CHIRANJIT RAY AND RUPAM BARMAN

ON ANDREWS INTEGER PARTITIONS WITH EVEN PARTS BELOW ODD PARTS arxiv:1812.08702v1 [math.nt] 20 Dec 2018 CHIRANJIT RAY AND RUPAM BARMAN Abstract. Recently, Andrews defined a partition function EOn) which counts the number of partitions of n in which every even part is less than each odd part. He also defined a partition function EOn) which counts the number of partitions of n enumerated by EOn) in which only the largest even part appears an odd number of times. Andrews proposed to undertake a more extensive investigation of the properties of EOn). In this article, we prove infinite families of congruences for EOn). We next study parity properties of EOn). We prove that there are infinitely many integers N in every arithmetic progression for which EON) is even; and that there are infinitely many integers M in every arithmetic progression for which EOM) is odd so long as there is at least one. Very recently, Uncu has treated a different subset of the partitions enumerated by EOn). We prove that Uncu s partition function is divisible by 2 k for almost all k. We use arithmetic properties of modular forms and Hecke eigenforms to prove our results. 1. Introduction and statement of results A partition of a nonnegative integer n is a nonincreasing sequence of positive integers whose sum is n. In a recent paper, Andrews [1] studied the partition function EOn) which counts the number of partitions of n where every even part is less than each odd part. He denoted by EOn), the number of partitions counted by EOn) in which only the largest even part appears an odd number of times. For example, EO8) = 12 with the relevant partitions being 8,6+2,7+1,4+4,4+2+ 2,5+,5+1+1+1,2+2+2+2,++2,++1+1,+1+1+1+1+1,1+ 1+1+1+1+1+1+1+1; and EO8) = 5, with the relevant partitions being 8,4+2+2,++2,++1+1,1+1+1+1+1+1+1+1. Andrews proved that the partition function EOn) has the following generating function [1, Eqn..2)]: 1.1) EOn)q n = q4 ;q 4 ) q 2 ;q 4 ) 2 = q4 ;q 4 ) q 2 ;q 2 ) 2, where a;q) := n 0 1 aqn ). In the same paper, he proposed to undertake a more extensive investigation of the properties of EOn). The objective of this paper is to study divisibility properties of EOn). To be specific, we use the theory of Hecke eigenforms to establish the following infinite family of congruences for EOn) modulo 2. Date: December 20, 2018. Key words and phrases. Partitions, congruences, modular forms. The first author acknowledges the financial support of Department of Atomic Energy, Government of India for supporting a part of this work under NBHM Fellowship. 1

2 CHIRANJIT RAY AND RUPAM BARMAN Theorem 1.1. Let k,n be nonnegative integers. For each i with 1 i k +1, if p i 5 is prime such that p i 2 mod ), then for any integer j 0 mod p k+1 ) EO p 2 1...p2 k+1 n+ p2 1...p2 k p ) k+1j +p k+1 ) 1 0 mod 2). Let p 5 be a prime such that p 2 mod ). By taking all the primes p 1,p 2,...,p k+1 to be equal to the same prime p in Theorem 1.1, we obtain the following infinite family of congruences for EOn): ) EO p 2k+1) n+p 2k+1 j + p2k+1) 1 0 mod 2), where j 0 mod p). In particular, for all n 0 and j 0 mod 5), we have 1.2) EO25n+5j +8) 0 mod 2). In [1], Andrews proved that, for all n 0 EO10n+8) 0 mod 5). In this article, we prove that the congruence 1.2) is also true modulo 4 if n 0 mod 5). To be specific, we prove the following result. Theorem 1.2. Let t {1,2,,4}. Then for all n 0 we have EO105n+t)+8) 0 mod 20). We note that Theorem 1.2 is not true if t = 0. For example, EO8) is not divisible by 4. For a nonnegative integer n, let pn) denote the number of partitions of n. In [10], Ono proved that there are infinitely many integers N in every arithmetic progression for which pn) is even; and that there are infinitely many integers M in every arithmetic progression for which pm) is odd so long as there is at least one. Ono s result gave affirmative answer to a well-known conjecture on parity of pn) in arithmetic progression. In this article, we prove that the partition function EOn) has this property as well. We note that EO2n+1) = 0 for all n 0. In the following theorem, we prove the parity of EO2n) in any arithmetic progression. Theorem 1.. For any arithmetic progression r mod t), there are infinitely many integers N r mod t) for which EO2N) is even. Also, for any arithmetic progression r mod t), there are infinitely many integers M r mod t) for which EO2M) is odd, provided there is one such M. Furthermore, if there does exist an M r mod t) for which EO2M) is odd, then the smallest such M is less than 2 9+j 7 t 6 1 1p ) 2 2 j, d 2 p 6t where d = gcd12r 1,t) and 2 j > t 12. We next prove that EO8n + 6) is almost always divisible by 8 from which it readily follows that the same is true for EOn). Theorem 1.4. Let n 0. Then EO8n+6) is almost always divisible by 8, namely, # { n X : EO8n+6) 0 mod 8) } lim = 1. X X

ON ANDREWS INTEGER PARTITIONS WITH EVEN PARTS BELOW ODD PARTS Recently, Uncu [14] has treated a different subset of the partitions enumerated by EOn). Alsosee [1, p. 45]. We denote by EO u n) the partition function defined by Uncu, and the generating function is given by EO u n)q n 1 1.) = q 2 ;q 4 ) 2. For any fixed positive integer k, Gordon and Ono [4] proved that the number of partitions of n into distinct parts is divisible by 2 k for almost all n. Bringmann and Lovejoy [] showed that this is also true for the number of overpartition pairs, and Lin [7] did the same for the number of overpartition pairs into odd parts. In [1], we proved that the number of overcubic partition pairs and overcubic partition have this property. In this article, we prove that the partition function EO u n) has this property as well. Theorem 1.5. Let k be a positive integer. Then EO u 2n) is almost always divisible by 2 k, namely, # { n X : EO u 2n) 0 mod 2 k ) } lim = 1. X X 2. Preliminaries In this section, we recall some definitions and basic facts on modular forms. For more details, see for example [9, 6]. We first define the matrix groups {[ ] } a b Γ := : a,b,c,d Z,ad bc = 1, c d {[ ] } 1 n Γ := : n Z, 0 1 {[ ] } a b Γ 0 N) := Γ : c 0 mod N), c d {[ ] } a b Γ 1 N) := Γ c d 0 N) : a d 1 mod N), where N is a positive integer. The index of Γ 0 N) in Γ is [Γ : Γ 0 N)] = N p N1+p 1 ), where p denotes a prime. Recall that Dedekind s eta-function ηz) is defined by ηz) := q 1/ q;q) = q 1/ 1 q n ), where q := e 2πiz and z is in the upper half complex plane. A function fz) is called an eta-quotient if it is of the form fz) = δ Nηδz) r δ, where N is a positive integer and r δ is an integer. For a positive integer l, the complex vectorspace of modular forms of weight l with respect to Γ 1 N) is denoted by M l Γ 1 N)). n=1

4 CHIRANJIT RAY AND RUPAM BARMAN Definition 2.1. [9, Definition 1.15] If χ is a Dirichlet character modulo N, then we say that a form fz) M l Γ 1 N)) has Nebentypus character χ if ) az +b f = χd)cz +d) l fz) cz +d [ ] a b for all z in the upper halfcomplex plane and all Γ c d 0 N). The space of such modular forms is denoted by M l Γ 0 N),χ). The correspondingspace of cusp forms is denoted by S l Γ 0 N),χ). If χ is the trivial character then we write M l Γ 0 N)) and S l Γ 0 N)) for short. We now recall two theorems from [9, p. 18] which will be used to prove our result. Theorem 2.2. [9, Theorem 1.64 and Theorem 1.65] If fz) = δ N ηδz)r δ is an eta-quotient with l = 1 2 δ N r δ Z, with the additional properties that δr δ 0 mod ) and δ N δ N N δ r δ 0 mod ), then fz) satisfies ) az +b f = χd)cz +d) l fz) cz +d [ ] a b 1) for every Γ c d 0 N). Here the character χ is defined byχd) := l δ N δr δ d In addition, if c,d, and N are positive integers with d N and gcdc,d) = 1, then the order of vanishing of fz) at the cusp c d is N δ N gcdd,δ) 2 r δ gcdd, N d )dδ. Suppose that fz) is an eta-quotient satisfying the conditions of the above theorem. If fz) is holomorphic at all of the cusps of Γ 0 N), then fz) M l Γ 0 N),χ). Definition2.. Letmbeapositiveintegerandfz) = an)qn M l Γ 0 N),χ). Then the action of Hecke operator T m on fz) is defined by fz) T m := d gcdn,m) In particular, if m = p is prime, we have 2.1) fz) T p := nm ) χd)d l 1 a q n d 2. apn)+χp)p l 1 a We note that an) = 0 unless n is a nonnegative integer. )) n q n. p Definition 2.4. A modular form fz) = an)qn M l Γ 0 N),χ) is called a Hecke eigenform if for every m 2 there exists a complex number λm) for which 2.2) fz) T m = λm)fz). ).

ON ANDREWS INTEGER PARTITIONS WITH EVEN PARTS BELOW ODD PARTS 5. Proof of Theorem 1.1 We use the theory of Hecke eigenforms to prove Theorem 1.1. We have EOn)q n = q4 ;q 4 ) q 2 ;q 2 ) 2 q;q) 8 mod 2). This gives EOn)q n+1 η 8 z) mod 2). Let η 8 z) = n=1 an)qn. Then an) = 0 if n 1 mod ) and for all n 0,.1) EOn) an+1) mod 2). By Theorem 2.2, we have η 8 z) S 4 Γ 0 9)). Since η 8 z) is a Hecke eigenform see, for example [8]), 2.1) and 2.2) yield )) n η 8 z) T p = apn)+p a q n = λp) an)q n, p which implies.2) n=1 apn)+p a ) n = λp)an). p Putting n = 1 and noting that a1) = 1, we readily obtain ap) = λp). Since ap) = 0 for all p 1 mod ), we have λp) = 0. From.2), we obtain ) n.) apn)+p a = 0. p From.), we derive that for all n 0 and p r,.4) and.5) ap 2 n+pr) = 0 ap 2 n) = p an) an) mod 2). Substituting n by n pr+1 in.4) and together with.1), we find that ) EO p 2 n+ p2 1.6) +pr 1 p2 0 mod 2). Substituting n by n+1 in.5) and using.1), we obtain ) EO p 2 n+ p2 1.7) EOn) mod 2). ) Since p 5 is prime, so 1 p 2 ) and gcd 1 p 2,p = 1. Hence when r runs over a residue system excluding the multiple of p, so does 1 p2 r. Thus.6) can be rewritten as ) EO p 2 n+ p2 1.8) +pj 0 mod 2), where p j. n=1

6 CHIRANJIT RAY AND RUPAM BARMAN Now, p i 5 are primes such that p i 1 mod ). Since p 2 1...p2 k n+ p2 1...p2 k 1 = p 2 1 p 2 2...p2 k n+ p2 2...p2 k 1 using.7) repeatedly we obtain that EO p 2 1...p 2 kn+ p2 1...p 2 k 1 ).9) EOn) mod 2). ) + p2 1 1, Let j 0 mod p k+1 ). Then.8) and.9) yield EO p 2 1...p 2 k+1n+ p2 1...p 2 k p ) k+1j +p k+1 ) 1 0 mod 2). This completes the proof of the theorem. 4. Proof of Theorem 1.2 We prove Theorem 1.2 using the approach developed in [11, 12]. To this end, we first recall some definitions and results from [11, 12]. For a positive integer M, let RM) be the set of integer sequences r = r δ ) δ M indexed by the positive divisors of M. If r RM) and 1 = δ 1 < δ 2 < < δ k = M are the positive divisors of M, we write r = r δ1,...,r δk ). Define c r n) by r n)q c n := q δ ;q δ ) r δ = 4.1) 1 q nδ ) r δ. δ M δ M n=1 The approach to proving congruences for c r n) developed by Radu [11, 12] reduces the number of coefficients that one must check as compared with the classical method which uses Sturm s bound alone. Let m be a positive integer. For any integer s, let [s] m denote the residue class of s in Z m := Z/mZ. Let Z m be the set of all invertible elements in Z m. Let S m Z m be the set of all squares in Z m. For t {0,1,...,m 1} and r RM), we define a subset P m,r t) {0,1,...,m 1} by P m,r t) := t : [s] m S m such that t ts+ s 1 δr δ mod m). Definition 4.1. Suppose m,m and N are positive integers, r = r δ ) RM) and t {0,1,...,m 1}. Let k = km) := gcdm 2 1,) and write δ rδ = 2 s j, δ M where s and j are nonnegative integers with j odd. The set consists of all tuples m,m,n,r δ ),t) satisfying these conditions and all of the following. δ M 1) Each prime divisor of m is also a divisor of N. 2) δ M implies δ mn for every δ 1 such that r δ 0. ) kn δ M r δmn/δ 0 mod ). 4) kn δ M r δ 0 mod 8). 5) m gcd kt k δ M δr δ,m) divides N. 6) If 2 m, then either 4 kn and 8 sn or 2 s and 8 1 j)n.

set and ON ANDREWS INTEGER PARTITIONS WITH EVEN PARTS BELOW ODD PARTS 7 Letm,M,N be positiveintegers. Forγ = p m,r γ) := min λ {0,1,...,m 1} p 1 r γ) := 1 δ N δ M r δ [ ] a b Γ, r RM)andr c d RN), r δ gcd 2 δa+δkλc,mc) δm gcd 2 δ,c). δ Lemma 4.2. [11, Lemma 4.5] Let u be a positive integer, m,m,n,r = r δ ),t) and r = r δ ) RN). Let {γ 1,γ 2,...,γ n } Γ be a complete set of representatives of the double cosets of Γ 0 N)\Γ/Γ. Assume that p m,r γ i )+p r γ i) 0 for all 1 i n. Let t min = min t P m,rt)t and ν := 1 r δ + δ M δ N r δ [Γ : Γ 0 N)] δ N δr δ 1 m δ M δr δ t min m. If the congruence c r mn+t ) 0 mod u) holds for all t P m,r t) and 0 n ν, then it holds for all t P m,r t) and n 0. To apply Lemma 4.2 we utilize the following result, which gives a complete set of representatives of the double cosets in Γ 0 N)\Γ/Γ. Lemma 4.. [15, Lemma 4.] If N or 1 2N is a square-free integer, then [ ] 1 0 Γ 0 N) Γ δ 1 = Γ. δ N Proof of Theorem 1.2. Due to 1.2) we need to prove our congruences modulo 4 only. We have EOn)q n = q4 ;q 4 ) q 2 ;q 2 ) 2 = q2 ;q 2 ) 2 q4 ;q 4 ) q 2 ;q 2 ) 4 q2 ;q 2 ) 2 q4 ;q 4 ) q 4 ;q 4 ) 2 mod 4) = q 2 ;q 2 ) 2 q4 ;q 4 ) mod 4). Letm,M,N,r,t) = 50,8,10,0,2,1,0),18). Itiseasytoverifythatm,M,N,r,t) {[ ] } 1 0 and P m,r t) = {18,28,8,48}. From Lemma 4. we know that : δ 10 δ 1 forms a complete set of double coset representatives of Γ 0 N)\Γ/Γ. Let r = 0,0,0,0,0,0) R10). We [ have ] used Sage to verify that p m,r γ δ ) + p r γ δ) 0 1 0 for each δ N, where γ δ =. We compute that the upper bound in Lemma δ 1 4.2 is ν = 1. Using Mathematica we verify that EO50n+t ) 0 mod 4) for n 1 and t P m,r t). Thus, by Lemma 4.2, we conclude that EO50n+t ) 0 mod 4) for any n 0, where t {18,28,8,48}. This completes the proof of the theorem.

8 CHIRANJIT RAY AND RUPAM BARMAN 5. Proof of Theorems 1., 1.4 and 1.5 We prove Theorem 1. by using the approach developed in [10]. Recently, Jameson and Wieczorek [5] have done a similar study for the generalized Frobenius partitions. To make this paper self-contained, we recall two results from [5]. Also see [10]. Let M! k Γ 0N 0 ),χ) denote the space of weakly holomorphic modular forms. Theorem 5.1. [5, Theorem 5] Let N 0,α,β,t be integers with N 0,α,t positive, and let cn)q αn+β M kγ! 0 N 0 ),χ), where cn) are algebraic integers in some number field. For any arithmetic progression r mod t), there are infinitely many integers N r mod t) for which cn) is even. Theorem 5.2. [5, Theorem 6] Let N 0,α,β,t be integers with N 0,α positive, and t > 1, and let cn)q αn+β M k! Γ 0N 0 ),χ), where cn) are algebraic integers in some number field. For any arithmetic progression r mod t), there are infinitely many integers M r mod t) for which cm) is odd, provided there is one such M. Furthermore, if there does exist an M r mod t) for which cm) is odd, then the smallest such M is less than C r,t for C r,t := 2j 12+k 12α [ Nα 2 t 2 ] 2 d p Nαt 1 1p 2 ) 2 j, where N := lcmαt,n 0 ), d := gcdαr +β,t), and j is a sufficiently large integer. Proof of Theorem 1.. We have EO2n)q n = q2 ;q 2 ) q;q) 2. We rewrite the above identity in terms of η-quotients, and then use the binomial theorem to obtain EO2n)q 6n+1 = η12z) η6z) 2 η12z)4 η6z) 4 mod 2). By Theorem 2.2, we have η12z) 4 η6z) 4 M! 0 Γ 072)). Let f t z) := η12z)4 η6z) 4 2j 6tz), where z) := η z). The cusps of Γ 0 72t) are represented by fractions c d where d 72t and gcdc,d) = 1. Now, f tz) vanishes at the cusp c d if and only if 4 gcdd,12)2 12 4 gcdd,6)2 6 + 2 jgcdd,6t)2 6t > 0.

ON ANDREWS INTEGER PARTITIONS WITH EVEN PARTS BELOW ODD PARTS 9 We have 4 gcdd,12)2 12 4 gcdd,6)2 6 + 2 jgcdd,6t)2 6t 2 j6 t 1 2. Hence, if j is an integer such that 2 j > t 12, then f tz) S 12 2 j Γ 0 72t)). Finally, our desired result follows immediately by applying Theorems 5.1 and 5.2 to EO2n)q6n+1. Proof of Theorem 1.4. We first recall the following 2-dissection formula from [2, Entry 25, p. 40]: 5.1) 1 q;q) 2 From 1.1), we have 5.2) q 8 ;q 8 ) 5 = q 2 ;q 2 ) 5 q 16 ;q 16 ) 2 EO2n)q n = q2 ;q 2 ) q;q) 2. +2q q4 ;q 4 ) 2 q 16 ;q 16 ) 2 q 2 ;q 2 ) 5 q 8 ;q 8 ). Combining 5.1) and 5.2), and then extracting the terms with odd powers of q, we deduce that EO4n+2)q n = 2 q2 ;q 2 ) 2 q8 ;q 8 ) 2 5.) q;q) 2 q 4 ;q 4. ) We again combine 5.1) and 5.), and then extract the terms with odd powers of q to obtain EO8n+6)q n = 4 q2 ;q 2 ) q 4 ;q 4 ) q 8 ;q 8 ) 2 q;q). Since q;q) 2 q 2 ;q 2 ) mod 2), we have EO8n+6)q n 4 q4 ;q 4 ) 5 q;q) mod 8). We rewrite the above equation in terms of η-quotients and obtain EO8n+6)q n+19 4 η5 96z) 5.4) mod 8). ηz) Let Az) = η2 z) η48z). Then, A2 z) 1 mod 4). Also, let Bz) = η5 96z)η z) η 2. 48z) Then we have 5.5) Bz) = η5 96z) ηz) A2 z) η5 96z) ηz) mod 4). The cusps of Γ 0 204) are represented by fractions c d where d 204and gcdc,d) = 1. By Theorem 2.2, Bz) is holomorphic at the cusp c d if and only if 5 gcdd,96)2 96 + gcdd,)2 2 gcdd,48)2 48 0.

10 CHIRANJIT RAY AND RUPAM BARMAN Now, 5 gcdd,96)2 + gcdd,)2 2 gcdd,48)2 96 48 = gcdd,48)2 5 gcdd,96) 2 ) 4gcdd,48) 2 +gcdd,)2 gcdd,48) 2 1 > 0. Hence, by Theorem 2.2, Bz) S Γ 0 204), ) 1 ). Let m be a positive integer. By a deep theorem of Serre [9, p. 4], if fz) M k Γ 0 N),χ) has Fourier expansion fz) = cn)q n Z[[q]], then there is a constant α > 0 such that ) X #{n X : cn) 0 mod m)} = O logx) α. Since Bz) S Γ 0 204), ) 1 ), the Fourier coefficients of Bz) are almost always divisible by m. Hence, using 5.5) and 5.4) we complete the proof of the theorem. Remark 5.. Let m be a fixed positive integer. If fz) M k Γ 0 N),χ) has Fourier expansion fz) = cn)qn Z[[q]], then using the result of Serre [9, p. 4] we also have that cn) 0 mod m) for almost all n in any given fixed arithmetic progression r mod t). Hence the even case of Theorem 1. also follows from Theorem 1.4. Proof of Theorem 1.5. The generating function of EO u 2n) is given by EO u 2n)q n 1 = q;q 2 ) 2 = q2 ;q 2 ) 2 5.6) q;q) 2. We note that ηz) = q n=1 1 qn ) is a power series of q. As in the proof of Theorem 1.4, let Az) = n=1 Then using binomial theorem we have 5.7) Define B k z) by 5.8) Modulo 2 k+1, we have 5.9) 1 q n ) 2 1 q 48n ) = η2 z) η48z). η2k+1 z) A 2k z) = 1 mod 2 k+1 ). η 2k 48z) B k z) = ) 2 η48z) A 2k z). ηz) B k z) = η2 48z) η 2 z) η2 48z) z) A2k η 2 z) = ;q 48 ) 2 q2q48 q ;q ) 2.

ON ANDREWS INTEGER PARTITIONS WITH EVEN PARTS BELOW ODD PARTS 11 Combining 5.6) and 5.9), we obtain 5.10) B k z) EO u 2n)q n+2 mod 2 k+1 ). The cusps of Γ 0 576) are represented by fractions c d where d 576and gcdc,d) = 1. By Theorem 2.2, it is easily seen that B k z) is a form of weight 2 k 1 on Γ 0 576). Therefore, B k z) M 2 k 1Γ 0 576)) if and only if B k z) is holomorphic at the cusp c d. We know that B kz) is holomorphic at a cusp c d if and only if Now, gcdd,) 2 2 k+1 2 ) + gcdd,48)2 1 2 k 1 ) 0. gcdd,) 2 2 k+1 2 ) +gcdd,48) 2 1 2 k 1) ) gcdd,) = gcdd,48) 2 2 gcdd,48) 22k+1 2)+1 2 k 1 ) 1 4 2k+1 2)+1 2 k 1 ) > 0. Hence, B k z) M 2 k 1Γ 0 576)). Now, using Serre s theorem [9, p. 4] as shown in the proof of Theorem 1.4, we arrive at the desired result due to 5.10). References [1] G. E. Andrews, Integer partitions with even parts below odd parts and the mock theta functions, Ann. Comb. 22 2018), 4 445. [2] B. C. Berndt, Ramanujan s Notebooks, Part III, Springer-Verlag, New York 1991). [] K. Bringmann and J. Lovejoy, Rank and congruences for overpartition pairs, Int. J. Number Theory 4 2008), 0 22. [4] B. Gordon and K. Ono, Divisibility of certain partition functions by powers of primes, Ramanujan J. 1 1997), 25 4. [5] M. Jameson and M. Wieczorek, Congruences for modular forms and generalized Frobenius partitions, arxiv:1809.00666v1. [6] N. Koblitz, Introduction to elliptic curves and modular forms, Springer-Verlag, New York 1991). [7] B. L. S. Lin, Arithmetic properties of overpartition pairs into odd parts, Electron. J. Combin. 19 2) 2012), #P17. [8] Y. Martin, Multiplicative η-quotients, Trans. Amer. Math. Soc. 48 12) 1996), 4825 4856. [9] K. Ono, The web of modularity: arithmetic of the coefficients of modular forms and q-series, CBMS Regional Conference Series in Mathematics, 102, Amer. Math. Soc., Providence, RI, 2004. [10] K. Ono, Parity of the partition function in arithmetic progression, J. Reine Angew. Math. 472 1996), 1 15. [11] S. Radu, An algorithmic approach to Ramanujan s congruences, Ramanujan J. 20 2) 2009), 295 02. [12] S. Radu and J. A. Sellers, Congruence properties modulo 5 and 7 for the pod function, Int. J. Number Theory 7 8) 2011), 29 2259. [1] C. Ray and R. Barman, Arithmetic properties of cubic and overcubic partition pairs, Ramanujan J. to appear). [14] A. Uncu, Countings on 4-decorated Ferrers diagrams, to appear). [15] L. Wang, Arithmetic properties of k,l)- regular bipartitions, Bull. Aust. Math. Soc. 95 2017), 5 64.

12 CHIRANJIT RAY AND RUPAM BARMAN Department of Mathematics, Indian Institute of Technology Guwahati, Assam, India, PIN- 78109 E-mail address: chiranjitray.m@gmail.com Department of Mathematics, Indian Institute of Technology Guwahati, Assam, India, PIN- 78109 E-mail address: rupam@iitg.ac.in