G I] F
, be the sequence defined by ,
)
Q
2
F D D
Q
D J
D
Q
H
(30)
(31)
F
relation (18). Recall
that
Proof. We only need to verify that the sequences
E8I F J is a sequence of structures of :and
satisfy
each
.
Let
be
G
G
G
G
b b
the corresponding
permutations (or words) underlying G G ! . Let! , for
each
I Q
,
2
2
g
that I]
F* . Suppose F , . Let F= G G I . Notice
F , and
E F=GI F J . Note that
G
G since I F .
Furthermore, let
E J be the structure of species obtained from by removing . For
each structure of a -species, we use to denote the underlying permutation of .
It is clear that
H H
and that
G
F
8
+
G
In order to form a
then form I
Consequently,
A
*,
A
D
A
A
%+B
A
%+B
I!
A
%+B
,
"
BC%
&%
`
-
D
( ( --
A
-
`
A
A
BC%
(
-
A
-
A
BC%
of E F=GIKJ ( /
E J and
!
I ),
E J.
/
( ( --
A
%+B
F
- D
, ( - D
, ( -
D
,
A
*,
ELI F J , we can first pick a -subset
, and finally concatenate any pair of
!
( ( --
`
D
`
D
( ( $--
D
-
( ( --
D
-
A
( (
--
( (
--
D.-
% *,
Example 3.8. Theorem 3.7 implies Theorem 5.2 in [13] and thus all its consequences as derived
by Gessel.
Example 3.9. Take F so that F in Theorem 3.7, we obtain
A
F
@
B,
where,
IFEN
Q
@ B
, ( F
D
@
Q
N
D
Q
2 /
( D D
@
Q
( -
2 /
0 2 D -
A
Q
*,
* *
D
+
G
(32)
In fact, when D
F , counts the number of sets of words on E F=GIKJ whose contents are disjoint
and whose union
of
contents
is exactly E$F=G IKJ . While, when D
F the right hand side of (32)
LC
goes
to XY[Z ( Q F. Q + . Thus, we could have proven easily identity (32) combinatorially when
D
F.
Following Gessel’s line of derivation we can generalize the previous example as follows.
8 %
Corollary 3.10. Let /
G
G be any permutation partition of number of words of size g of / . Define a weight function for / by /
Then,
)
D
Q32
A
F
Q
F
( /
@
B
where
A
@
B
D
-
F
, ( F
A
F) D D
3*,
,
9
( D
ELI
F JN
Q
. Let
b 3 b
#
b(
( -
/
be the
, and let
(33)
Q
Q
+
G
(34)
Example
3.11.
Write /
3
Q
F D , so that
%
0
if / 0
and hence
!
and all words of / are of size at most . Set d
F
/
( 3
A
F
D
2
A
( if /
0 * otherwise,
43
Q
<
3
( -
D
-
( Corollary 3.10 gives
)
3
GQ 2D
@
A
B
,
#3
D
2
@
Q
IFEN
@
where
?b7T
*
(
*
N
N and Thus,
A
2
A
( D
F
3
<
N
F%
!D
Q
D
%
* * G
D
Q
(35)
*
F D . We now get
g Q
C3
F D D
, F.
%+B
F
, F.
B
F6 D
g
( - D
-
( -
P
*
2
J
A
: \
M
g
3
F D
D
_
(36)
F D
: ( : - 3 ( :
D
G
(37)
an interesting combinatorial identity.
3.3 A 1 -analogue of the bicolored -involutionary graphs
The previous section gives a combinatorial view of identity (18). As outlined at the end of section 3.1, the
next step in
the proof of Theorem 3.2 is to find a -species whose -generating
)
Q.2 D
series Q
is such that the right hand side of (19) is the same as that of (23). The
)
)
) -species
is a D -analogue of the bicolored I -involutionary graphs. We actually
will start defining first.
Definition 3.12. A graph
G
satisfies the following conditions:
1.
2.
is called an ordered bicolored I -involutionary graph if
has I vertices labeled by I distinct positive integers in .
has no multiple edges, but can have loops.
3. The I vertices of line up on a horizontal line, so that we can speak of a vertex being
on the-left
or right
of another,
and so that the vertices of forms a permutation /
V
/ / *
/
.
4. Each edge of
5. Each vertex of
is colored either red or blue.
N
is incident to exactly edges of different colors.
6. A non-loop edge of
can only connect some /
10
b
to /
b *,
unless it completes a cycle of
.
7. Let G
G be the connected components of from left
to right. Let denote
must satisfy the
the largest vertex number
in a connected
component of , then /
5 5
condition that .
-
8. For each connected component , the vertex numbered
blue edge incident to it.
has to be on the left of the
9. If a connected component is a cycle, then the vertex numbered has to be the
left most vertex among all vertices of . It is not difficult to see that the connected
components of can only be in one of forms as shown in Figure 2. In the figure, the
bold lines represent blue edges and the thin lines represent red edges.
9
9
(
;
;
(-
(-
(-
(
;
;
Figure 2: Possible connected component types of an ordered bicolored I -involutionary graph.
Let denote the set of all ordered bicolored I -involutionary graphs on , where is
be the set of all graphs in which have exactly one
an I -set of positive integers. Let
connected component.
When E F=GIMJ , and shall be used for convenience.
9
Let E F=GIMJ be
the trivial one-to-one
correspondence between and E$FHGIKJ which
> preserves order. For each , let D
denote the graph obtained from by renumber
to denote the set of vertices of
ing each vertex of by . Conversely, we also use .
Definition 3.13. A weight function defined on with values over some commutative algebra over the rationals is said to be multiplicative if it satisfies the following conditions:
(i) F
-
9
D
>
H
.
%
(ii) If G
are ordered bicolored involutionG are the connected components
%
F of (which
ary graphs themselves), then .
The following theorem is obviously a very special case of Theorem 3.7 applied to the ordered bicolored involutionary graphs.
Theorem
3.14. Supposed is a multiplicative function on
@ B
Let
,
@ B
, be the sequence defined by ,
%+*,
A
A
D
( ( --
-
, and
11
D
-
( (--
. For I
, define a sequence
!
for
. Then,
@
A
B
,
Q
A@ 9
ISEN
Q
ISEN
B,
9
9
9
Definition
3.15. Let be a graph in
. For each edge (respectively
vertex g ) of , let
9
connected component containing (respectively g ). Define
(respectively g ) denote the
a weight function on each edge of
D
9
D
as follows:
9
V
is a non-loop red edge G
is non-loop, blue and to the left of
is non-loop, blue and to the right of
is a red loop G
is a blue loop
if 9
if 9
if 9
if 9
if
F
3
9
H
9
G
H
G
Let be a weight function defined on by:
9
G
( -
9 then obviously is multiplicative.
We
call an ordered bicolored I -involutionary graphs with the weight associated a bicolored
D4G I -involutionary graph. Figure 3 shows an example of such a graph. In the figure, the largest
;
N
N
B
(
;
N
>
N
<
N
W W (W W , > < * <
N
*
;
(
%
N
*.-
-
"!$#%'&)(+*,-*/.0'*1*
( -
B
*
Figure 3: An example of a bicolored D GI -involutionary graph.
vertex number
in each component has been put in bold face.
@ B
, be a sequence defined by ,
Lemma 3.16. Let be the function defined above, and
and
A
D - ( (- - G when I] F
Moreover, let
D
Q32
Then,
Q)2
D
F
F.
Q
* *
D
B
Q
D
*
F)
@
A
Q
IFEN
,
F
* *
Q D
12
Q
* * $3^V
D
*Q
F
D Q
D
43
*
V
*
(38)
Proof. Firstly, we claim that
*
%+*,
!
8
!
E J& E
FJ D
%
*
!
N
3^V
N
E
%+*,
! the paths which
To see this, let us consider Figure 2. The components in *
can only be
N
F . Summing
start and end with different colored loops, and have largest vertex number
( (- Dover all components which start with a blue loop and end with a red loop we get
!
!
%
the term
* N 3^V
E JD
N
G
E
while the components which start red and end blue introduce the term
!
E
F JD
%
*
!
N
N
E
3^V
The details are easy to be verified and hence omitted here.
Secondly, we claim that
%+*
*
% *
*
% *
*
D !
N
*
!
N
!
F E N
E
*
F JD
% *,
!
N
!
!
% *, E
Here, the term D F EN is from the cycle components,
N
*
paths which start and end with a blue loop, and E F JD
start and end with a red loop.
By definition,
D
Q32
@
A
B
A
ISEN
@
A
!
E J[ E
,
%+B
!
N
!
%+B
+
3 *
F J D V *
F EN from the
F E N from the paths which
! *
*
V
%+*,
Q
,
@
*
43
F E N
,
%+*
*
(D *
FJ D
%
N
!
N
!
3^V
N
E
!Q
N
F E N
E
F JD
F E N
!
%+*,
*
*
%+*,
!
N
F E N
*
3
*
V
+
N
% *
*
!Q
*
N?
N
E
Hence,
Q42
D
A
F
@
!
%+B
!
E J& E
,
*
FJ D
%
3^V
Q
*
%
@
A
%+*
%+B
* *
D ,
%+*,
Q
@
A
!
%+B
E
,
F JD
%+*,
*
*
3
*
V
Q
*
%+*,
(39)
Now, we calculate each term of (39) separately as follows:
3^V
A
@
%+B
,
!
E J& E
!
FJ D
*
%
Q
*
%
3^V
Q
* *
D
A
Q
* *
D
A
Q
* *
D
A
F
3^V
F
3^V
F
F
13
F
!
E
%+B
,
A
b B
cHB
,
@
Q
Q
Q
b
* *
D
*(
D
,
b
,
%
*
%
@
%+B
%
*
FJ D
* * $3^V
D
* * *
D
F Q D Q
@
A
@
c B
% c
,
Q
*
-
c
*
c
c
Q
D *
%
(40)
Similarly,
@
A
%+*
%+B
D ,
%+*,
* *
Q
*
D
Q
F
Q
*
G
D (41)
and
*
3
*
V
@
A
!
% B
E
,
F JD
*
%+*,
%+*,
Q
*
*
83
*
83
*
V
F
A
%+B
*
=
Q
*
,
V
* *
D
b
A
F
*
Q
,
*
Q
*
-
c
c
%=c
*(
Q
*
- *
c
(42)
c
D D @
c B
% c
%
*
Q
*(
D
,
b
b B
cHB
,
@
A
D
Q
* *
D
Q
D
,
%
@
A
D
cHB
,
% *Oc
*
A
%+B
Q
43
D
Q
D
Q
*
V
*
83
*
V
%
@
A
Combining (40), (41) and (42) yields (38).
Corollary 3.17. Let be the function defined above, and
A
( ( --
D.-
@ B
, be a sequence defined by
(43)
Then,
Q)2
D
F
@ B
%+
, F
Q
* *
D
% *
**
where
D%
* *
Q
@
F D D ( F
Q
Q32
D
A
* *
D
%+*
*
$3^V
Q
D
%+*,
3
*
V
*
+
G
@
B
Q
,
Q
ISEN
Proof. This is straightforward from Theorem 3.14, Lemma 3.16 and equation (19).
3.4 A bijection
This section
completes
the last
step of the proof of Theorem 3.2. We are left to demonstrate
2 CV
2 C3
2
2
. We shall show this relation combinatorially as formally put in
that
D
D
the following theorem.
Theorem 3.18. Let
Then,
2
be defined combinatorially by equation (22), and
2
43
2
D
2
CWV
2
D
F
by equation (43).
[
*,
1
`
Proof.
We want to finda*, weight-preserving
bijection which maps
a pair*,7 G7
*,
*,
. Let 7 G7 be a pair
1
1
`01
to a graph of
matchings
in
,
where
the
fixed
3
V
GI0 F of 7
points of 7 are weighted by and of 7 by . As before, we view the vertices F=G
and 7 as lying on a horizontal line from left to right in that order, with the edges drawn on the
14
upper half plane. Let G
G , ( / I F ) be the sequence of vertices
of 7 starting from the
-
9
9
9
right which are not left end-points
of 7 ’s edges. Similarly,
let G
G , be9 the
corresponding
-
:
sequence for 7 . Notice that
I F . Let G
G (respectively G
G : ) be the
set of edges of 7 (respectively 7 ) ordered by their right end-points
starting from the right.
Our idea is to start from the right, look simultaneously at and , * and * , ... determine
the “right place” to stop and build up the right most connected component of based on the
relative distribution of edges and points of 7 and 7 seen so far. Then, remove certain points
and edges from 7 and 7 to get and respectively, and re-apply the method to get the next
(from the right) connected
component of , and so on.
Looking at and , * and * , ... there will roughly be situations as follows.
!
b
1. At some F , all
of !
i0
respectively, and
!
b
9
and , F / g /
F , are right end-points
of edges in 7 and 7
cF
.
F is the least integer such that
*,
!
%+*,
9 of 7
of 7 and then a fixed point
where /
. For this
2. We meet a fixed point
9
case to be disjoint from
1, it is necessary that all edges of 7 whose right end-points
*, case
.
are on the right of
have 3. We meet a fixed point
*,
of 7
% *,
strictly before a fixed point
of 7 .
4. Two fixed points of 7 are met before any fixed points of 7 .
5. Two fixed points of 7
are met before any fixed points of 7 .
Note that similar to case 2, the cases 3, 4, and 5 need to be defined so that they are disjoint from
case 1. These cases determine our “right place” to stop as mentioned above.
Formally, we consider cases as follows.
!
!
Case 1. There exists a , 0/
(i)
i
!
F
(ii) For all
i
(
/
*
-
, such that
is the smallest
!
9 integer where
FHG
G
F,
c
9
c
has right end-point
The situation is depicted in Figure 4. Let
c
9
. (i.e.9 and
c
(respectively
c-
for all
!
i
.)
/
has right end-point
c
.
) be the matching obtained
9
!#"
Figure 4: Illustration of case 1.
9
%+*,
9
9
%+*,
G
by removing G
and their end-points (respectively
end
G 7 G7 G suchandthattheir
points) from 7 (respectively 7 ). We shall
construct
the
last
%+*, *
I %+FH*,G g - G GHg , and that G forms
connected component of is a
cycle Rg *
G
G g be the set of the%+rest
the rest of the components of . Let a
of the
*,, points
on
the
cycle
as
shown.
To
do
this,
we
need
to
pick
a
permutation
g *
g
V
%+*, *
, where a
a
Rg
G
G g is a set of distinct integers in E F=GIKJ , such that the
9 of this cycle to the weight of is exactly equal 9to the contribution
9
9
contribution
of G - G %+*, to the weight of 7 plus the contribution of G G %+*, to the
15
9
9
c
c
weight of 7 . Notice that removing the edges and
total weights of the rest of edges of 7 and 7 . Let
number
of inversions created by pairs of numbers in
GD `]a
such that D .
does not have any effect on the
E F=GIMJ5 a , and
G+a be the
`a , namely the number of pairs
As each red edge on is weighted D and each blue edge weighted F , it is easy to see that
D -
* *
( - ( - *
%+*,
D
%+*,
D - ( 9 - G
D $ ( 9 $ $
$
%+*,
D
%+*,
D
G
(44)
(45)
(46)
Hence, we need to pick such that
%+*,
G+a
A
F
%+*,
9
c B,
A
c
9
c B,
!
c (47)
Observe that
9
F/
%+*, 9
c
9
c /
Q
d
I
I
g
N
G
a
the
Q
if Q
if Q
c % *,
Gg*
Q
A
I
/
i
2
T B,
A
2
I
(
T B,
Q
A
i
N-i
N-i
i0
G
FHG
T
Q
F
N-i
%+*,
Q
cHB,
/
F
T
jg ,
G! G
-
G
F
!
F=G
G G
HF G
G
i0
!
F
Rg
F=G
G
-
N
G
g
, working
TU?
GHg
(51)
!
T
F so that g is well
9
c
A
cHB,
* , and an , 9
/
16
I
h
N-i#
+
A
cHB,
(
I
!
c G
F 2
%+*,
i
+
%+*,
A
!
i
preceeds g ,
I
(
cHB,
Case 2. There exists a , 0/
F=G
=F G
F for all Q
%
!
, element by element starting from
%+*,
*
which is exactly (47).
i
(48)
(49)
(50)
!
-
%+*,
by
c-
9
G
9
F
i
th smallest number in E$F=G IKJ
/
*
g
!
N
G
F
Q It is easy to check that F
defined. Moreover,
-
G
N-i
F
G
N-i
F
I
/
Then, recursively
determine
% *,
toward g *
as follows:
T I
/
Q G
on F=G
Now, define a function
/
!
such that
N-i
NO
N-i
F +
i
9
c
c
(i) For all
F=G
G , is the right end-point
of
3
9
is a fixed point, which
is weighted by . And, for all
c+
right end-point of ! .
9
i
F=G
(ii) For all
V
weighted by .
c
G ,
9
, and
i0
c
is the right end-point of
c
N
. And,
G
%+*,
The situation is depicted in Figure 5. This time, the last component
*,
!
. Moreover,
c
G F , is the
is a fixed point
of
starts with
Figure 5: Illustration of case 2.
N
F st point from the
a red loop and ends with a blue loop. The point I F is the
right. Let , a , be defined as in the previous case, then the corresponding , and
are as follows:
* *
( - ( - *
- D -%
(9 D % D $ $
(9 D D $
$
3
D
V
%=
*
3^V
G
(52)
G
(53)
(54)
Hence, we need to pick so that
%
For Q
F=G
G+a
A
F
, the corresponding
Q
F
I
I
I
N
Q Q Q Q
9
c B,
!
N
G
%
F. F. 9
9
9
c B,
c ]
(55)
is:
c
if Q
if Q
if Q
c
9
A
c
c N i
N i
N i
i0
,
i0
,
F,
i
T
FHG
G G !
=F G
G! G
F=G
G
(56)
T
As in the previous case, g is defined by (51). To show that g is well defined and that they
satisfy (55), we only need to observe that
F
/
/
/
(ii) For all
i
9
c FHG
/
!
G ,
G
F
N-i
N-i
/
I
F
/
I
F
/
F=G
I
/
c
!
Case 3. There exists a , F
i0
c
9
!
(i) For
all
3
.
9
* , and an , c
,
i
N-i
/
G
G
F G
i
F=G
c
.
17
G
i0
-
F=G
G
!
G
F=G
!
G G
(57)
(58)
(59)
!
/
is the right end-point of
9
-
9
c
F , such that
, and
%+*,
is a fixed point weighted
Figure 6: Illustration of case 3.
i
(iii) For
all
F=Gi
V
, and for all
-
G
9
c
,
*,
9
is-the
right end-point
of ,
is a fixed
point weighted
c
c
G
G F
is the right end-point of
.
N
c
!
* *
( - ( - *
D -%
- (9 D % D $ $
(9 D D $
$
(
The situation is depicted in Figure 6. In this case, we have
Hence, we need to pick so that
For Q
F=G
G+a
I
Q
F
I
I
I
N
Q Q V,
F.
9
F. if Q
if Q
if Q
if Q
c 9
-
3^V
G
9
c
c c
(60)
G
(61)
(62)
A
9
c B,
c ]
is:
c
F. Q 9
9
Q
c B,
Q 3
*,
%
A
F
, the corresponding
D
% *
%
!
N
G
*,
N i
N i
N i
N i
i0
,
i0
,
F,
F,
F=G
i
T
G G !
-
=F G
G G
- F=G
G G !
-
F=G
G .
i
(63)
(64)
T
As in the previous case, g is defined by (51). To show that g is well defined and that they
satisfy (63), we only need to observe that
F
/
/
/
/
9
I
F
/
I
F
c -
*
N-i
F
, and an
G
i0
, 0/
F=G
(ii) For all
FHG
/
c
G
F,
c
9
c
* *
%
( - ( - *
*
- D
(9 - 3 *
D %+*, D $
G
$ ( 9 - D
D
$
$
-%
18
9
, and N
9
c
*,(
c
-
G
is the right end-point of
D
, such that
G
!
*,
. Moreover,
c
F , is the right
.
The situation is depicted in Figure 7. In this case, we have
!
(65)
(66)
(67)
(68)
!
(i) For all%+*
F=G
G , is the right 3 end-point i of
* are 9 fixed points weighted . For all
and
c
end-point of
.!
i
G G !
N-i
i0
-
G
=F G
G G
N-i
i0
G
F=G
G G
N-i
i0
-
F G
F=G
G
F
I
/
(
/
I
/
c !
/
c
9
Case 4. There exists a , 0/
i
c
9
!
9
3
*
G
(69)
(70)
(71)
! " Figure 7: Illustration of case 4.
Hence, we need to pick so that
For Q
F=G
G+a
Q
F
I
I
I
N
Q Q Q 9
F F
c A
c
9
c B,
c ]
is:
if Q
if Q
if Q
c
9
Q
c
9
9
c B,
F , the corresponding
%+*,
A
F
!
N
G
%
N-i
N-i
N-i
i0
,
i0
,
i
F,
G G !
=F G
G! G
F=G
G F
T
-
F=G
(72)
(73)
T
As in the previous case, g is defined by (51). To show that g is well defined and that they
satisfy (72), we only need to observe that
9
F
c
/
9
/
/
!
c
9
!
/
-
I
/
c Case 5. There exists a , 0/
i
I
/
(
-
*
N-i
F
N-i
, and an
i
G
N-i
F
I
/
F
F=G
F G
i
G
F=G
-
i
(ii) For all
(iii) For all
FHG
HF G
G
G
,
F 9,
G
G
F
9
c
!
(74)
(75)
(76)
!
G G
*,
%+*
* are
. Moreover,
and
c
F , is the right end-point of
9
is the right end-point of
c
F=G
!
, such that
/
c
c
i0
(i) For all
F=G
G , V is the right
end-point
of
i N
9
G
G
fixed
points weighted . For all
c
.
!
i
G
!
, 0/
-
c
.
.
The situation is depicted in Figure 8. In this case, we have
"
Figure 8: Illustration of case 5.
D
*,
D
%
D
* *
( - ( - *
D.-%
D
-
$ ( 9 $
$
19
*,
(9 $
D
V
G
*
*
% *,(
(
*,
V
*
G
(77)
(78)
(79)
Hence, we need to pick so that
%+*,
For Q
F=G
G
N
F
I
F
I
I
Q Q Q Q I
N
9
9
9
F.
F. 9
Q
if Q
if Q
if Q
if Q
c F. c
c c
9
c B,
F , the corresponding
Q
!
G+a
%
A
c
A
9
c B,
c ]
(80)
is:
N i
N i
N i
i0
,
i0
,
F,
N i
F,
F=G
G G !
=F G
G G
F=G
G G
-
F=G
G
i0
i0
T
(81)
!
F
T
As in the previous case, g is defined by (51). To show that g is well defined and that they
satisfy (80), we only need to observe that
9
F
c
/
9
/
/
/
9
I
F
/
I
F
c
/
9
c c I
/
/
F
I
N-i
i0
N-i
G
N-i
N-i
F
F=G
i0
G
G
i0
F G
G G
-
=F G
G
F=G
G G
i0
!
F=G
F=G
-
G
!
(82)
(83)
(84)
(85)
Acknowledgements
I would like to thank Prof. Dennis Stanton at the School of Mathematics, University of Minnesota, U.S.A., for many helpful discussions and insightful suggestions, which made this work
possible. This problem was suggested by him as part of my mathematics Masters thesis at the
University of Minnesota. I would also like to thank Prof. Dominique Foata at the Département
de mathématique, Université Louis Pasteur, France, for various detailed comments which have
greatly improved the presentation of this paper.
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