Infinitely Many Congruences for k -Regular Partitions with Designated Summands
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Infinitely Many Congruences for k-Regular Partitions with Designated Summands Robson da Silva1
· James A. Sellers2
Received: 2 November 2018 / Accepted: 13 June 2019 © Sociedade Brasileira de Matemática 2019
Abstract Andrews et al. (Acta Arith. 105:51–66, 2002) introduced and studied the partition function P D(n), the number of partitions of n with designated summands. Recently, congruences involving the number of -regular partitions with designated summands, denoted by P D (n), have been explored for specific fixed values of . In this paper, we provide several families containing infinitely many congruences for P Dk (n) for various values of k. Keywords Regular partition · Designated summands · Congruence · Generating function Mathematics Subject Classification 11P83 · 05A17
1 Introduction A partition of an integer n ≥ 0 is a non-increasing sequence of positive integers, λ1 ≥ · · · ≥ λs , such that n = λ1 + · · · + λs . The λi s are called the parts of the partition. Andrews et al. (2002) introduced and studied many properties of a new class of objects called partitions with designated summands. These partitions are constructed by taking ordinary partitions and tagging exactly one of each part size. For instance, the ten partitions of 4 with designated summands are:
B
Robson da Silva [email protected] James A. Sellers [email protected]
1
Universidade Federal de São Paulo-UNIFESP, Av. Cesare M. G. Lattes, 1201, São José dos Campos, SP 12247-014, Brazil
2
Department of Mathematics, Penn State University, 104 McAllister Building, University Park, PA 16802, USA
123
R. da Silva, J. A. Sellers
4 , 3 + 1 , 2 + 2, 2 + 2 , 2 + 1 + 1, 2 + 1 + 1 , 1 + 1 + 1 + 1, 1 + 1 + 1 + 1, 1 + 1 + 1 + 1, 1 + 1 + 1 + 1 . The arithmetic aspects of the number of partitions with designated summands have been studied in Andrews et al. (2002), Baruah and Ojah (2015), Chen et al. (2013), Hemanthkumar et al. (2017) and Xia (2016). Let S be a set of positive integers. By Theorem 1 in Andrews et al. (2002), we know that the generating function for the number of partitions with designated summands whose parts belong to S, P D S (n), is given by n∈S
(1 − q 6n ) . (1 − q n )(1 − q 2n )(1 − q 3n )
Thus, by taking Sk as the set of positive integers not divisible by k, we deduce that the generating function for the number of partitions with designated summands whose parts belong to Sk , denoted by P Dk (n), is ∞
P Dk (n)q n =
n=0
n∈Sk
=
(1 − q 6n ) (1 − q n )(1 − q 2n )(1 − q 3n )
∞ k−1 n=0 j=1
=
(1 − q 6kn+6 j ) (1 − q kn+ j )(1 − q 2kn+2 j )(1 − q 3kn+3 j )
(q k ; q k )∞ (q 2k ; q 2k )∞ (q 3k ; q 3k )∞ (q 6 ; q 6 )∞ , (q; q)∞ (q 2 ; q 2 )∞ (q 3 ; q 3 )∞ (q 6k ; q 6k )∞
(1)
where we use the following standard q-series notation: (a; q)0 = 1, (a; q)n = (1 − a)(1 − aq) · · · (1 − aq n−1 ), ∀n ≥ 1, (a; q)∞ = limn→∞ (a; q)n , |q| < 1. In Andrews et al. (2002), the authors also presented some arithmetic properties of P D2 (n), the number of 2-regular partitions of n with designated summands (a partiti
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