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On weighted signed color partitions V GUPTA, M RANA∗ and S SHARMA School of Mathematics, Thapar Institute of Engineering and Technology, Patiala 147 004, India *Corresponding author. E-mail: [email protected]; [email protected]; [email protected]

MS received 30 May 2019; revised 22 August 2019; accepted 8 September 2019 Abstract. In this paper, we provide combinatorial interpretations of certain proved Rogers–Ramanujan type identities using signed color partitions with attached weights. The approach of using the signed color partitions is interesting since negative exponents do not make an explicit appearance in these identities. Keywords. (n + t)-color partitions; signed partitions; combinatorial interpretation; Rogers–Ramanujan type identities. 2010 Mathematics Subject Classification.

05A17, 05A19, 11P84.

1. Introduction Euler included a result in the chapter titled “De Partitio Numerorum”, given in [5], that every positive integer is uniquely represented as the sum or difference of distinct power of 3. He wrote this in terms of the generating function ∞ 

xν =

ν=−∞

∞ 

(x −3 + x 3 + 1), n

n

(1.1)

ν=0

which converges for no values of x. But in [2], Andrews treated (1.1) as an identity in formal Laurent series. He showed great interest in this section and asked the question: Why have we thought so little about partition generating functions in which some of the partitions might have some negative parts? He also explored Euler’s eye-catching identity (1.1) and found some new and appealing results. He called such partitions as ‘signed partitions’ in which parts may appear with + or − sign. DEFINITION 1.1 In [8], a signed partition  of an integer ν, denoted by   ν, is a partition pair (θ1 , θ2 ), where ν = θ1 + θ2 , © Indian Academy of Sciences 0123456789().: V,-vol

10

Page 2 of 10

Proc. Indian Acad. Sci. (Math. Sci.)

(2020) 130:10 l(θ1 )

θ1 (resp. θ2 ) is the positive (resp. negative) subpartition of  and θ11 , θ12 , . . . , θ1 l(θ ) θ21 , θ22 , . . . , θ2 2 ) are the positive (resp. negative) parts of .

(resp.

Remark 1. The unrestricted signed partitions of an integer are infinitely many. We are particularly interested in finite sets of signed partitions and this can be done by imposing some suitable restrictions on parts. Example 1.1. Let θ1 = 6 + 3 + 3 and θ2 = −3 − 2 − 1. Then  = (6 + 3 + 3, −3 − 2 − 1) is a signed partition of 6. Signed partitions fit naturally while interpreting many classical q-series identities. For instance, refer [2] for interpretations of Göllnitz–Gordon identities using signed partition. Further in [9], Sills provided a bijection between the ordinary and signed partitions for the Göllnitz–Gordon identity. In [7], Keith explored four combinatorial theorems by presenting bijections between restricted signed partitions and ordinary partitions. He also studied the behavior of signed partitions of zero in arithmetic progression. In [8], McLaughlin and Sills provided interpretations of Rogers–Ramanujan type identities, which belonged to the family