An analogue of k -marked Durfee symbols for strongly unimodal sequences
aa r X i v : . [ m a t h . N T ] S e p AN ANALOGUE OF k -MARKED DURFEE SYMBOLS FOR STRONGLYUNIMODAL SEQUENCES SAVANA AMMONS, YOUNG JIN KIM, LAURA SEABERG, AND HOLLY SWISHER
Abstract.
In a seminal 2007 paper, Andrews introduced a class of combinatorial objects thatgeneralize partitions called k -marked Durfee symbols. Multivariate rank generating functions forthese objects have been shown by many to have interesting modularity properties at certain vectorsof roots of unity. Motivated by recent studies of rank generating functions for strongly unimodalsequences, we apply methods of Andrews to define an analogous class of combinatorial objects called k -marked strongly unimodal symbols that generalize strongly unimodal sequences. We establisha multivariate rank generating function for these objects, which we study combinatorially. Weconclude by discussing potential quantum modularity properties for this rank generating functionat certain vectors of roots of unity. Introduction and Statement of Results
Partitions and k -marked Durfee symbols. A partition of a positive integer n is anynonincreasing sequence of positive integers called parts that sum to n ; we further define the emptyset to be the sole partition of 0. Partitions have been a rich source of study from many mathematicaland physical perspectives, which is partly due to their powerful connection to the theory of modularforms. One can see this connection immediately due to the following relationship between thegenerating function for the partition counting function p p n q and Dedekind’s eta function η p τ q ,(1.1) ÿ n ě p p n q q n “ q η p τ q ´ . Here p p n q counts the number of partitions of n , and η p τ q “ q ś n “ p ´ q n q is a weight 1 { q “ e πiτ , τ P H . However, the combinatorial rank function for partitions demonstratesperhaps an even more striking relationship between partitions and modularity. Dyson [14] definedthe rank of a partition to be its largest part minus its number of parts, and conjectured that therank could be used to combinatorially explain Ramanujan’s famous partition congruences modulo5 and 7. This was later proved by Atkin and Swinnerton-Dyer [5].The partition rank function N p m, n q counts the number of partitions of n with rank equalto m . The two variable generating function for N p m, n q may be expressed as the following q -hypergeometric series ÿ m P Z ÿ n ě N p m, n q z m q n “ ÿ n “ q n p zq ; q q n p z ´ q ; q q n “ : R p z ; q q , (1.2) Mathematics Subject Classification.
Key words and phrases. partitions, k -marked Durfee symbols, strongly unimodal sequences, rank generating func-tions, quantum modular forms.This work was partially supported by the National Science Foundation REU Site Grant DMS-1757995, and OregonState University. here N p m, q “ δ m , in terms of the Kronecker delta function δ ij , and the q -Pochhammer symbol p a ; q q n is defined by p a ; q q n : “ n ź j “ p ´ aq j ´ q , for n either a nonnegative integer or infinity.Specializing R p z ; q q at z “ R p q q “ ÿ n ě p p n q q n from which we can observe modularity via (1.1). Letting z “ ´ R p z ; q q gives(1.3) R p´ q q “ ÿ n “ q n p´ q ; q q n , which is one of Ramanujan’s original third order mock theta functions. Bringmann and Ono [12]showed that letting z “ ω , for any nonidentity root of unity ω , yields that R p ω ; q q is a weight 1 { Ferrers diagrams , where each part is represented bya horizontal row of dots, which are left-justified and decreasing from top down. The largest squareof dots within the Ferrers diagram of a partition is called the
Durfee square of the partition. Ina 2007 paper, Andrews [3] defined Durfee symbols, which provide an alternate representation ofa partition. For each partition, the Durfee symbol encodes the side length of the Durfee squareand in addition the lengths of the columns to the right of as well as the rows beneath the Durfeesquare. For example, below we show the 5 partitions of 4, followed by the Ferrers diagrams withhighlighted Durfee squares, and then the associated Durfee symbols.
Figure 1.
Partitions, Ferrers Diagrams, and Durfee Symbols for n “
54 3 ` ` ` ` ` ` ` ‚ ‚ ‚ ‚ ‚ ‚ ‚‚ ‚ ‚‚ ‚ ‚ ‚‚‚ ‚ ‚‚‚ ˆ ˙ ˆ ˙ ` ˘ ˆ
11 1 ˙ ˆ ˙ We can think of the parts on the top or bottom row as a partition α or β , respectively. Let ℓ p α q denote the number of parts in a partition α . The rank of a Durfee symbol is(1.4) ℓ p α q ´ ℓ p β q , the length of the partition in the top row, minus the length of the partition in the bottom row,which gives Dyson’s rank of the corresponding partition.Andrews [3] modifies Durfee symbols by defining objects called k -marked Durfee symbols using k copies of the integers. He further defines k different rank statistics for the k -marked Durfeesymbols. Moreover, letting D k p m , m , . . . , m k ; n q denote the number of k -marked Durfee symbolsof n with i th rank equal to m i , Andrews establishes a k ` R k p x , . . . , x k ; q q which may be expressed in terms of q -hypergeometric series, analogous to (1.2).Complete definitions of these objects are given in § ringmann [6] showed that the function R p , q q is a quasimock modular form. Bringmann,Garvan, and Mahlburg [9] expanded on this by showing that R k p , ..., q q is a quasimock modularform for all k ě
2. In 2013, Folsom and Kimport [17] went on to prove that for more generalvectors of roots of unity, R k p ω , . . . , ω k ; q q with k ě R k p ω , . . . , ω k ; q q with k ě Strongly Unimodal Sequences.
There is a related combinatorial object which has alsoexhibited interesting connections to modular forms theory. A strongly unimodal sequence of size n is a list of positive integers a , . . . , a s that sum to n such that a ă a ă ¨ ¨ ¨ ă a k ą a k ` ą ¨ ¨ ¨ ą a s , where a k is called the peak . We write u p n q to count the number of strongly unimodal sequences ofsize n .As with partitions, strongly unimodal sequences have been a fruitful source of study from avariety of perspectives. Similar to Ferrers diagrams for partitions, strongly unimodal sequences canbe visualized graphically by representing each part a i as a column of dots, ordering by index fromleft to right. For each strongly unimodal sequence we can define a symbol analogous to the Durfeesymbol of a partition that encodes the size of the peak of the strongly unimodal sequence, as well asthe length of the columns to the right and left of the peak. We call such symbols strongly unimodalsymbols in this paper. For example, below we show the 4 strongly unimodal sequences of size 4with highlighted peaks, followed by their diagrams and associated strongly unimodal symbols. Figure 2.
Strongly Unimodal Sequences, Diagrams, and Symbols for n “
44 1, 3 3, 1 1, 2, 1 ‚‚‚‚ ‚‚ ‚ ‚ ‚‚‚ ‚ ‚ ‚ ‚ ‚ ` ˘ ˆ ˙ ˆ ˙ ˆ ˙ From a partition theoretic perspective (see [11] for instance), it is natural to define the rank ofa strongly unimodal sequence of size n to be the number of terms to the right of the peak minusthe number of terms to the left of the peak. We write u p m, n q to count the number of stronglyunimodal sequences of size n with rank m . Similarly to (1.2), the generating function for u p m, n q can be expressed as a q -hypergeometric series ÿ m P Z ÿ n ě u p m, n q z m q n “ ÿ n ě p´ zq ; q q n p´ zq ´ ; q q n q n ` “ : U p z ; q q . Specializing U p z ; q q at z “ U p q q “ ÿ n ě u p n q q n , which Andrews [4] showed could be expressed in terms of two mock theta functions. In a beau-tiful paper by Bryson, Ono, Pitman, and Rhoades [13], they show that U p´ q q is a mock andquantum modular form which has a duality relationship with Kontsevich’s “strange” function,one of Zagier’s original examples of a quantum modular form [20]. Moreover, U p˘ i ; q q is a mock heta function. There has also been related work connecting rank generating functions for stronglyunimodal sequences with mock and quantum modular or Jacobi forms (see [8, 18, 7] for example).1.3. Combinatorial work of Andrews and our analogous results.
When Andrews [3] intro-duced k -marked Durfee symbols, he demonstrated a variety of interesting combinatorial propertiesthat they satisfy, including a combinatorial explanation for congruences of the symmetrized k thmoment functions for partition ranks. Motivated by the above discussion of combinatorial rankgenerating functions in the context of modularity, our project was to construct objects analogousto k -marked Durfee symbols in the setting of strongly unimodal sequences and study their combi-natorial properties with an eye toward connections to modularity.Andrews [3] established the following combinatorial rank generating function for k -marked Durfeesymbols. Theorem 1.1 (Andrews [3, Thm. 10]) . Let D k p m , m , . . . , m k ; n q count the number of k -markedDurfee symbols of n with i th rank equal to m i . Then for k ě , ÿ m i P Z ÿ n ą D k p m , m , ..., m k ; n q x m x m ...x m k k q n “ R k p x , x , ..., x k ; q q , where R p x ; q q is defined in (1.2) and for k ě , R k p x , . . . , x k ; q q : “ ÿ m ą m ,...,m k ě q M k `p M `¨¨¨` M k ´ q p x q ; q q m ´ qx ; q ¯ m p x q M ; q q m ` ´ q M x ; q ¯ m ` ¨ ¨ ¨ p x k q M k ´ ; q q m k ` ´ q Mk ´ x k ; q ¯ m k ` , where M j : “ m ` m ` ¨ ¨ ¨ ` m j for each ď j ď k . In §
2, we review the full definition of k -marked Durfee symbols, and provide our construction ofanalogous k -marked strongly unimodal symbols. Our first result is to establish a rank generatingfunction for k -marked strongly unimodal symbols analogous to Theorem 1.1. Theorem 1.2.
Let U k p m , m , ..., m k ; n q count the number of k -marked strongly unimodal symbolsof size n with i th rank equal to m i . Then for k ě , ÿ m i P Z ÿ n ě U k p m , m , ..., m k ; n q x m x m ...x m k k q n “ U k p x , x , ..., x k ; q q , where U k p x , x , ..., x k ; q q : “ ÿ m ,...,m k ě q M `¨¨¨` M k ¨ p ` x ´ q M qp ` x ´ q M q ¨ ¨ ¨ p ` x ´ k ´ q M k ´ q¨ p´ x q ; q q m ´ p´ x ´ q ; q q m ´ p´ x q M ` ; q q m ´ p´ x ´ q M ` ; q q m ´ ¨ ¨ ¨ p´ x k q M k ´ ` ; q q m k ´ p´ x ´ k q M k ´ ` ; q q m k ´ , and M j : “ m ` m ` ¨ ¨ ¨ ` m j for each ď j ď k . We note here that this is not the first generalization of k -marked Durfee symbols. In work ofBringmann, Lovejoy, and Osburn [10], they construct a generalization related to overpartition pairs.Further, Alfes, Bringmann, and Lovejoy [1] have considered this same generalization in the oddsetting. In both cases, automorphic properties of the generating functions were demonstrated.Some of the combinatorics Andrews [3] pursued for k -marked Durfee symbols was from theperspective of self-conjugation. The conjugate of a partition can be obtained from its Ferrersdiagram by simply constructing parts from the columns rather than the rows. When the resultingpartition is unchanged we call it self-conjugate . From this we can observe that self-conjugate artitions must have rank 0. As Andrews describes in [3], Sylvester and Durfee [19] were the firstto study self-conjugate partitions, and it was this which originally led to the idea of Durfee squaresas well as the proof that self-conjugate partitions are in bijection with partitions into distinct oddparts. Andrews [3] defines a self-conjugate k -marked Durfee symbol to be one in which the top andbottom rows are identical, and proves the following combinatorial result. Theorem 1.3 (Andrews [3, Thm. 14]) . The number of self-conjugate k -marked Durfee symbolsof n is equal to the number of partitions of n into distinct unmarked odd parts, as well as k ´ differently marked p k ´ q -marked even parts each at most twice the number of odd parts. We similarly define a self-conjugate k -marked strongly unimodal symbol to be one in which thetop and bottom rows are identical. Let SCU k p n q denote the number of self-conjugate k -markedstrongly unimodal symbols of n , and write the generating function as SCU k p q q : “ ÿ n ě SCU k p n q q n . When k “ SCU p q q “ ÿ n ě q n p´ q ; q q n ´ . Remark 1.4.
The set of self-conjugate strongly unimodal sequences is in bijection with the set ofpartitions into odd parts where each odd part of size up to the largest part must occur at least once.This can be realized by reading across the rows of the diagram for a self-conjugate strongly unimodalsequence to construct an appropriate partition into odd parts. This explains combinatorially thefollowing interpretation of Ramanujan’s rd order mock theta function ψ p q q as stated in Bryson etal [13] , ψ p q q “ ÿ n ě q n p q ; q q n “ ÿ n ě q n p´ q ; q q n ´ “ SCU p q q , and shows that SCU p n q can be interpreted combinatorially as the number of partitions of n intoodd parts where each odd part of size up to the largest part must occur at least once. Our next result provides a more general combinatorial interpretation for
SCU k p q q when k ě Definition 1.5.
Let ω k p n q count the number of partitions of n into at least k unmarked odd partssuch that every odd part less than the largest part appears at least once, as well as k ´ differentlymarked and distinctly valued p k ´ q -marked even parts (which may repeat) such that each even partis less than twice the number of odd parts and the total number of even parts is odd. Similarly, let ǫ k p n q count the same as above, except where the total number of even parts is even. Theorem 1.6.
For k ě , SCU k p q q “ ÿ n ě p´ q k p ω k p n q ´ ǫ k p n qq q n , where ω k p n q and ǫ k p n q are defined in Definition 1.5. The rest of the paper is organized as follows. In §
2, we review the combinatorial constructionof k -marked Durfee symbols and define k -marked strongly unimodal symbols. In §
3, we proveTheorems 1.2 and 1.6. In § . Combinatorial Constructions
Recall from § n encodes the side length of the Durfeesquare and in addition the lengths of the columns to the right of as well as the rows beneath theDurfee square, as is demonstrated in Figure 1. We now give the full definition of k -marked Durfeesymbol for a positive integer k . Definition 2.1 (Andrews [3]) . A k -marked Durfee symbol is a Durfee symbol using k copies ofpositive integers, denoted t , , . . . u , . . . , t k , k , . . . u , for the parts in both rows. Additionally,when k ě the following are required. (1) In each row the sequence of parts and the sequence of subscripts are nonincreasing. (2)
In the top row, each of , , . . . , k ´ appears as the subscript of some part. (3) If M j is the largest part with subscript ď j ď k ´ in the top row, then all parts in thebottom row with subscript lie in the interval r , M s , with subscript lie in r M , M s , ...with subscript k ´ lie in r M k ´ , M k ´ s , and with subscript k lie in r M k ´ , M k s , where M k is the side length of the Durfee square of the corresponding partition. When we write a k -marked Durfee symbol it is convenient to separate the parts with distinctsubscripts with vertical lines. We can think of the collective parts with given subscript j on thetop or bottom row as a partition α j or β j , respectively. In addition, one way to visualize k -markedDurfee symbols is through a Ferrers diagram in which parts corresponding to different subscriptshave different colors. We demonstrate this below with an example of a 3-marked Durfee symbol of55. Figure 3.
A 3-marked Durfee symbol of 55 ˆ ˙ “ : ˆ α α α β β β ˙ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚‚ ‚ ‚‚ ‚‚ ‚ Using k copies of the integers naturally allows for the definition of k rank statistics on k -markedDurfee symbols, generalizing the Durfee symbol rank given in (1.4). Definition 2.2 (Andrews [3]) . Let γ be a k -marked Durfee symbol and let α j , β j denote thepartitions corresponding to subscript j in the top and bottom rows, respectively. The j th rank of γ ,denoted ρ j p γ q , is defined by ρ j p γ q “ ℓ p α j q ´ ℓ p β j q ´ j ă kℓ p α n q ´ ℓ p β n q j “ k. We note here that the extra 1 is subtracted when j ‰ k because in Definition 2.1 it is requiredthat each subscript 1 , , . . . , k ´ k “ his recovers Dyson’s rank of a partition. In our example from Figure 3, we see the 3rd rank is1, the 2nd rank is 0, and the 1st rank is ´
1. As in § D k p m , m , . . . , m k ; n q denote thenumber of k -marked Durfee symbols of n with i th rank equal to m i .We make the following definition for k -marked strongly unimodal symbols, analogous to Defini-tion 2.1. Definition 2.3. A k -marked strongly unimodal symbol is a strongly unimodal symbol using k copiesof positive integers (denoted with a subscript) for parts in both rows. Additionally, when k ě thefollowing are required. (1) In each row the parts are strictly decreasing and the subscripts are nonincreasing. (2)
In the top row, each of , , . . . , k ´ appears as the subscript of some part. (3) If M j is the largest part with subscript ď j ď k ´ in the top row and M : “ , then allparts in the bottom row with subscript ď j ď k ´ lie in the interval r M j ´ ` , M j s , andthose with subscript k lie in r M k ´ ` , M k ´ s , where M k is the size of the peak of thecorresponding strongly unimodal sequence. The k -marked strongly unimodal symbols can be represented analogously to that of k -markedDurfee symbols. We demonstrate this below with an example of a 3-marked strongly unimodalsymbol of 18. Figure 4.
A 3-marked strongly unimodal symbol of 18 ˆ ˙ “ : ˆ α α α β β β ˙ ‚‚ ‚‚ ‚ ‚ ‚‚ ‚ ‚ ‚ ‚ ‚‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ We can define k ranks on k -marked strongly unimodal symbols exactly as in the Durfee case.Since the definition is the same, we use the notation ρ j p γ q in both settings. Definition 2.4.
Let γ be a k -marked strongly unimodal symbol and let α j , β j denote the partitionscorresponding to subscript j in the top and bottom rows, respectively. The j th rank of γ , denoted ρ j p γ q , is defined by ρ j p γ q “ ℓ p α j q ´ ℓ p β j q ´ j ă kℓ p α n q ´ ℓ p β n q j “ k. We can observe that when k “
1, this recovers the rank of a strongly unimodal sequence. In ourexample from Figure 4, we see the 3rd rank is 0, the 2nd rank is 0, and the 1st rank is ´
1. Welet U k p m , m , . . . , m k ; n q denote the number of k -marked strongly unimodal symbols of n with j thrank equal to m j . 3. Proofs of Results
In this section we prove our two main theorems, beginning with Theorem 1.2. roof of Theorem 1.2. The proof is analogous to that of Andrews [3, Theorem 10]. Consider anarbitrary k -marked strongly unimodal symbol D . As in Definition 2.3, let M k be the size of thepeak of the corresponding strongly unimodal sequence, let M j be the largest part with subscript1 ď j ď k ´ D , and set M : “
0. We define positive integers m , . . . , m k associated to D by setting for each 1 ď j ď k ,(3.1) m j : “ M j ´ M j ´ . Let α j , β j denote the partitions corresponding to subscript j in the top and bottom rows,respectively, as in Definition 2.4. We next show how to generate the pairs α j , β j beginning with j “ α and β , we observe that by condition 3 of Definition 2.3 the parts of α and β must lie in r , M s and be distinct. Also, the part M must exist in α . Furthermore, we needto track the 1st rank by counting the number of parts other than M in α , and subtracting thenumber of parts in β . Thus since M “ m , the parts in α and β are generated by q M p´ x q ; q q m ´ p´ x ´ q ; q q m “ q M p ` x ´ q M qp´ x q ; q q m ´ p´ x ´ q ; q q m ´ . To generate α and β , we observe that by condition 3 of Definition 2.3 the parts must lie in r M ` , M s and be distinct. Also, the part M must exist in α . Furthermore, we need to trackthe 2nd rank. Thus, the parts in α and β are generated by q M p ` x ´ q M qp´ x q M ` ; q q m ´ p´ x ´ q M ` ; q q m ´ . For general 1 ď j ď k ´
1, we have that the parts are distinct and lie in r M j ´ ` , M j s , andthe part M j must exist in α j . Thus to track the j th rank we see that the parts in α j and β j aregenerated by q M j p ` x ´ j q M j qp´ x q M j ´ ` ; q q m j ´ p´ x ´ q M j ´ ` ; q q m j ´ . The parts in α k and β k are distinct and lie in r M k ´ ` , M k ´ s . Since α k is allowed to beempty, the parts in α k and β k are generated more simply by p´ x k q M k ´ ` ; q q m k ´ p´ x ´ k q M k ´ ` ; q q m k ´ . Finally, the peak is generated by q M k .As all of these factors generate their respective parts of D , their product will generate the entiretyof D . To account for all possible values for each m i and the size of U , we sum over all m i ’s with m i ě ď i ď k as well as over all n ě
1. The result follows. (cid:3)
We now prove Theorem 1.6.
Proof of Theorem 1.6.
This proof follows the method of Andrews [3, Theorem 14]. We first derivethe generating function for
SCU k p n q using a similar argument to that in the proof of Theorem 1.2.Consider an arbitrary self-conjugate k -marked strongly unimodal symbol D , letting α j denotethe partition corresponding to subscript j that occurs in both the top and bottom rows of D . Recallthe definition of M k , for 0 ď j ď k given in Definition 2.3, and define the positive integers m j for1 ď j ď k by m j : “ M j ´ M j ´ .The parts in α are distinct, lie in r , M s , and the part M must occur. Thus the parts in α aregenerated by q M p´ q ; q q m ´ . To generate both copies of α (those from both the top and bottomrow of D ), we need two copies of each part; thus q M p´ q ; q q m ´ generates both copies of α . For general 1 ď j ď k ´
1, the two copies of α j are generated by q M j p´ q p M j ´ ` q ; q q m j ´ , hile the two copies of α k which are not required to have a part are generated by p´ q p M k ´ ` q ; q q m k ´ . Lastly, the peak is generated by q Mk . Together, this gives that SCU k p q q “ ÿ m ,...,m k ě q p M `¨¨¨` M k ´ q` M k p´ q ; q q m ´ p´ q p M ` q ; q q m ´ ¨ ¨ ¨ p´ q p M k ´ ` q ; q q m k ´ . We next observe that p´ q ; q q m ´ p´ q p M ` q ; q q m ´ ¨ ¨ ¨ p´ q p M k ´ ` q ; q q m k ´ “ p q ; q q M k ´ p ` q M q ¨ ¨ ¨ p ` q M k ´ q , which allows us to rewrite SCU k p q q as(3.2) SCU k p q q “ ÿ m ,...,m k ě q p M `¨¨¨` M k ´ q` M k p q ; q q M k ´ p ` q M q ¨ ¨ ¨ p ` q M k ´ q“ ÿ M k ě k q M k p´ q ; q q M k ´ ÿ ď M ă M 㨨¨ă M k q p M `¨¨¨` M k ´ q p ` q M q ¨ ¨ ¨ p ` q M k ´ q . We now interpret the right hand side of (3.2) combinatorially. From the perspective of Remark1.4, we see that for each choice of M k ě k , the term q M k p´ q ; q q M k ´ in (3.2) generates self-conjugate strongly unimodal sequences with peak of size M k , or equivalently partitions into M k odd parts where each odd part at most the size of the largest part must occur at least once. Then,in the inner sum, expand each factor as q M j p ` q M j q “ q M j ´ q p M j q ` q p M j q ´ q p M j q ` ¨ ¨ ¨ , and interpret ÿ ď M ă M 㨨¨ă M k q p M `¨¨¨` M k ´ q p ` q M q ¨ ¨ ¨ p ` q M k ´ q as generating the difference in the number of partitions of some n into an odd number of evenparts 2 M , . . . , M k ´ minus the number of partitions of n into an even number of even parts2 M , . . . , M k ´ , where each part 2 M j is j -marked, and 1 ď M ă M ă ¨ ¨ ¨ ă M k . Putting theseinterpretations together gives the result. (cid:3) Concluding Remarks
There are many potential directions to pursue in the study of U k p x , x , ..., x k ; q q . For one, itis natural in the context of our discussions in § § U k p x , x , ..., x k ; q q possesses modularity properties of mock and/or quantum type when p x , . . . , x k q is specialized atvectors of roots of unity. As part of our REU project, we focused on potential quantum modularityproperties. We were able to determine a rational domain for U k p x , x , ..., x k ; q q for certain vectorsof roots of unity, and establish a trivial transformation property (see [2, Section 4]). However, abarrier for us to make more substantial headway is that we only have the multi-sum generatingfunction for U k p x , x , ..., x k ; q q given in Theorem 1.2 to work with. This is in stark contrast tothe situation for k -marked Durfee symbols, for which a q -hypergeometric transformation yieldsa beautiful single sum generating function for R k p x , x , ..., x k ; q q . Despite our efforts, we werenot able to produce a single sum generating function for U k p x , x , ..., x k ; q q . It would be of greatinterest if a single sum generating function for U k p x , x , ..., x k ; q q were discovered. eferences [1] Claudia Alfes, Kathrin Bringmann, and Jeremy Lovejoy. Automorphic properties of generating functions forgeneralized odd rank moments and odd Durfee symbols. Math. Proc. Cambridge Philos. Soc. , 151(3):385–406,2011.[2] Savana Ammons, Young Jin Kim, and Laura Seaberg. An analogue of k -marked durfee sym-bols for strongly unimodal sequences. Oregon State University 2019 REU Proceedings, 2019. http://sites.science.oregonstate.edu/~math_reu/proceedings/REU_Proceedings/Proceedings2019/SavanaYoungLaura.pdf .[3] George E. Andrews. Partitions, Durfee symbols, and the Atkin-Garvan moments of ranks. Invent. Math. ,169(1):37–73, 2007.[4] George E. Andrews. Concave and convex compositions.
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Harvey Mudd College
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