THE FIRST POSITIVE RANK AND CRANK MOMENTS FOR OVERPARTITIONS

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1 THE FIRST POSITIVE RANK AND CRANK MOMENTS FOR OVERPARTITIONS GEORGE ANDREWS, SONG HENG CHAN, BYUNGCHAN KIM, AND ROBERT OSBURN Abstract. In 2003, Atkin Garvan initiated the study of rank crank moments for ordinary partitions. These moments satisfy a strict inequality. We prove that a strict inequality also holds for the first rank crank moments of overpartitions consider a new combinatorial interpretation in this setting.. Introduction A partition of a non-negative integer n is a non-increasing sequence of positive integers whose sum is n. For example, the 5 partitions of 4 are 4, 3 +, 2 + 2, 2 + +, In 944, Dyson introduced the rank of a partition as the largest part minus the number of parts [8]. In 988, the first author Garvan defined the crank of a partition as either the largest part, if does not occur as a part, or the difference between the number of parts larger than the number of s the number of s, if does occur [4]. These two statistics give a combinatorial explanation of Ramanujan s congruences for the partition function modulo 5, 7. Let Nm, n) denote the number of partitions of n whose rank is m Mm, n) the number of partitions of n whose crank is m. A recent development in the theory of partitions has been the study of rank crank moments as initiated by Atkin Garvan [6]. For k, the kth rank moment N k n) the kth crank moment M k n) are given by N k n) : m Z m k Nm, n).) M k n) : m Z m k Mm, n)..2) Date: September 2, Mathematics Subject Classification. Primary: P8, 05A7. Key words phrases. overpartitions, ranks, cranks, positive moments, positivity. The first author was partially supported by National Security Agency Grant H The second author was partially supported by Nanyang Technological University Academic Research Fund, project number RG68/0. The third author was supported by the Basic Science Research Program through the National Research Foundation of Korea NRF) funded by the Ministry of Education, Science Technology NRF ), the TJ Park Science Fellowship from the POSCO TJ Park Foundation.

2 2 GEORGE ANDREWS, SONG HENG CHAN, BYUNGCHAN KIM, AND ROBERT OSBURN As N m, n) Nm, n) [8] M m, n) Mm, n) [4], we have N k n) M k n) 0 for k odd. The even moments are of considerable interest as they have been the subject of a number of works [, 2, 7, 8,, 3, 4, 7, 9, 20, 2, 27]. In particular, Garvan [20] conjectured that M 2j n) > N 2j n).3) for all j, n. In [3],.3) was proved for fixed j sufficiently large n. Garvan proved.3) for all j n via symmetrized rank crank moments Bailey pairs [2]. Recently, the first three authors gave an elementary proof of.3) by considering modified versions of.).2). Namely, consider the positive rank crank moments In [3], it was proved that N + k n) : m k Nm, n) M + k n) : m k Mm, n). M + k n) > N + k n).4) for all k, n by a careful study of the decomposition of the generating function for the difference M + k n) N + k n). For a discussion concerning the asymptotic behavior of these moments, see [2]. Inequality.4) combined with the fact that N 2j n) 2N 2j + n) M 2jn) 2M 2j + n) imply.3). Our interest in this paper is to consider an analogue of.4) for overpartitions. Recall that an overpartition [25] is a partition in which the first occurrence of each distinct number may be overlined. For example, the 4 overpartitions of 4 are 4, 4, 3 +, 3 +, 3 +, 3 +, 2 + 2, 2 + 2, 2 + +, 2 + +, 2 + +, 2 + +, + + +, These combinatorial objects have recently played an important role in the construction of weight 3/2 mock modular forms [9], in Rogers-Ramanujan Gordon type identities [5] in the study of Jack superpolynomials in supersymmetry quantum mechanics [6]. Let Nn, m) denote the number of overpartitions of n whose rank is m Mn, m) the number of overpartitions of n whose first residual) crank is m. Here, Dyson s rank extends easily to overpartitions the first residual crank of an overpartition is obtained by taking the crank of the subpartition consisting of the non-overlined parts [0]. It is now natural to consider the rank crank overpartition moments N k n) : m Z m k Nm, n)

3 THE FIRST POSITIVE RANK AND CRANK MOMENTS FOR OVERPARTITIONS 3 M k n) : m Z m k Mm, n). Via the symmetries N m, n) Nm, n) [23] M m, n) Mm, n) [0], we have N k n) M k n) 0 for k odd. Thus, to obtain non-trivial odd moments, we consider N + k n) : m k Nm, n) M + k n) : m k Mm, n). The main result in this paper is an analogue of.4) for overpartitions in the case k. Theorem.. For all n, we have M + n) > N + n)..5) The paper is organized as follows. In Section 2, we prove Theorem.. In Section 3, we give a combinatorial interpretation of M + n) N + n). In Section 4, we conclude with some remarks regarding future directions. 2. The proof of Theorem. For k, we define the generating functions M k q) M + k n)q n R k q) N + k n)q n compute their explicit expressions for k. Throughout, we use the stard q-hypergeometric notation, valid for n N { }. Proposition 2.. We have a) n a; q) n R q) 2 q) n aq k ), k ) n+ qnn+) q 2n 2.)

4 4 GEORGE ANDREWS, SONG HENG CHAN, BYUNGCHAN KIM, AND ROBERT OSBURN M q) q) ) n+ qnn+)/2 q n. 2.2) Proof. We begin with the generalized Lambert series representation of the two-variable generating function for Dyson s rank for overpartitions, Rz, q) : Nm, n)z m q n n0 m Z ) n q nn+)/2 zq) n q/z) n + 2 n0 q) q) ) z) /z) ) n q n2 +n zq n ) q n /z) + 2 ) n q n2 2 ) n q n2 q n ) + q n z m q mn + m0 z m q )). mn 2.3) For the second third equalities in 2.3), see the proof of Proposition 3.2 in [23]. Here, we have used the identity z) /z)q n zq n ) q n /z) ) qn + q n z m q mn + z m q mn m0 for the last equality in 2.3). We now apply the differential operator z z to both sides of 2.3) to obtain z ) Rz, q) z q) 2 ) n+ q n2 q n ) + q n mz m q mn + 2 ) n q n2 q n ) + q n Only the first term on the right side of 2.4) contributes to positive powers of z so mz m q ). mn 2.4) 2 q) R q) lim z 2 q) 2 q) ) n+ q n2 q n ) + q n ) n+ q n2 q n ) + q n ) n+ q nn+) q 2n, which is 2.). In the last equality of 2.5), we applied the identity mz m q mn mq mn 2.5)

5 THE FIRST POSITIVE RANK AND CRANK MOMENTS FOR OVERPARTITIONS 5 mq mn q n q n ) 2. For the two-variable generating function for the first residual crank for overpartitions, we have Cz, q) : Mm, n)z m q n q) Cz, q) 2.6) n0 m Z where Cz, q) is the two-variable generating function for the crank for partitions. Thus, by the proof of Theorem in [3], we obtain 2.2). We now require the following two Lemmas for the proof of Theorem.. Lemma 2.2. If then hq) : ) n+ q nn+)/2 q n, hq) q j2 + 2q j + 2q 2j + + 2q j2 j + q j2 ). Proof. Setting b c in [5, Theorem 2.3], we have the Bailey pair α n qn2 aq 2n )a) 2 n a)q) 2 n 2.7) n0 β n q) ) n Substituting 2.7) 2.8) into [5, Corollary 2.] with ρ, ρ 2, we find that ) q n2 a n q) 2 q 2n2 a n aq 2n )a) 2 n + n aq) a)q) ) n Next, we apply d a. The left side of 2.9) becomes da n0 nq n2 q) 2 n ) n+ q n+ 2 ) q n

6 6 GEORGE ANDREWS, SONG HENG CHAN, BYUNGCHAN KIM, AND ROBERT OSBURN after invoking [3, Eq..3)], while the right side of 2.9) becomes q j q j q j q j q 2n2 q 2n )q) 2 n q) 2 n q 2n2 + q n ) q n. ) Multiplying both sides by, we obtain hq) q j q j q j2 + q j ) q j q 2n2 +q n) q n ) q 2j2 +q j) q j q j2 + q j ) q j2 ) q j q j2 + q j ) + q j + + q jj ) ) q j2 + 2q j + 2q 2j + + 2q j2 j + q j2 ). 2.0) Note that in the second equality of 2.0), we used the elementary manipulation q j q j k q jk q jk + q jk k j k j>k q j2 + q j ) q j. Lemma 2.3. hq) 2hq 2 ) ) n+ q n2 2q n + 2q 2n + + ) n 2q n2 n + ) n q n2). 2.)

7 THE FIRST POSITIVE RANK AND CRANK MOMENTS FOR OVERPARTITIONS 7 Proof. Exping the right side of 2.) according to the parity of n then separating the positive terms from the negative terms, we find that ) n+ q n2 2q n + 2q 2n + + ) n 2q n2 n + ) n q n2) ) q 2n )2 + 2q 4n 2 + 2q 8n q 4n2 6n+2 + q 2n )2 2q 2n + 2q 6n q 4n2 8n+3 + q 2n )2) ) q 2n)2 2q 2n + 2q 6n + + 2q 4n2 2n q 2n)2 + 2q 4n + + 2q 4n2 4n + q 2n)2). 2.2) Using Lemma 2.2, we compute a similar expansion for hq), then compare with 2.2) in order to see that it suffice to prove hq 2 ) q 2n2 + 2q 2n + 2q 4n + + 2q 2n2 2n + q 2n2) q 2n )2 2q 2n + 2q 6n q 4n2 8n+3 + q 2n )2) + q 2n)2 + 2q 4n + + 2q 4n2 4n + q 2n)2). 2.3) Subtracting show that q 2n2 + q 2n2) from both sides of 2.3) then dividing by 2, it remains to ) q 2n2 q 2n + q 4n + + q 2n2 2n n2 ) ) q 2n )2 q 2n + q 6n q 4n2 8n+3 + q 4n2 q 4n + + q 4n2 4n. 2.4) n2 n2

8 8 GEORGE ANDREWS, SONG HENG CHAN, BYUNGCHAN KIM, AND ROBERT OSBURN Define fn, j) q 2n2 +2nj. Substituting fn, j) into the left side of 2.4) making a change of summation index k n + j, we find that ) n 2n q 2n2 q 2n + q 4n + + q 2n2 2n fn, j) fn, k n) n2 l2 2l 2 nl fn, 2l n) + 2m m2 nm+ n2 q 4l2 2l + q 4l2 +2l q 8l2 2l+4 + l2 fn, 2m n) n2 kn+ q 4l2 +4l + q 4l2 +8l + + q 8l2 4l, where in the penultimate equality, we rearranged the order of summation separated the terms into odd even values of k via k 2l k 2m. We see that these are equal to the right side of 2.4) this completes the proof. We can now prove Theorem. Proof of Theorem.. By Proposition 2., we have M q) R q) q) ) hq) 2hq 2 ). 2.5) Thus, it suffices to prove that the right side of 2.5) has positive power series coefficients for all positive powers of q. By Lemma 2.3, ) hq) 2hq 2 ) ) n+ q n2 + 2 ) n q n2 q n q 2n + + ) n 2 q n2 n For the sum A, note that Hence Similarly, for the sum A 4, Therefore, n2 m2 2 q 22n)2 ) n+ q 2n2 : A + 2A 2 2A 3 A A 2 k ) n q n2. 2 q) q) A q) 2 2. q 2 ; q 2 ) q 2 ; q 2 ) A 4 q2 ; q 2 ) 2q 2 ; q 2 ) 2. q) A A 4 ) q) 2 2 q; q2 ) q; q 2 ) q 2 ; q 2 ) 2q 2 ; q 2 ) q; q2 ) ) 2 2q; q 2 ) 2,

9 THE FIRST POSITIVE RANK AND CRANK MOMENTS FOR OVERPARTITIONS 9 which has positive power series coefficients for all positive powers of q. Next, we examine A 2 A 3. We define gn, j) ) n+j q n2 +jn. Then A 2 A 3 n ) n q n2 ) j q jn n2 n2 n gn, j) + g2n, 2n). q 22n)2 We now rearrange the series A 2 A 3 into several sums. Note that for j 0 n 2j + 2, g2n, 4j + 3) + g2n +, 4j + 3) + g2n +, 4j + ) + g2n + 2, 4j + ) ) 2n+4j+2 q 4n2 +4j+3)2n q 4n+4j+4 q 4j+2 + q 4n+8j+6) q 4n2 +4j+3)2n q 4j+2) q 4n+4j+4), for j 0 n 2j + 2, g2n +, 4j + 4) + g2n + 2, 4j + 4) + g2n + 2, 4j + 2) + g2n + 3, 4j + 2) ) 2n+4j+4 q 2n+)2 +4j+4)2n+) q 4n+4j+7 q 4j+3 + q 4n+8j+0) q 2n+)2 +4j+4)2n+) q 4j+3) q 4n+4j+7). These take care of all the terms except, for all integers n 0, g4n + 2, 4n + ) + g4n + 3, 4n + ) + g4n + 4, 4n + ) + g4n + 2, 4n + 2) + g4n + 3, 4n + 2) + g4n + 4, 4n + 2) + g4n + 5, 4n + 2) + g4n + 4, 4n + 4) [g4n + 2, 4n + ) + g4n + 3, 4n + ) + g4n + 4, 4n + ) + g4n + 2, 4n + 2) + g4n + 3, 4n + 2) g4n + 3, 4n + 4)] + [g4n + 3, 4n + 4) + g4n + 4, 4n + 4) + g4n + 4, 4n + 2) + g4n + 5, 4n + 2)]. Note that g4n + 2, 4n + ) + g4n + 3, 4n + ) + g4n + 4, 4n + ) + g4n + 2, 4n + 2) + g4n + 3, 4n + 2) g4n + 3, 4n + 4) q 4n+2)2 +4n+)4n+2) [ q 2n+6 + q 24n+4 q 4n+2 + q 6n+9 q 24n+5] [ q 4n+2)2 +4n+)4n+2) q 4n+2 ) q 8n+7 ) q 2n+6 ) ] + q 2n+8 q) q 4n ) + q 8n+7 q 4n+ ) q 2n+6 ) while g4n + 3, 4n + 4) + g4n + 4, 4n + 4) + g4n + 4, 4n + 2) + g4n + 5, 4n + 2) q 4n+2)2 +4n+)4n+2)+24n+5 q 4n+3 ) q 2n+ ).

10 0 GEORGE ANDREWS, SONG HENG CHAN, BYUNGCHAN KIM, AND ROBERT OSBURN These sums show that q) A 2 A 3 ) q) + q) + q) j0 n2j+2 j0 n2j+2 q 4n2 +4j+3)2n q 4j+2) q 4n+4j+4) q 2n+)2 +4j+4)2n+) q 4j+3) q 4n+4j+7) [ q 4n+2)2 +4n+)4n+2) q 4n+2 ) q 8n+7 ) q 2n+6 ) n0 ] + q 2n+8 q) q 4n ) + q 8n+7 q 4n+ ) q 2n+6 ) + q) q 4n+2)2 +4n+)4n+2)+24n+5 q 4n+3 ) q 2n+ ). n0 For positive integers a, b, c d with b < c < d, expressions of the form q) q a q b ) q c ) q) q a q b ) q c ) q d ) have nonnegative coefficients so q) A 2 A 3 ) has nonnegative power series coefficients. Since q) A A 4 ) has positive power series coefficients for all positive powers of q, we conclude that the power series expansion of q) hq) 2hq 2 )) has positive coefficients for all q n, n. This proves.5). Corollary 2.4. hq) 2hq 2 )) has positive power series coefficients for all q n with n 6. Proof. From the proof of Theorem. by invoking the elementary identity q) /q; q 2 ), we see that q; q 2 ) A A 4 ) q) 2 q; q 2 ) ) 2 2 q; q2 ) q; q 2 ) ), which has positive power series coefficients for all odd positive powers of q the terms with even powers of q vanishes). Again, from the proof of Theorem., it is easy to see that A 2 A 3 ) has nonnegative power series coefficients. Since one of the terms in the corresponding expression

11 of A 2 A 3 ) is THE FIRST POSITIVE RANK AND CRANK MOMENTS FOR OVERPARTITIONS q 6 q 2 ) q 6 ) q 8 ) q 6 k k 2,6,8 q k, the coefficients of q n for n 6 in the power series expansion of A 2 A 3 ) are all positive. 3. A combinatorial interpretation In [3], the first three authors defined a new counting function osptn) as osptn) M + n) N + n) provided its combinatorial interpretation. The function osptn) is an interesting companion of sptn) in sense of that sptn) M + 2 n) N + 2 n). Here, sptn) is the number of smallest parts in the partitions of n [2]. In this section, we discuss an overpartition analogue of osptn) its combinatorial meaning. Let us define osptn) M + n) N + n). Before giving a combinatorial interpretation for osptn), we first recall the description of osptn). An even string in the partition λ is a sequence of the consecutive parts starting from some even number 2k + 2 where the length is an odd number greater than or equal to 2k + 2k + 2 plus the length of the string the number of consecutive parts) do not appear as a part. An odd string in λ is a sequence of the consecutive parts starting from some odd number 2k + where the length is greater than or equal to 2k + such that the part 2k + appears exactly once 2k + 2 plus the length of the string does not appear as a part. By consecutive parts, we allow repeated parts. With these notions in mind, we have the following. Theorem 3.. [3, Theorem 4] For all positive integers n, osptn) λ n STλ), where the sums runs over the partitions of n STλ) is the number of even odd strings in the partition λ. The function osptn) now counts the number of certain stings in the overpartitions of n, but the difference is that we have a weighted count of strings. We start by defining f k q) as f k q) ) n+ q nn+)/2+nk ). By Proposition 2. exchanging the order of summation, we have M + n) N + n))q n q) f k q) 2f k q 2 )). k

12 2 GEORGE ANDREWS, SONG HENG CHAN, BYUNGCHAN KIM, AND ROBERT OSBURN Note that for a fixed k, q) f 2k q) + f 2k q) 2f k q 2 )) q) q 2n2 5n+4nk 2k+2 q 2n2 n ) q 4n+2k 2 ) q 2n2 3n+4nk q 2n2 +n ) q 4n+2k ). Now we define A k n) resp. B k n)) to be the number of overpartitions of n counted by the first resp. second) sum. By noting that 2n 2 n + 4nk + 2k 2) k 2) + + 2n ) + 2k 2) + 2n + 2k 2), we define an odd string starting from 2k in an overpartition as ) 2k, 2k,..., 2l + 2k 3 appears at least once, i.e. there are 2l consecutive parts starting from 2k. 2) There is no other part of size 2l 2 l 4l + 2k 2. Similarly, we define an even sting starting from 2k in an overpartition as ) 2k, 2k,..., 2l+2k 2 appears at least once, i.e. there are 2l consecutive parts starting from 2k. 2) There is no other part of size 2l 2 + l 4l + 2k. As with the osptn) function, A k n) is now the number of odd strings starting from 2k along the overpartitions of n, B k n) is the number of even strings starting from 2k along the overpartitions of n. Then we have M + n) N + n))q n We have thus proven the following. n+)/2 k Theorem 3.2. For all positive integers n, we have A k n) B k n))q n osptn) ST o n) ST e n), osptn)q n. where ST o n) resp. ST e n)) is the number of odd resp. even) strings along the overpartitions of n. Let us illustrate the above discussion for n 5. From Table, we see that ST o 5) 8 ST e 5) 4, so ospt5) 4. This matches with M + 5) 24 N + 5) 20. We have numerically observed that 4. Concluding Remarks M + k n) > N + k n) 4.) for all k, n. Inequality 4.) the fact that N 2j n) 2N + 2jn) M 2j n) 2M + 2jn) implies that a complete analogue of.3) should hold. It would be interesting to see if the techniques in [3] can be used to prove 4.) discover a combinatorial meaning for M + k n) N + k n). Moreover, there is an inequality of note which has a similar flavor to.3). If we consider the rank moment

13 THE FIRST POSITIVE RANK AND CRANK MOMENTS FOR OVERPARTITIONS 3 Overpartitions of 5 The number of odd strings The number of even strings Table. The number of strings in the overpartitions of 5. N2 k n) : m Z m k N2m, n) where N2m, n) is the number of overpartitions of n with M 2 -rank m [24], then Mao [26] has proven that N 2j n) > N2 2j n) 4.2) for all j, n 2. Another proof of 4.2) using the similarly defined positive rank moment N2 + k n) can be found in [22]. It is still not known what N + k n) N2 + k n) counts. One could also compute asymptotics in the spirit of [, 2, 3, 4, 27]. Finally, while proving Corollary 2.4 Theorem 3.2, we observed the following. First, it appears that for all integers m 3. hq) mhq m )) has positive power series coefficients for all positive powers of q. Second, numerical computations suggest that A k n) B k n) for all n, k. We leave these questions to the interested reader. References [] G.E. Andrews, Partitions, Durfee symbols, the Atkin-Garvan moments of ranks, Invent. Math ), no., [2] G.E. Andrews, The number of smallest parts in the partitions of n, J. Reine Angew. Math ), [3] G.E. Andrews, S.H. Chan, B. Kim, The odd moments of ranks cranks, J. Combin. Theory Ser. A ), no., [4] G.E. Andrews, F.G. Garvan, Dyson s crank of a partition, Bull. Amer. Math. Soc. N.S.) 8 988), no. 2, [5] G.E. Andrews, D. Hickerson, Ramanujan lost notebook. VII. The sixth order mock theta functions, Adv. Math ), no., [6] A.O.L. Atkin, F.G. Garvan, Relations between the ranks cranks of partitions, Ramanujan J ), no. -3, [7] K. Bringmann, On the explicit construction of higher deformations of partition statistics, Duke Math. J ), no. 2,

14 4 GEORGE ANDREWS, SONG HENG CHAN, BYUNGCHAN KIM, AND ROBERT OSBURN [8] K. Bringmann, F. Garvan, K. Mahlburg, Partition statistics quasiharmonic Maass forms, Int. Math. Res. Not. IMRN 2009, no., Art. ID rnn24, [9] K. Bringmann, J. Lovejoy, Overpartitions class numbers of binary quadratic forms, Proc. Natl. Acad. Sci. USA ), no. 4, [0] K. Bringmann, J. Lovejoy, R. Osburn, Rank crank moments for overpartitions, J. Number Theory ), no. 7, [] K. Bringmann, K. Mahlburg, Inequalities between ranks cranks, Proc. Amer. Math. Soc ), no. 8, [2] K. Bringmann, K. Mahlburg, Asymptotic inequalities for positive crank rank moments, Trans. Amer. Math. Soc., to appear. [3] K. Bringmann, K. Mahlburg, R. Rhoades, Asymptotics for rank crank moments, Bull. Lond. Math. Soc ), no. 4, [4] K. Bringmann, K. Mahlburg, R. Rhoades, Taylor coefficients of mock-jacoi forms moments of partition statistics, Math. Proc. Camb. Phil. Soc., to appear. [5] W.Y.C. Chen, D. Sang, D. Shi, The Rogers-Ramanujan-Gordon theorem for overpartitions, Proc. London. Math. Soc., to appear. [6] P. Desrosiers, L. Lapointe, P. Mathieu, Evaluation normalization of Jack superpolynomials, Int. Math. Res. Not. IMRN 202, no. 23, [7] P. Diaconis, S. Janson, R. Rhoades, Note on a partition limit theorem for the rank crank, Bull. London Math. Soc ), no. 3, [8] F.J. Dyson, Some guesses in the theory of partitions, Eureka 8 944), 0 5. [9] A. Folsom, K. Ono, The spt-function of Andrews, Proc. Natl. Acad. Sci. USA ), no. 5, [20] F.G. Garvan, Congruences for Andrews smallest parts partition function new congruences for Dyson s rank, Int. J. Number Theory 6 200), no. 2, [2] F. G. Garvan, Higher order spt-functions, Adv. Math ), no., [22] A. Larsen, A. Rust, H. Swisher, Inequalities for positive rank crank moments of overpartitions, in preparation. [23] J. Lovejoy, Rank conjugation for the Frobenius representation of an overpartition, Ann. Comb ), [24] J. Lovejoy, Rank conjugation for a second Frobenius representation of an overpartition, Ann. Comb ), no., 0 3. [25] J. Lovejoy, S. Corteel, Overpartitions, Trans. Amer. Math. Soc ), no. 4, [26] R. Mao, Inequalities between rank moments of overpartitions, J. Number Theory, to appear. [27] R. Rhoades, Families of quasimodular forms Jacobi forms: the crank statistic for partitions, Proc. Amer. Math. Soc ), no., Department of Mathematics, The Pennsylvania State University, University Park, PA 6802, USA Division of Mathematical Sciences, School of Physical Mathematical Sciences, Nanyang Technological University, 2 Nanyang link, Singapore 63737, Republic of Singapore School of Liberal Arts Institute of Convergence Fundamental Studies, Seoul National University of Science Technology, 72 Gongreung 2 dong, Nowongu, Seoul , Republic of Korea School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Irel address: rews@math.psu.edu address: chansh@ntu.edu.sg address: bkim4@seoultech.ac.kr address: robert.osburn@ucd.ie