On a certain vector crank modulo 7

Transcription

1 On a certain vector crank modulo 7 Michael D Hirschhorn School of Mathematics and Statistics University of New South Wales Sydney, NSW, 2052, Australia mhirschhorn@unsweduau Pee Choon Toh Mathematics & Mathematics Education, National Institute of Education, Nanyang Technological University Nanyang Walk, Singapore peechoontoh@nieedusg January 8, 205 Abstract We define a vector crank to provide a combinatorial interpretation for a certain Ramanujan type congruence modulo 7 Keywords: partitions; congruences; crank Introduction In [7], one of the authors established several new Ramanujan type identities and congruences modulo 3, 5 and 7 for certain types of partition functions For example, define Q po,p (n) as the number of partitions of n into two colors, where the red colored parts form a partition into odd parts and the blue Supported by the NIE Academic Research Fund RI 3/2 TPC

2 colored parts form an overpartition Using the standard notation n (a; q) n ( aq j ), j0 (a; q) lim n (a; q) n, (a,, a m ; q) (a ; q) (a m ; q), for q < and a, a,, a m 0, we can write the generating function of Q po,p (n) as n0 Toh [7] proved that Q po,p (n)q n (q; q 2 ) ( q; q) (q; q) ( q, q; q) (q; q) Q po,p (7n + 2)q n 0 (mod 7) () n0 Zhou [9] subsequently provided alternative proofs of all of the congruences in [7] with the exception of () She re-interpreted these partition functions as partitions into multi-colors, introduced what she termed as multiranks which are essentially vector cranks as defined by Garvan [4] and proved that these vector cranks divided the partitions into equinumerous parts The aim of this article is to define a vector crank that will explain () combinatorially 2 A vector crank If λ is a partition, we define σ(λ) and n(λ) as the sum of the parts and the number of parts of λ respectively Let D, O, P denote the sets of partitions into distinct parts, partitions into odd parts, and unrestricted partitions respectively Define the cartesian product V D D O O P P For a vector partition λ (λ, λ 2, λ 3, λ 4, λ 5, λ 6 ) V define a sum of parts s, a weight w and a crank r by s( λ) 2σ(λ ) + σ(λ 2 ) + σ(λ 3 ) + σ(λ 4 ) + 2σ(λ 5 ) + 2σ(λ 6 ), (2a) w( λ) ( ) n(λ ), (2b) r( λ) 2n(λ 3 ) 2n(λ 4 ) + n(λ 5 ) n(λ 6 ) (2c) 2

3 The weighted count of vector partitions of n with crank m, denoted by N V (m, n), is given by N V (m, n) w( λ) (3) λ V s( λ)n r( λ)m We also define the weighted count of vector partitions of n with crank congruent to k modulo t by N V (k, t, n) m N V (mt + k, n) λ V s( λ)n r( λ) k (mod t) Finally, we have the following generating function for N V (m, n), m n0 N V (m, n)z m q n w( λ) (4) (q 2 ; q 2 ) ( q; q) (z 2 q; q 2 ) (z 2 q; q 2 ) (zq 2 ; q 2 ) (z q 2 ; q 2 ) Theorem The following equation holds for all nonnegative integers n N V (0, 7, 7n + 2) N V (, 7, 7n + 2) N V (6, 7, 7n + 2) Q po,p(7n + 2) 7 The main ingredient in the proof of the theorem is Winquist s identity [8], which is a variant of the B 2 case of the Macdonald identities [5] We state the identity in the following symmetric form [6, Eq (3)] If we define we have F (x) F 2 (x) ( ) j q 3j2 (x 3j + x 3j ), j k F (x)f 2 (y) F (y)f 2 (x) 2 x ( ) k q 3k2 +2k (x 3k+ + x 3k ), (xq, (5) (6a) (6b) qx qy q2, yq,, xy, xy, x ) y, yq2 x, q2, q 2 ; q 2 (6c) 3

4 Proof of Theorem If we set ζ exp(2πi/7) in (5), we obtain 6 t0 ζ t n0 m n0 N V (t, 7, n)q n N V (m, n)ζ m q n (q 2 ; q 2 ) (q, ζ 2 q, q/ζ 2, ζq 2, q 2 /ζ; q 2 ) (ζq, q/ζ, ζ3 q, q/ζ 3 ; q 2 ) (q 7 ; q 4 ) (q2, q 2, ζ 2 q 2, q 2 /ζ 2, ζ 3 q 2, q 2 /ζ 3 ; q 2 ) (q 4 ; q 4 ) F (ζ 3 )F 2 (ζ) F (ζ)f 2 (ζ 3 ) 2ζ( ζ 2 )( ζ 3 )(q 7 ; q 7 ), where we used (6) with x ζ 3 and y ζ Since 3j 2 0, 3, 5, 6 (mod 7) and 3k 2 +2k 0,, 2, 5 (mod 7), the power of q in q 3j2 +3k 2 +2k is congruent to 2 modulo 7 exactly when j 0 (mod 7) and k 2 (mod 7) This means that the coefficient of q 7n+2 in is zero since F (ζ 3 )F 2 (ζ) F (ζ)f 2 (ζ 3 ) ( ) j+k (ζ 9j + ζ 9j )(ζ 3k+ + ζ 3k ) ( ) j+k (ζ 3j + ζ 3j )(ζ 9k+3 + ζ 9k 3 ) 0 when j 0 (mod 7) and k 2 (mod 7) Thus 6 N V (t, 7, 7n + 2)ζ t 0 (7) t0 Since the minimal polynomial for ζ over the rational numbers is we conclude that p(x) + x + x x 6, N V (0, 7, 7n + 2) N V (, 7, 7n + 2) N V (6, 7, 7n + 2) 4

5 We end by indicating how one may prove () directly as the details were omitted in [7] This can be done by observing that n0 Q po,p (n)q n (q2 ; q 2 ) 2 (q2 ; q 2 ) 9 (q 4 ; q 4 ) (mod 7) (8) Thus () is equivalent to proving the coefficients of q 7n+2 in (q 2 ; q 2 ) 9 are all divisible by 7 We offer three alternative ways of doing this The easiest way is to appeal directly to [3, Th 2] Alternatively, we can use one of the Macdonald identities associated with the C 2 root system [5, p 37] or [6, Eq 32], to express (q 2 ; q 2 ) 9 α (mod 8) β 3 (mod 8) 8 (β2 α 2 )q α 2 +β If the exponent of q is congruent to 2 modulo 7, we have α 2 + β 2 6(2) (mod 7) Since is a quadratic nonresidue modulo 7, 7 must divide both α and β The third way is to apply the Hecke operator T 7 to η(6τ)9, a weight 3 cusp η(8τ) 3 form of level 28 One can refer to [] for examples of how this may be done Acknowledgements We thank the anonymous referee for helpful comments that improved the article In particular, the referee pointed out that the coefficients N V (m, n) are nonnegative which can be proved using the following corollary [2, Eq (4)] of the q-binomial theorem Proposition 2 If q, t < then (at; q) (a; q) (t; q) (a; q) + 5 n t n (aq n ; q) (q; q) n

6 References [] S Ahlgren Multiplicative relations in powers of Euler s product J Number Theory, 89: , 200 [2] A Berkovich, and F G Garvan K Saito s Conjecture for nonnegative eta products and analogous results for other infinite products J Number Theory, 28:73 748, 2008 [3] S Cooper, M D Hirschhorn, and R Lewis Powers of Euler s product and related identities Ramanujan J, 4:37 55, 2000 [4] F G Garvan New combinatorial interpretations of Ramanujan s partition congruences mod 5, 7 and Trans Amer Math Soc, 305:47 77, 988 [5] I G Macdonald Affine root systems and Dedekind s η-function Invent Math, 5:9 43, 972 [6] P C Toh Generalized mth order Jacobi theta functions and the Macdonald identities Int J Number Theory, 4:46 474, 2008 [7] P C Toh Ramanujan type identities and congruences for partition pairs Discrete Math, 32: , 202 [8] L Winquist An elementary proof of p(m + 6) 0 (mod ) J Combin Theory, 6:56 59, 969 [9] R R Zhou Multiranks for partitions into multi-colors Electron J Combin, 9:#P8, 202 6