= (q) M+N (q) M (q) N

Transcription

1 A OVERPARTITIO AALOGUE OF THE -BIOMIAL COEFFICIETS JEHAE DOUSSE AD BYUGCHA KIM Abstract We define an overpartition analogue of Gaussian polynomials (also known as -binomial coefficients) as a generating function for the number of overpartitions fitting inside the M rectangle We call these new polynomials over Gaussian polynomials or over -binomial coefficients We investigate basic properties and applications of over -binomial coefficients In particular, via the recurrences and combinatorial interpretations of over -binomial coefficients, we prove a Rogers-Ramauan type partition theorem 1 introduction Gaussian polynomials are defined by M + G(M, ) () M+ () M (), where (a) n (a; ) n : n k1 (1 ak 1 ) for n 0 { } These polynomials (also known as -binomial coefficients) have played many roles in combinatorics and number theory For example, Gaussian polynomials serve as generating functions for the number of inversions in permutations of a multi-set, the number of restricted partitions and the number of dimensional subspaces of M + dimensional vector spaces over F Our interest in this paper is to study an overpartition analogue of Gaussian polynomials Recall G(M, ) is the generating function for the number of partitions of n fitting inside an M rectangle, ie with largest part M and number of parts (For example, see [1]) In this light, we define our overpartition analogue of Gaussian polynomials, which we will call over -binomial coefficients, as the generating function for the number of overpartitions fitting inside an M rectangle An overpartition [9] is a partition in which the last occurrence of each distinct number Date: April 29, Mathematics Subect Classification 05A17, 11P81, 11P84 This research was supported by the International Research & Development Program of the ational Research Foundation of Korea (RF) funded by the Ministry of Education, Science and Technology(MEST) of Korea (RF-2014K1A3A1A ), and the STAR program number 32142ZM 1

2 2 JEHAE DOUSSE AD BYUGCHA KIM may be overlined For example, the 8 overpartitions of 3 are 3, 3, 2 + 1, 2 + 1, 2 + 1, 2 + 1, , Overpartitions have recently played an important role in the construction of weight 3/2 mock modular forms [5] and in the combinatorial proof of the 1 ψ 1 summation formula [17], and their arithmetic properties have been of great interest [4, 6, 14, 16] Our first result is an expression for over -binomial coefficients M+ Theorem 11 For positive integers M and, min{m,} M + k(k+1) () M+ k 2 () k () M k () k k0 Remarks 1 (i) The above expression can be rewritten by employing -trinomial coefficients a + b + c () a+b+c a, b, c () a () b () c (ii) We have an obvious symmetry M + M + M (iii) We will omit from the notation if it is clear from the context that the base is For example, from Theorem 11 we find that , 3 and we can check that there are 12 overpartitions of 5 fitting inside a 3 3 rectangle as follows 3 + 2, 3 + 2, 3 + 2, 3 + 2, , , , , , , Just as -binomial coefficients satisfy simple recurrences, which are the -analogues of Pascal s identity [M ] + M + 1 M M, 1 M + M + 1 M + 1 +, 1 over -binomial coefficients also satisfy similar recurrences Theorem 12 For positive integers M and, we have

3 A OVERPARTITIO AALOGUE OF THE -BIOMIAL COEFFICIETS 3 (i) M + M (ii) M + M + 1 M + 1 M (11) 1 M + 1 M M + M (12) 1 1 By employing over -binomial coefficients, we can establish various identities We discuss these applications in Section 3 Here, we highlight that over -binomial coefficients can be used to derive a Rogers-Ramanuan type theorem for overpartitions The first Rogers-Ramanuan identity is given by n0 n2 () n 1 (; 5 ) ( 4 ; 5 ) The left-hand side can be interpreted as the generating function for partitions with a gap 2 between two successive parts, and the right-hand side as the generating function for partitions into parts 1, 4 (mod 5) Rogers-Ramanuan identities have been proved via various methods Among them, one of the most elementary and beautiful is a proof by Andrews which uses recurrence relations of -binomial coefficients [2, 3] Motivated by this proof, we find a Rogers-Ramanuan type identity for overpartitions Before stating the result, we define three partition functions Let A(n) be the number of overpartitions λ λ l of n satisfying the following gap conditions { 1, if λ i is not overlined, λ i λ i+1 2, if λ i is overlined (If there are l parts in the overpartition, we define λ l+1 0 for convenience, thus 1 cannot be a part) We define B(n) as the number of overpartitions of n with nonoverlined parts 2 (mod 4) and C(n) as the number of partitions into parts 0 (mod 4), ie the number of 4-regular partitions of n Theorem 13 For all non-negative integers n, A(n) B(n) C(n) Remark 1 This is a special case of [12, Theorem 12], which is generalized by Chen, Sang, and Shi [8] While the previous results are obtained by employing Bailey chain machinery, we use the recurrence formulas for over -binomial coefficients The euality B(n) C(n) is clear from Euler s partition theorem (the number of partitions into odd parts euals the number of partitions into distinct parts), thus

4 4 JEHAE DOUSSE AD BYUGCHA KIM the important euality is A(n) B(n) Here we illustrate Theorem 13 for the case n 8 There are 16 overpartitions satisfying the gap conditions: 8, 8, 7 + 1, 7 + 1, 6 + 2, 6 + 2, 6 + 2, 6 + 2, 5 + 3, 5 + 3, 5 + 3, 5 + 3, , , , , and there are also 16 overpartitions satisfying the congruence conditions: 8, 7 + 1, 6 + 2, 6 + 2, 6 + 2, 6 + 2, 5 + 3, , , , , , , , , The rest of paper is organized as follows In Section 2, we prove Theorem 11 and the recurrence formulas of Theorem 12 In Section 3, we give several applications of over -binomial coefficients In Section 4, by using recurrence formulas we prove a Rogers-Ramauan type identity for overpartitions 2 Basic Properties of over -binomial coefficients We start with the proof of Theorem 11 Proof of Theorem 11 Let G(M,, k) be the generating function for overpartitions fitting inside an M rectangle and having exactly k overlined parts Such an overpartition can be decomposed as a partition into k distinct parts, each of which is at most M, and a partition fitting inside an M ( k) box By appending a partition fitting into an (M k) k box (generated by M ) to the right of the k staircase partition (k, k 1,, 1) (generated by k(k+1) 2 ), we see that k(k+1) M 2 k generates partitions into k distinct parts M As [ +M k k fitting inside M ( k) box, we see that G(M,, k) k(k+1) 2 [ M k ] + M k k k(k+1) 2 ] generates the partitions () M+ k () k () M k () k Since G(M,, k) is non-zero if and only if 0 k min{m, }, we have + M min{m,} k0 G(M,, k) ow we turn to proving the recurrences min{m,} k0 k(k+1) 2 () M+ k () k () M k () k

5 A OVERPARTITIO AALOGUE OF THE -BIOMIAL COEFFICIETS 5 Combinatorial proof of Theorem 12 Let O(M,, n) denote the number of overpartitions of n fitting inside an M rectangle ote that O(M,, n) O(M, 1, n) is the number of overpartitions of n fitting inside an M rectangle having exactly parts Let λ be such an overpartition If the smallest part of λ is 1, then by removing 1 from every part we obtain an overpartition of n fitting inside (M 1) ( 1) rectangle If the smallest part of λ is different from 1, by removing 1 from every part we arrive at an overpartition of n fitting inside an (M 1) rectangle Therefore, we find that O(M,, n) O(M, 1, n) O(M 1, 1, n ) + O(M 1,, n ) By rewriting the above identity in terms of generating functions we obtain the first recurrence The second recurrence follows from a similar argument by tracking the size of the maximum part instead of the number of parts ote that O(M,, n) O(M 1,, n) is the number of overpartitions of n fitting inside an M rectangle with largest part eual to M If the largest part is overlined, then by removing it we obtain an overpartition of n M fitting inside a (M 1) ( 1) rectangle If the largest part is not overlined, then by removing it we obtain an overpartition of n M fitting inside a M ( 1) rectangle Therefore, we find that O(M,, n) O(M 1,, n) O(M 1, 1, n M) + O(M, 1, n M) By rewriting the above identity in terms of generating functions we obtain the second recurrence Analytic Proof of Theorem 12 We first note that -trinomial coefficients satisfy the following recurrence a + b + c a + b + c 1 a + b + c 1 a + b + c 1 + a + a+b a, b, c a 1, b, c a, b 1, c a, b, c 1 Therefore, we find that min{m,} M + ( M + k 1 k(k+1)/2 k 1, k, M k k0 ) M + k 1 + k, k, M k 1 min{m, 1} M + k 1 k(k+1)/2 k, M k, 1 + k0 min{m 1, 1} k0 [ ] k(k+1)/2+ M + k 2 k, M 1 k, 1 k M + k 1 + k k, k 1, M k

6 6 JEHAE DOUSSE AD BYUGCHA KIM min{m 1,} M + k 1 + k(k+1)/2+ k, M 1 k, 1 k0 M + 1 M + 2 M , 1 1 where we have made a change of variable k k + 1 in the second sum The second recurrence can be proved similarly Throughout the paper, we use the following asymptotic behaviour freuently Proposition 21 For a non-negative integer, lim ( ) () Proof When goes to the infinity, the restriction on the number of parts disappears Proposition 21 is useful to obtain new identities For example, by taking a limit in Theorem 11, we find that k(k+1)/2 ( ), () k () k () k0 which gives an alternative generating function for overpartitions into parts This identity is also a special case of the finite -binomial theorem [11, Exer 12(vi)] 3 Applications By tracking the number of parts in the overpartitions, we prove the following identity Proposition 31 For a positive integer, ( z) (z) 1 + k 1 ( [ ] + k 1 z k k k + ) + k 2 k 1 Proof Let p (n, k) be the number of overpartitions of n into parts with k parts Then, it is not hard to see that ( z) p (z) (n, k)z k n n 0 k 0 Let λ be an overpartition counted by p (n, k) Discussing whether the smallest part of λ is eual to 1 and removing 1 from each part as in the proof of Theorem 12, we have p (n, k) O( 1, k, n k) + O( 1, k 1, n k)

7 A OVERPARTITIO AALOGUE OF THE -BIOMIAL COEFFICIETS 7 Thus n 0 The claimed identity follows ( [ ] + k 1 p (n, k) n k + k ) + k 2 k 1 By taking the limit as in the above proposition, we find the following generating function Corollary 32 Let p(n, k) be the number of overpartitions of n with k parts Then, n 0 k 0 p(n, k)z k n ( z) (z) k 1 z k k ( ) k 1 () k Remark 2 The above identity is a special case of the -binomial theorem [11, (II3)] ote that k 1 k ( ) k 1 () k k 1 k k 1 n 1 τ(n) n (mod 2), where τ(n) is the number of divisors of n This recovers a well known congruence Corollary 33 For all non-negative integers n, p(n) 2τ(n) (mod 4) Our next application is finding an analogue of Sylvester s identity [15]: ( x; ) 1 x 3( 1)/2 1 + ( x; ) 1 x +1 3(+1)/2 1 We define S(; x; ) as S(; x, ) : Then we have the following identity Theorem 34 For a positive integer, ( [ ] ) 1 1 x x (x) 1 2 (x) S(; x; ) (x)

8 8 JEHAE DOUSSE AD BYUGCHA KIM Proof Let us consider an overpartition into parts, generated by (x) The variable counts the size of the Durfee suare of the overpartition The Durfee suare is generated x 2 Then either the corner at the bottom right of the Durfee suare is overlined or it is not If it is overlined, then we have an overpartition generated by [ 1 1 ] at the right of the Durfee suare, and an overpartition generated by 1 (x) 1 under it If it is not overlined, then we have an overpartition generated by to the right of the Durfee suare, and an overpartition generated by (x) two cases correspond to the two sums in S(; x; ) under it These By taking a limit and using Proposition 21, we obtain the following identity Corollary 35 We have (x) ( ( ) 1 () 1 1 x 2 + ( ) (x) 1 () ) x 2 (x) In particular, by setting x 1 we obtain a well known theta function identity () n n2 ( ) n Z As another application, we obtain the overpartition rank generating function To explain Ramanuan s famous three partition congruences, Dyson [10] introduced the rank for the partition as the difference between the size of the largest part and the number of parts For an overpartition, we can define a rank in the same way [7] Let (m, n) be the number of overpartitions of n with rank m Then, we can express the generating function in terms of over -binomial coefficients Theorem 36 For a non-negative integer m, m () : n 0 (m, n) n 2 1+m + k 2 2k + m 3 + k 2 ( [2k ] + m 2 2k+m 1 + k 1 ) 2k + m 4 k 2 2k + m 3 + k 1 Proof If there is only one part in the overpartition, m + 1 and m + 1 are the only two such overpartitions with rank m, which corresponds to 2 m+1 ow we assume that an overpartition has at least two parts and the rank of the overparition is m

9 A OVERPARTITIO AALOGUE OF THE -BIOMIAL COEFFICIETS 9 Under this assumption, the largest part would be m + k and the number of part is k, this corresponds to m+2k 1 inside the summation ow the first sum counts the case where the largest part is not overlined and there is no 1 The second sum counts the case where the largest part is overlined and there is no 1 The third sum counts the case where the largest part is not overlined and the smallest part is 1 The last sum corresponds to the case where the largest part is overlined and the smallest part is 1 By comparing the known generating function for m () [13, Proposition 32] m () 2 ( ) () we derive the following identity n 1 Corollary 37 For a non-negative integer m, 2 ( ) n 1 n2 + m n (1 n ) () 1 + n n 1 n 1 n2 + m n (1 n ) 1 + n, ( [2k ] + m 2 2k+m 1 + k m + k 2 2k + m k 2 2k + m 3 k 1 ) 2k + m 4 k 2 4 Proof of a Rogers-Ramanuan type identity We first define two functions D(, x; ) : [ 0 ] x (+1)/2 and C(, x; ) : 0 [ ] (x) x 2 ( (2+1) x (+1)(2+1)) The following observation is the key for obtaining a Rogers-Ramanuan type identity Theorem 41 For a positive integer, (x) D(, x; ) C(, x; ) x 2 +3 Z[[x, ]] By taking the limit as, we obtain the following corollary

10 10 JEHAE DOUSSE AD BYUGCHA KIM Corollary 42 We have (x) D(, x; ) C(, x; ) In particular, the case x 1 is a Rogers-Ramanuan type identity, where we applied Lemma 21 to evaluate the limit Corollary 43 We have () D(, 1; ) n n(2n+1) (, 3, 4 ; 4 ) ( ) n Z Proof of Theorem 13 After multiplying ( ) () we obtain that k0 k(k+1)/2 ( ) k () k ( ) ( 2 ; 4 ) to both sides and from the definitions, 1 (, 2, 3 ; 4 ) A basic partition theoretic interpretation of the above identity gives the desired result ow we turn to proving Theorem 41 Let g(, x) : (x) D(, x) The key idea of the proof is that g(, x) and C(, x) satisfy the same recurrence (up to a high power of times a polynomial in x and ) as follows Lemma 44 Lemma 45 g(, x) (1 x)g( 1, x) + (x) 2 2 x 2 g( 2, x 2 ) C(, x) (1 x)c( 1, x) (x) 2 2 x 2 C( 2, x 2 ) x 2 +3 Z[[x, ]] (41) Theorem 41 follows immediately from these two recurrences and an induction over We now need to prove these lemmas Proof of Lemma 44 By applying the first recurrence in Theorem 12, we find that D(, x) ( [ ] ) x 1 (+1)/2 0 1 x +1 (+1)(+2)/2 + D( 1, x) + 2 x +1 (+1)(+4)/2 0 0 (1 + x)d( 1, x) + x 2 D( 2, x 2 ), where we replace 1 by in the first and the third sum for the second identity After multiplying by (x) we get the desired recurrence

11 A OVERPARTITIO AALOGUE OF THE -BIOMIAL COEFFICIETS 11 Proof of Lemma 45 We calculate each term in (41) By the definition of C, we find that (1 x)c( 1, x) (1+x) 1 (x)+1 ( ) x x Expanding and making the change of variable 1 in the fourth sum, we get (1 x)c( 1, x) 1 (x)+1 x (x) x (x)+1 x (42) (x) 1 x (x) By the change of variable 1 in the definition of 2 2 x 2 C( 2, x 2 ), we obtain (x) 2 x 2 C( 2, x 2 ) 2 (x) x (43) 2 (x)+1 1 x Using the first recurrence (11) on the first sum in (42) and the second sum in (43) and extracting the term 0 in the first and third sums of (42) leads to (1 x)c( 1, x) + (x) 2 x 2 C( 2, x 2 ) 2 1 x + (x)+1 x (x) x (x)+1 +1 x (44)

12 12 JEHAE DOUSSE AD BYUGCHA KIM (x)+1 x (x) 1 x (x) x ow we want to write both C(, x) and (1 x)c( 1, x)+ (x) 2 2 x 2 C( 2, x 2 ) as sums involving the product to be able to make cancellations We have (x) +1 C(, x) (x) ( ) 1 + x +1 x (x) ( ) 1 + x x Extracting the terms 0 of each sum and expanding, we get C(, x) 1 x + (x) x (x) x (x) +1 x (x) +1 x (45) Rewriting all the sums in (44) except the third one in terms of (x) +1 leads to (1 x)c( 1, x) + (x) 2 x 2 C( 2, x 2 ) (46) 2 1 x + (x) x (x) +1 x

13 A OVERPARTITIO AALOGUE OF THE -BIOMIAL COEFFICIETS (x) 1 +1 x (x) 1 x (x)+1 +1 x (47) (x) x (x) +1 x (x) 1 x (x) 1 x (x) 1 +1 x (x) 1 x Subtracting (45) from (47) and noting that the third sum of (45) cancels with the second sum of (47) we obtain C(, x) (1 x)c( 1, x) (x) 2 x 2 C( 2, x 2 ) 2 (x) x (x) x (x) +1 x (48) (49) (410)

14 14 JEHAE DOUSSE AD BYUGCHA KIM (x) +1 x (x) 1 x (x) 1 +1 x (x)+1 x (x) +1 x (x) x (x) 1 +1 x (x) 1 +1 x (x) 1 x (411) (412) (413) (414) (415) (416) (417) (418) (419) (x) 1 +1 x (420) +1 By the second recurrence (12), we observe that the sum (413) is eual to 2 (x) 1 +1 x O ( x 3 +5), (421) +1 where we define f(x, ) O(x k l ) to mean that f(x, ) x k l Z[[x, ]] Thus by the first recurrence (11), the sum of (49), (413), (415) and (418) is eual to O ( x 3 +5) Furthermore, by the second recurrence (12), the sum of (410) and (416) is O ( x 4 +7) Finally again by the second recurrence (12), the sum (411) is eual to 1 (x) +1 x O ( x 2 +3) +1 1

15 A OVERPARTITIO AALOGUE OF THE -BIOMIAL COEFFICIETS 15 Thus by the first recurrence (11), the sum of (48), (411), (417) and (420) is eual to O ( x 2 +3) Hence we are left with the following C(, x) (1 x)c( 1, x) (x) 2 x 2 C( 2, x 2 ) 2 1 (x) 1 x (x)+1 x (x) 1 x O ( x 2 +3) By the second recurrence (12), the third sum is eual to 1 (x) 1 x O ( x 3 +7) +1 1 Factorising it with the first sum we get C(, x) (1 x)c( 1, x) (x) 2 x 2 C( 2, x 2 ) 2 1 (x)+1 x (x) 1 x O ( x 2 +3) ow by a simple change of variable 1 we see that the two sums are cancelled, and this completes the proof 5 Concluding Remarks In this paper, we introduced a polynomial version of the overpartition generating function and its applications in the theory The main theme is emphasizing combinatorial motivations and roles of recurrence formulas to derive results In this sense, the finite form of the identities are the main obects of this paper The limiting version of the identities can also be proven using well-known transformation formulas in the theory of basic hypergeometric series We can recover Corollary 35 by setting a x,

16 16 JEHAE DOUSSE AD BYUGCHA KIM b, and c, d in the very-well-poised 6 φ 5 summation [11, (II20)] Corollary 42 can also be proven via employing 8 φ 7 summation and Heine s transformation By setting a x, b, and c, d, e, f in 8 φ 7 summation [11, (III23)], we find that C(, x; ) (x) lim e,f 2 φ 1 (e, f; x;, x 2 /ef) By employing Heine s transformation [11, (III2)], we can derive that the limit in the above euation is the same as 1 D(, x; ) Acknowledgement The authors thank Jeremy Loveoy for the valuable discussions and comments at every stage of this paper References [1] G E Andrews, The Theory of Partitions, Addison Wesley, Reading, MA, 1976; reissued: Cambridge University Press, Cambridge, 1998 [2] G E Andrews, J J Sylvester, Johns Hopkins and partitions, in A Century of Mathematics in America, Part I, P Duren, ed, American Mathematical Society, Providence, RI, 1988, [3] G E Andrews, On the proofs of the Rogers Ramanuan identities, in -Series and Partitions, D Stanton, ed, Springer-Verlag, ew York, 1989, 1âĂŞ14 [4] GE Andrews, SH Chan, B Kim, R Osburn, The first positive rank and crank moments for overpartitions, preprint [5] K Bringmann, J Loveoy, Overpartitions and class numbers of binary uadratic forms, Proc atl Acad Sci USA 106 (2009), no 14, [6] K Bringmann, J Loveoy, Rank and congruences for overpartition pairs, Int J umber Theory 4 (2008), [7] K Bringmann, J Loveoy, Dyson s rank, overpartitions, and weak Maass forms Int Math Res ot (2007), rnm063 [8] WYC Chen, DDM Sang, and DYH Shi, The Rogers-Ramanuan-Gordon theorem for overpartitions, Proc London Math Soc 106 (2013) [9] S Corteel, J Loveoy Overpartitions, Trans Amer Math Soc 356 (2004), no 4, [10] FJ Dyson, Some guesses in the theory of partitions, Eureka 8 (1944), [11] G Gasper and M Rahman, Basic Hypergeometric Series, 2nd Edition, Cambridge Univ Press, Cambridge, 2004 [12] J Loveoy, Gordon s theorem for overpartitions, J Combin Theory Ser A 103 (2003), [13] J Loveoy, Rank and conugation for the Frobenius representation of an overpartition, Ann Combin 9 (2005) [14] K Mahlburg, The overpartition function modulo small powers of 2, Discrete Mathematics 286 (2004), [15] J J Sylvester, A constructive theory of partitions, arranged in three acts, an interact, and an exodion, in The Collected Papers of J J Sylvester, Vol 3, Cambridge University Press, London, 1 83; reprinted by Chelsea, ew York, 1973

17 A OVERPARTITIO AALOGUE OF THE -BIOMIAL COEFFICIETS 17 [16] S Treneer, Congruences for the coefficients of weakly holomorphic modular forms, Proc London Math Soc (3) 93 (2006), [17] AJ Yee, Combinatorial proofs of Ramanuan s 1 ψ 1 summation and the -Gauss summation, J Combin Theory Ser A, 105 (2004), LIAFA, Universite Paris Diderot - Paris 7, Paris Cedex 13, FRACE address: ehannedousse@liafauniv-paris-diderotfr School of Liberal Arts, Seoul ational University of Science and Technology, 232 Gongreung-ro, owon-gu, Seoul, , Korea address: bkim4@seoultechackr