Báo cáo toán học: "A q-analogue of some binomial coefficient identities of Y. Sun" ppt

A q-analogue of some binomial coefficient
identities of Y. Sun
Victor J. W. Guo
1
and Dan-Mei Yang
2
Department of Mathematics, East China Normal University
Shanghai 200062, People’s Republic of China
1
,
2
plain
Submitted: Dec 1, 2010; Accepted: Mar 24, 2011; Published: Mar 31, 2011
Mathematics Subject Classifications: 05A10, 05A17
Abstract
We give a q-analogue of some binomial coefficient identities of Y. Sun [Electron.
J. Combin. 17 (2010), #N20] as follows:
⌊n/2⌋

k=0

m + k
k

q
2

m + 1
n − 2k

q

q
(
n−2k
2
)
=

m + n
n

q
,
⌊n/4⌋

k=0

m + k
k

q
4

m + 1
n − 4k

q
q
(
n−4k
2

)
=
⌊n/2⌋

k=0
(−1)
k

m + k
k

q
2

m + n − 2k
n − 2k

q
,
where

n
k

q
stands for the q-binomial coefficient. We provid e two proofs, one of
which is comb inatorial via partitions.
1 Introduction
Using the Lagra nge inversion fo r mula, Mansour and Sun [2] obtained the following two
binomial coefficient identities:

⌊n/2⌋

k=0
1
2k + 1

3k
k

n + k
3k

=
1
n + 1

2n
n

, (1.1)
⌊(n−1)/2⌋

k=0
1
2k + 1

3k + 1
k + 1

n + k

3k + 1

=
1
n + 1

2n
n

(n  1). (1.2)
the electronic journal of combinatorics 18 (2011), #P78 1
In the same way, Sun [3] derived the following binomial coefficient identities
⌊n/2⌋

k=0
1
3k + a

3k + a
k

n + a + k − 1
n − 2k

=
1
2n + a

2n + a
n


, (1.3)
⌊n/4⌋

k=0
1
4k + 1

5k
k

n + k
5k

=
⌊n/2⌋

k=0
(−1)
k
n + 1

n + k
k

2n − 2k
n

, (1.4)
⌊n/4⌋


k=0
n + a + 1
4k + a + 1

5k + a
k

n + a + k
5k + a

=
⌊n/2⌋

k=0
(−1)
k

n + a + k
k

2n + a − 2k
n + a

. (1.5)
It is not hard t o see that both (1.1) and (1.2) are special cases of (1.3), and (1.4) is
the a = 0 case of (1.5 ) . A bijective proo f of (1.1) and (1.3) using binary trees and colored
ternary trees has been given by Sun [3 ] himself. Using the same model, Yan [4] presented
an involutive proof of (1.4) and (1.5), answering a question of Sun.
Multiplying both sides of (1.3) by n +a and letting m = n + a − 1, we may write it as

⌊n/2⌋

k=0

m + k
k

m + 1
n − 2k

=

m + n
n

, (1.6)
while letting m = n + a, we may write (1.5) as
⌊n/4⌋

k=0

m + k
k

m + 1
n − 4k

=
⌊n/2⌋


k=0
(−1)
k

m + k
k

m + n − 2k
m

. (1.7)
The purpose of this paper is to give a q-analogue of (1.6) and (1.7) as follows:
⌊n/2⌋

k=0

m + k
k

q
2

m + 1
n − 2k

q
q
(
n−2k
2

)
=

m + n
n

q
, (1.8)
⌊n/4⌋

k=0

m + k
k

q
4

m + 1
n − 4k

q
q
(
n−4k
2
)
=
⌊n/2⌋


k=0
(−1)
k

m + k
k

q
2

m + n − 2k
n − 2k

q
, (1.9)
where the q-binomial coefficient

x
k

q
is defined by

x
k

q
=






k

i=1
1 − q
x−i+1
1 − q
i
, if k  0,
0, if k < 0.
We shall give two proofs of (1.8) and (1.9). One is combinatorial and the other algebraic.
the electronic journal of combinatorics 18 (2011), #P78 2
2 Bijective proof of (1.8)
Recall that a partition λ is defined as a finite sequence of nonnegative integers (λ
1
, λ
2
,
. . . , λ
r
) in decreasing order λ
1
 λ
2
 · · ·  λ
r
. A nonzero λ
i

is called a part of λ. The
number of parts of λ, denoted by ℓ(λ), is called the length of λ. Write |λ| =

m
i=1
λ
i
,
called the weight of λ. The sets of all partitions and partitions into distinct parts ar e
denoted by P and D respectively. For two partitions λ and µ, let λ ∪ µ be the partition
obtained by putting all parts of λ and µ together in decreasing order.
It is well known that (see, f or example, [1, Theorem 3.1])

λ
1
m+1
ℓ(λ)=n
q
|λ|
= q
n

m + n
n

q
,

λ∈D
λ

1
m+1
ℓ(λ)=n
q
|λ|
=

m + 1
n

q
q
(
n+1
2
)
.
Therefore,

µ∈D
λ
1
,µ
1
m+1
2ℓ(λ)+ℓ(µ)=n
q
2|λ|+|µ|
= q
n

⌊n/2⌋

k=0

m + k
k

q
2

m + 1
n − 2k

q
q
(
n−2k
2
)
,
where k = ℓ(λ). Let
A = {λ ∈ P : λ
1
 m + 1 and ℓ(λ) = n},
B = {(λ, µ) ∈ P × D : λ
1
, µ
1
 m + 1 and 2ℓ(λ) + ℓ(µ) = n}.
We shall construct a weight-preserving bijection φ from A to B. For any λ ∈ A , we

associate it with a pair (λ, µ) as follows: If λ
i
appears r times in λ, then we let λ
i
appear
⌊r/2⌋ times in λ and r −2⌊r/2⌋ times in µ. For example, if λ = (7, 5, 5, 4, 4, 4, 4, 2, 2, 2, 1),
then λ = (5, 4, 4, 2) and µ = (7, 2, 1). Clearly, (λ, µ) ∈ B and | λ | = 2|λ| + |µ|. It is easy
to see that φ : λ → (λ, µ) is a bijection. This proves that

λ∈A
q
|λ|
=

(λ,µ)∈B
q
2|λ|+|µ|
.
Namely, the identity (1.8) holds.
the electronic journal of combinatorics 18 (2011), #P78 3
3 Involutive proof of (1.9)
It is easy to see that
q
n
⌊n/2⌋

k=0
(−1)
k


m + k
k

q
2

m + n − 2k
n − 2k

q
=
⌊n/2⌋

k=0
(−1)
k

λ
1
m+1
ℓ(λ)=k
q
2|λ|

µ
1
m+1
ℓ(µ)=n−2k
q
|µ|

=

λ
1
,µ
1
m+1
2ℓ(λ)+ℓ(µ)=n
(−1)
ℓ(λ)
q
2|λ|+|µ|
, (3.1)
and
q
n
⌊n/4⌋

k=0

m + k
k

q
4

m + 1
n − 4k

q

q
(
n−4k
2
)
=

µ∈D
λ
1
,µ
1
m+1
4ℓ(λ)+ℓ(µ)=n
q
4|λ|+|µ|
. (3.2)
Let
U = {(λ, µ) ∈ P × P : λ
1
, µ
1
 m + 1 and 2ℓ(λ) + ℓ(µ) = n},
V = {(λ, µ) ∈ U : each λ
i
appears an even number of times and µ ∈ D }.
We shall construct an involution θ on the set U \ V with the properties that θ preserves
2|λ| + |µ| and reverses the sign (−1)
ℓ(λ)
.

For any (λ, µ) ∈ U \ V , notice that either some λ
i
appears an odd number of times
in λ, or some µ
j
is repeated in µ, or bot h are true. Choose the largest such λ
i
and µ
j
if
they exist, denoted by λ
i
0
and µ
j
0
respectively. Define
θ((λ, µ)) =

((λ \ λ
i
0
), µ ∪ (λ
i
0
, λ
i
0
)), if λ
i

0
 µ
j
0
or µ ∈ D ,
((λ ∪ µ
j
0
), µ \ (µ
j
0
, µ
j
0
)), if λ
i
0
< µ
j
0
or λ
i
0
does not exist.
For example, if λ = (5, 5, 4, 4, 4, 3, 3, 3, 1, 1) and µ = (5, 3, 2, 2, 1), then
θ(λ, µ) = ((5, 5, 4, 4, 3, 3, 3, 1, 1), ( 5 , 4, 4, 3, 2 , 2, 1)).
It is easy t o see that θ is an involution on U \ V with the desired properties. This proves
that

(λ,µ)∈U

(−1)
ℓ(λ)
q
2|λ|+|µ|
=

(λ,µ)∈V
(−1)
ℓ(λ)
q
2|λ|+|µ|
=

µ∈D
τ
1
,µ
1
m+1
4ℓ(τ)+ℓ(µ)=n
q
4|τ|+|µ|
, (3.3)
where λ = τ ∪ τ. Combining (3.1)–(3.3), we complete the proof of (1.9).
the electronic journal of combinatorics 18 (2011), #P78 4
4 Generating function proof of (1.8) and (1.9)
Recall that the q-shifted factorial is defined by
(a; q)
0
= 1, (a; q)

n
=
n−1

k=0
(1 − aq
k
), n = 1, 2, . . . .
Then we have
1
(z
2
; q
2
)
m+1
(−z; q)
m+1
=
1
(z; q)
m+1
, (4.1)
1
(z
4
; q
4
)
m+1

(−z; q)
m+1
=
1
(z; q)
m+1
1
(−z
2
; q
2
)
m+1
. (4.2)
By the q-binomial theorem (see, for example, [1, Theorem 3.3]), we may expand (4 .1 ) and
(4.2) respectively a s f ollows:

∞

k=0

m + k
k

q
2
z
2k

m+1


k=0

m + 1
k

q
q
(
k
2
)
z
k

=
∞

k=0

m + k
k

q
z
k
, (4.3)

∞


k=0

m + k
k

q
4
z
4k

m+1

k=0

m + 1
k

q
q
(
k
2
)
z
k

=

∞


k=0

m + k
k

q
z
k

∞

k=0

m + k
k

q
2
(−1)
k
z
2k

. (4.4)
Comparing the coefficients of z
n
in both sides of (4.3) and (4.4), we obtain (1.8) and (1.9)
respectively.
Finally, we give the following special cases of (1.8):
⌊n/2⌋


k=0

n + k
k

q
2

n + 1
2k + 1

q
q
(
n−2k
2
)
=

2n
n

q
, (4.5)
⌊n/2⌋

k=0

n + k

k + 1

q
2

n
2k + 1

q
q
(
n−2k−1
2
)
=

2n
n − 1

q
. (4.6)
When q = 1, the identities ( 4.5) a nd ( 4.6) reduce to (1.1) and (1 .2 ) respectively.
Acknowledgments. This work was partially support ed by the Fundamental Research
Funds for the Cent r al Universities, Shanghai Rising-Star Program (#09QA1401700 ) ,
Shanghai Leading Academic Discipline Project (#B407), and the National Science Foun-
dation of China (#10801054).
the electronic journal of combinatorics 18 (2011), #P78 5
References
[1] G. E. Andrews, The Theory of Partitions, Cambridge University Press, Cambridge, 1998.
[2] T. Mansour and Y. Sun, Bell polynomials and k-generalized Dyck paths, Discrete Appl.

Math. 156 (2008), 2279–2292.
[3] Y. Sun, A simple bijection between binary trees and colored ternary trees, Electron. J.
Combin. 17 (2010), #N20.
[4] S. H. F. Yan, Bijective proofs of identities from colored binary trees, Electron. J. Combin.
15 (2008), #N20.
the electronic journal of combinatorics 18 (2011), #P78 6