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 jwguo@math.ecnu.edu.cn, 2 plain dan2004@126.com 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 . 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. 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. 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