A q-analogue of some binomial coefficient identities of Y. Sun

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1 A -aalogue of some biomial coefficiet idetities of Y. Su arxiv: v2 [math.co] 5 Apr 20 Victor J. W. Guo ad Da-Mei Yag 2 Departmet of Mathematics, East Chia Normal Uiversity Shaghai , People s Republic of Chia jwguo@math.ecu.edu.c, 2 plai da2004@26.com Submitted: Dec, 200; Accepted: Mar 24, 20; Published: Mar 3, 20 Mathematics Subject Classificatios: 05A0, 05A7 Abstract We give a -aalogue of some biomial coefficiet idetities of Y. Su [Electro. J. Combi. 7 (200, #N20] as follows: /2 0 /4 0 [ ] m+ [ ] m+ 2 4 [ ] [ m+ ( 2 2 m+ 2 0 ], [ ] /2 m+ [ ] ( 4 2 m+ ( 4 2 [ ] m+ 2 2 where [ ] stads for the -biomial coefficiet. We provide two proofs, oe of which is combiatorial via partitios. Itroductio, Usig the Lagrage iversio formula, Masour ad Su [2] obtaied the followig two biomial coefficiet idetities: ( /2 0 / ( 3 ( ( ( ( 2, (. ( 2 (. (.2 the electroic joural of combiatorics 8 (20, #P78

2 I the same way, Su [3] derived the followig biomial coefficiet idetities /2 0 / a ( ( 3 +a +a+ 2 /4 0 +a+ 4 +a+ 4 + ( 5 +a ( 5 ( + 5 ( +a+ 5 +a 2+a /2 0 /2 0 ( 2+a ( +, (.3 ( ( ( ( +a+ ( 2+a 2, (.4 +a. (.5 It is ot hard to see that both (. ad (.2 are special cases of (.3, ad (.4 is the a 0 case of (.5. A bijective proof of (. ad (.3 usig biary trees ad colored terary trees has bee give by Su [3] himself. Usig the same model, Ya [4] preseted a ivolutive proof of (.4 ad (.5, aswerig a uestio of Su. Multiplyig both sides of (.3 by +a ad lettig m +a, we may write it as /2 0 ( m+ ( m+ 2 while lettig m +a, we may write (.5 as /4 0 ( m+ ( m+ 4 /2 0 ( m+ ( ( m+, (.6 ( m+ 2 The purpose of this paper is to give a -aalogue of (.6 ad (.7 as follows: /2 0 /4 0 [ ] m+ [ ] m+ 2 4 [ ] 2 m+ [ ] m+ ( 2 2 ] /2 2 [ m+ 4 ( 4 where the -biomial coefficiet [ ] x [ ] x 0 is defied by i m. (.7, (.8 ] [ ] m+ 2 [ m+ ( 2 x i+ i, if 0, 0, if < 0. 2, (.9 We shall give two proofs of (.8 ad (.9. Oe is combiatorial ad the other algebraic. the electroic joural of combiatorics 8 (20, #P78 2

3 2 Bijective proof of (.8 Recall that a partitio λ is defied as a fiite seuece of oegative itegers (λ,λ 2,...,λ r i decreasig order λ λ 2 λ r. A ozero λ i is called a part of λ. The umber of parts of λ, deoted by l(λ, is called the legth of λ. Write λ m i λ i, called the weight of λ. The sets of all partitios ad partitios ito distict parts are deoted by P ad D respectively. For two partitios λ ad µ, let λ µ be the partitio obtaied by puttig all parts of λ ad µ together i decreasig order. It is well ow that (see, for example, [, Theorem 3.] λ m+ l(λ λ D λ m+ l(λ λ [ m+ [ ] m+ λ ], (+ 2. Therefore, µ D λ,µ m+ 2l(λ+l(µ /2 2 λ + µ 0 [ ] m+ 2 [ ] m+ ( 2 2, 2 where l(λ. Let A {λ P: λ m+ ad l(λ }, B {(λ,µ P D: λ,µ m+ ad 2l(λ+l(µ }. We shall costruct a weight-preservig bijectio φ from A to B. For ay λ A, we associate it with a pair (λ,µ as follows: If λ i appears r times i λ, the we let λ i appear r/2 times i λ ad r 2 r/2 times i µ. For example, if λ (7,5,5,4,4,4,4,2,2,2,, the λ (5,4,4,2 ad µ (7,2,. Clearly, (λ,µ B ad λ 2 λ + µ. It is easy to see that φ : λ (λ,µ is a bijectio. This proves that λ 2 λ + µ. Namely, the idetity (.8 holds. λ A (λ,µ B the electroic joural of combiatorics 8 (20, #P78 3

4 3 Ivolutive proof of (.9 It is easy to see that [ ] m+ ( /2 0 2 [ ] m+ 2 2 /2 ( 0 λ,µ m+ 2l(λ+l(µ λ m+ l(λ 2 λ µ m+ l(µ 2 µ ( l(λ 2 λ + µ, (3. ad /4 0 [ ] m+ 4 [ ] m+ ( µ D 4 λ + µ. (3.2 λ,µ m+ 4l(λ+l(µ Let U {(λ,µ P P: λ,µ m+ ad 2l(λ+l(µ }, V {(λ,µ U : each λ i appears a eve umber of times ad µ D}. We shall costruct a ivolutio θ o the set U \ V with the properties that θ preserves 2 λ + µ ad reverses the sig ( l(λ. For ay (λ,µ U \ V, otice that either some λ i appears a odd umber of times i λ, or some µ j is repeated i µ, or both are true. Choose the largest such λ i ad µ j if they exist, deoted by λ i0 ad µ j0 respectively. Defie { ((λ\λ i0,µ (λ i0,λ i0, if λ i0 µ j0 or µ D, θ((λ,µ ((λ µ j0,µ\(µ j0,µ j0, if λ i0 < µ j0 or λ i0 does ot exist. For example, if λ (5,5,4,4,4,3,3,3,, ad µ (5,3,2,2,, the θ(λ,µ ((5,5,4,4,3,3,3,,,(5,4,4,3,2,2,. It is easy to see that θ is a ivolutio o U \V with the desired properties. This proves that ( l(λ 2 λ + µ ( l(λ 2 λ + µ (λ,µ U (λ,µ V µ D τ,µ m+ 4l(τ+l(µ where λ τ τ. Combiig (3. (3.3, we complete the proof of (.9. 4 τ + µ, (3.3 the electroic joural of combiatorics 8 (20, #P78 4

5 4 Geeratig fuctio proof of (.8 ad (.9 Recall that the -shifted factorial is defied by (a; 0, (a; ( a,,2,... 0 The we have ( z; (z 2 ; 2 m+, m+ (z; m+ (4. ( z; (z 4 ; 4 m+. m+ (z; m+ ( z 2 ; 2 m+ (4.2 By the -biomial theorem (see, for example, [, Theorem 3.3], we may expad (4. ad (4.2 respectively as follows: ( [ ] ( m+ m+ [ ] m+ [ ] z 2 2 ( z m+ z, ( ( [ ] ( m+ m+ [ ] m+ z 4 2 ( z ( [ ] ( m+ [ ] m+ z ( z 2. ( Comparig the coefficiets of z i both sides of (4.3 ad (4.4, we obtai (.8 ad (.9 respectively. Fially, we give the followig special cases of (.8: /2 0 /2 0 [ ] + [ + + ] 2 2 [ [ ] [ ] + ( 2 2 2, ( ] ] 2 + ( 2 [ 2 2. (4.6 Whe, the idetities (4.5 ad (4.6 reduce to (. ad (.2 respectively. Acowledgmets. This wor was partially supported by the Fudametal Research Fuds for the Cetral Uiversities, Shaghai Risig-Star Program (#09QA40700, Shaghai Leadig Academic Disciplie Project (#B407, ad the Natioal Sciece Foudatio of Chia (# the electroic joural of combiatorics 8 (20, #P78 5

6 Refereces [] G. E. Adrews, The Theory of Partitios, Cambridge Uiversity Press, Cambridge, 998. [2] T. Masour ad Y. Su, Bell polyomials ad -geeralized Dyc paths, Discrete Appl. Math. 56 (2008, [3] Y. Su, A simple bijectio betwee biary trees ad colored terary trees, Electro. J. Combi. 7 (200, #N20. [4] S. H. F. Ya, Bijective proofs of idetities from colored biary trees, Electro. J. Combi. 5 (2008, #N20. the electroic joural of combiatorics 8 (20, #P78 6