Two Families of Buffered Frobenius Representations of Overpartitions
TTWO FAMILIES OF BUFFERED FROBENIUSREPRESENTATIONS OF OVERPARTITIONS
THOMAS MORRILL
Abstract.
We generalize the generating series of the Dyson ranks and M -ranks of overpartitions to obtain k -fold variants, and give a combinatorial inter-pretation of each. The k -fold generating series correspond to the full ranks oftwo families of buffered Frobenius representations , which generalize Lovejoy’sfirst and second Frobenius representations of overpartitions, respectively. Introduction and Statement of Results A partition of n is a nonincreasing sequence of integers λ = ( (cid:96) , (cid:96) , . . . , (cid:96) k ) suchthat the sum of the (cid:96) i equals n . Each of the (cid:96) i is called a part of λ . We use theterm partition statistic loosely to refer to any integer valued function on the set ofpartitions. For example, the weight of an arbitrary partition λ is the sum of itsparts, | λ | := k (cid:88) i =0 (cid:96) i . We use (cid:96) ( λ ) to denote the largest part of λ , and λ ) to denote the number ofparts of λ .Historically, the theory of partition ranks was developed to give combinatorialevidence for the Ramanujan congruences, which state that for all n ≥ p (5 n + 4) ≡ , (1.1) p (7 n + 5) ≡ , (1.2) p (11 n + 6) ≡ , (1.3)where p ( n ) denotes the number of partitions of n . Given a partition λ , Dyson [9]defined the rank of λ to be r ( λ ) := (cid:96) ( λ ) − λ ) , that is, the largest part of λ minus the number of parts of λ . For example, thepartitions of 4 are given with their ranks in Table 1. Note that p (4) = 5, whichagrees with (1.1).Moreover, each equivalence class of Z / Z appears exactly once in the second rowof Table 1. Atkin and Swinnerton-Dyer [5] proved that for all n ≥ i, j ∈ Z , N ( i, n + 4 ,
5) = N ( j, n + 4 , , (1.4) Date : October 12, 2018.1991
Mathematics Subject Classification.
Primary 11P81; Secondary 05A17.
Key words and phrases. basic hypergeometric series, overpartitions, rank, conjugation, Frobe-nius symbols. a r X i v : . [ m a t h . N T ] F e b MORRILL λ (4) (3 ,
1) (2 ,
2) (2 , ,
1) (1 , , , r ( λ ) 3 1 0 − − Table 1.
Ranks of the partitions of 4.where N ( m, n, k ) denotes the number of partitions of n with rank m modulo k .Consequently, the set of partitions of 5 n + 4 can be separated into five classes ofequal size by their ranks, which proves (1.1) via a counting argument. Atkin andSwinnerton-Dyer also proved that N ( i, n + 5 ,
7) = N ( j, n + 5 , , which treats (1.2) similarly. However, it is easy to confirm that N ( i, n + 6 ,
11) = N ( j, n + 6 , n = 0. A counting argument for (1.3) was later found byusing the partition crank function, which was predicted by Dyson [9] and laterdefined by Andrews and Garvan [4].We now generalize. An overpartition is a nonincreasing sequence of positive in-tegers λ = ( (cid:96) , (cid:96) , . . . , (cid:96) k ), where the first occurrence of each part may be overlined.For example, the fourteen overpartitions of 4 are given by(4) (4) (3 ,
1) (3 ,
1) (3 , ,
1) (2 ,
2) (2 ,
2) (2 , ,
1) (2 , , , ,
1) (2 , ,
1) (1 , , ,
1) (1 , , , . Since every partition is an overpartition, we retain the notation | λ | , (cid:96) ( λ ), and λ )for the weight, largest part, and number of parts of an overpartition λ , respectively.It is useful to represent partitions or overpartitions graphically as arrays of boxes.The Young tableau of a partition or overpartition λ = ( (cid:96) , (cid:96) , . . . , (cid:96) k ) is a left alignedarray where the i th row of the array consists of (cid:96) i boxes. For overpartitions, if thefirst occurrence of the integer (cid:96) is overlined in λ , then we mark the last row of (cid:96) boxes with a dot . An example is given in Figure 1. Figure 1.
The Young tableau for (4 , , ,
1) and its conjugate, (4 , , , This convention ensures that mirroring the diagram across its main diagonal will produce theYoung tableau of another overpartition, more commonly known as conjugating the overpartition.
UFFERED FROBENIUS REPRESENTATIONS 3
The
Dyson rank of an overpartition λ is defined to be r D ( λ ) = (cid:96) ( λ ) − λ ) , an extension of Dyson’s rank function for ordinary partitions. For example, if λ = (4 , , , r D ( λ ) = 0. We see the generating series for the Dyson ranksof overpartitions in the following theorem. Theorem 1.1 (Lovejoy [12]) . The coefficient of z m q n in the series R [1]( z, q ) := ( − q ; q ) ∞ ( q ; q ) ∞ (cid:88) n ≥ (1 − z )(1 − z − )( − n q n + n (1 − zq n )(1 − z − q n ) (1.5) is equal to the number of overpartitions λ with | λ | = n and r D ( λ ) = m . Lovejoy also developed an M -rank for overpartitions [13], which expands onBerkovich and Garvan’s M -rank for ordinary partitions whose odd parts cannotrepeat [6]. Given an overpartition λ = ( (cid:96) , (cid:96) , . . . , (cid:96) k ), the M -rank of λ is definedto be r M ( λ ) := (cid:24) (cid:96) ( λ )2 (cid:25) − λ ) + λ o ) − χ ( λ ) , where λ o is the subpartition of λ consisting of all non-overlined odd parts of λ , and χ ( λ ) is defined to be χ ( λ ) := (cid:40) , if the largest part of λ is both odd and non-overlined0 , otherwise.For example, let λ = (2 , , λ o = (1 , r M ( λ ) =1 − − M -ranks of overpartitionsin the following theorem. Theorem 1.2 (Lovejoy [13]) . The coefficient of z m q n in the series R [2]( z ; q ) := ( − q ; q ) ∞ ( q ; q ) ∞ (cid:88) n ≥ (1 − z )(1 − z − )( − n q n +2 n (1 − zq n )(1 − z − q n ) (1.6) is equal to the number of overpartitions λ with | λ | = n and r M ( λ ) = m . The proofs of these theorems are based on Lovejoy’s first and second Frobeniusrepresentations for overpartitions [12] [13], which we summarize in Section 2. Notethe similarity in the summands in (1.5) and (1.6); they are identical apart from theexponents of q in the summation.We now continue this pattern. For k ≥
1, define the series R [ k ]( z, q ) := ( − q ; q ) ∞ ( q ; q ) ∞ (cid:18) ∞ (cid:88) n =1 (1 − z )(1 − z − )( − n q n + kn (1 − zq kn )(1 − z − q kn ) (cid:19) . (1.7)It is natural to ask is if R [ k ]( z, q ) can be interpreted as the generating series of anoverpartition rank. In this paper we give a partial answer in terms of Frobeniusrepresentations. We may think of a Frobenius representation as an array (cid:18) a a . . . a k b b . . . b k (cid:19) , MORRILL where λ = ( a , a , . . . , a k ) and µ = ( b , b , . . . , b k ) are partitions or overpartitions.As we will see in Section 2, certain Frobenius representations correspond bijectivelyto overpartitions.In Section 3, we introduce buffered Frobenius representations , which are arraysof the form (cid:18) α α . . . α k β β . . . β k (cid:19) , where each of the entries α i and β i are partitions or overpartitions. A bufferedFrobenius representation can be interpreted as an exploded Young tableau for anordinary Frobenius representation ( λ, µ ) T . Thus, every overpartition admits mul-tiple buffered Frobenius representations.We now present our first main result, which interprets R [ k ]( z, q ) in terms ofbuffered Frobenius representations. Theorem 1.3.
Let ζ k be a primitive k th root of unity. The coefficient of z mk q n in R [ k ]( z, q ) is equal to the weighted count of buffered Frobenius representations of thefirst kind ν with at most k columns, | ν | = n , and full rank m , where the count isweighted by ( − h ( ν ) k (cid:89) i =1 ζ ( i − ρ i ( ν ) k . In particular, the count vanishes for buffered Frobenius representations whose fullrank is not a multiple of k . Following Lovejoy’s work on the M -rank and the second Frobenius representa-tion of an overpartition [13], our second main result interprets R [2 k ]( z, q ) in termsof a second family of buffered Frobenius representations. Theorem 1.4.
Let ζ k be a primitive k th root of unity. The coefficient of z mk q n in R [2 k ]( z, q ) is equal to the weighted count of buffered Frobenius representationsof the second kind ν with at most k columns, | ν | = n , and full rank m , where thecount is weighted by ( − h ( ν ) k (cid:89) i =1 ζ ( i − ρ i ( ν ) k . In particular, the count vanishes for buffered Frobenius representations whose fullrank is not a multiple of k . Each of these families is equipped with k rank functions, ρ i ( ν ) and ρ i ( ν ), respec-tively, and k rank-reversing conjugation maps, which are developed in Sections 4and 5. The observant reader will note that R [ k ]( z, q ) and R [2 k ]( z, q ) are generatingseries for the ranks of buffered Frobenius representations, rather than for the ranksof overpartitions. We discuss this gap and the potential for improvement in Section6. The organization of this paper is as follows. In Section 2, we outline our q -series techniques and summarize the motivating results for the Dyson rank and M -rank. In Section 3, we define a generic buffered Frobenius representation andgive a combinatorial map from buffered Frobenius representations to generalizedFrobenius representations. This allows us to construct our first family of buffered UFFERED FROBENIUS REPRESENTATIONS 5
Frobenius representations and prove Theorem 1.3 in Section 4. Then, in Section5, we construct our second family of buffered Frobenius representations and proveTheorem 1.4. Finally, we give our closing remarks in Section 6.2.
Preliminaries
The q -Pochhammer Symbol and q -Hypergeometric Series. We beginwith the definition of the q -Pochhammer symbol and its conventional shorthandnotations. For a ∈ C , define( a ; q ) n := n − (cid:89) i =0 (1 − aq i )(2.1) ( a ; q ) ∞ := ∞ (cid:89) i =0 (1 − aq i )(2.2) ( a , a , . . . , a k ; q ) n := ( a ; q ) n ( a ; q ) n · · · ( a k ; q ) n (2.3) ( a , a , . . . , a k ; q ) ∞ := ( a ; q ) ∞ ( a ; q ) ∞ · · · ( a k ; q ) ∞ . (2.4)Manipulating q -Pochhammer symbols typically entails expanding the product andcanceling individual factors, as seen in the following lemma. Lemma 2.1.
For all nonnegative integers m and n , ( a ; q ) m ( aq ; q ) m + n = (1 − a )( aq m ; q ) n +1 Proof.
The case m = 0, 1( aq ; q ) n = (1 − a )( a ; q ) n +1 , is trivial.Next, consider m >
0. By expanding the q -Pochhammer symbol and cancelinglike terms, we have( a ; q ) m ( aq ; q ) m + n = (1 − a ) · · · (1 − aq m − )(1 − aq ) · · · (1 − aq m − )(1 − aq m ) · · · (1 − aq m + n )= (1 − a )(1 − aq m ) · · · (1 − aq m + n ) = (1 − a )( aq m ; q ) n +1 . (cid:3) The q -Pochhammer symbol is necessary for the definition of the q -hypergeometricseries, r Φ r − (cid:20) a , a , a , . . . , a r b , b , . . . , b r − ; q ; z (cid:21) := (cid:88) n ≥ ( a , a , . . . , a r ; q ) n z n ( b , b , . . . , b r − , q ; q ) n . (2.5)These series admit many beautiful transformation formulas; see Gasper and Rah-man [10] for examples. In this paper, we only require Andrews’ k -fold generalizationof the Watson-Whipple transformation. MORRILL
Theorem 2.2 (Andrews [1]) . Let a, b , c , b , c , . . . , b k , c k be complex numbers, andlet k ≥ and N ≥ . Then, (2.6) k +4 Φ k +3 a, a q, − a q, b , c , b , c , . . . , b k , c k , q − N a , − a , aqb , aqc , . . . , aqb k , aqc k , aq N +1 ; q ; a k q k + N b c · · · b k c k = ( aq, aqb k c k ; q ) N ( aqb k , aqc k ; q ) N (cid:88) n ,...,n k − ≥ ( aqb c ; q ) n ( q ; q ) n · · · ( aqb k − c k − ; q ) n k − ( q ; q ) n k − × ( b , c ; q ) N ( aqb , aqc ; q ) N ( b , c ; q ) N ( aqb , aqc ; q ) N · · · ( b k , c k ; q ) N k − ( aqb k − , aqc k − ; q ) N k − × ( q − N ; q ) N k − ( b k c k q − N a ; q ) N k − ( aq ) N + N + ··· + N k − q N k − ( b c ) N ( b c ) N · · · ( b k − c k − ) N k − , where we write N = 0 and N i = n + n + · · · + n i for all i ≥ . Observe that the left hand side of (2.6) is a symmetric function in the variables b , c , b , c , . . . , b k , c k . Thus, we may permute the indices of b i and c i on the righthand side while leaving the corresponding indices fixed on the left hand side. Wemap 1 (cid:55)→ ( k − , (cid:55)→ ( k − , . . . , ( k − (cid:55)→ , k (cid:55)→ k, which gives the following corollary to Theorem 2.2. Corollary 2.3.
Let a, b , c , b , c , . . . , b k , c k be complex numbers, and let k ≥ and N ≥ . Then, k +4 Φ k +3 a, a q, − a q, b , c , b , c , . . . , b k , c k , q − N , a , − a , aqb , aqc , . . . , aqb k , aqc k , aq N +1 ; q ; a k q k + N b c · · · b k c k = ( aq, aqb k c k ; q ) N ( aqb k , aqc k ; q ) N (cid:88) n ,...,n k − ≥ ( aqb k − c k − ; q ) n ( q ; q ) n · · · ( aqb c ; q ) n k − ( q ; q ) n k − × ( b k − , c k − ; q ) N ( aqb k − , aqc k − ; q ) N ( b k − , c k − ; q ) N ( aqb k − , aqc k − ; q ) N · · · ( b , c ; q ) N k − ( aqb , aqc ; q ) N k − ( b k , c k ; q ) N k − ( aqb , aqc ; q ) N k − × ( q − N ; q ) N k − ( b k c k q − N a ; q ) N k − ( aq ) N + N + ··· + N k − q N k − ( b k − c k − ) N ( b k − c k − ) N · · · ( b c ) N k − , where we write N = 0 and N i = n + n + · · · + n i for all i ≥ . We now summarize Lovejoy’s work on the Dyson rank and M -rank.2.2. Summary of Lovejoy’s Work.
In this context, it is convenient to allow par-titions and overpartitions to contain 0 as a part, such as λ = (3 , , , , partitions into nonnegative parts and overpartitions into nonnegative parts ,respectively . The reader may consider this approach as a way for shorter parti-tions and overpartitions to attain a longer length requirement. For example, we canadmit (3 ,
3) in contexts where a partition with exactly five parts is required. This When unspecified, the terms partition and overpartition should be taken to mean partitionsand overpartitions into positive parts.
UFFERED FROBENIUS REPRESENTATIONS 7 is a common technique when working with generalized Frobenius representations,which we now define.
Definition 2.4 (Andrews [3]) . Let A and B be sets of partitions or overpartitions,possibly into nonnegative parts. A generalized Frobenius representation is a tworowed array ν = (cid:18) a a . . . a k b b . . . b k (cid:19) where ( a , a , . . . , a k ) ∈ A , and ( b , b , . . . , b k ) ∈ B .We define the weight of a generalized Frobenius representation to be the sum ofits entries , | ν | := k (cid:88) i =1 ( a i + b i ) . For example, (cid:18) (cid:19) is a generalized Frobenius representation with weight 28. The top row is an ordinarypartition, and the bottom row is an overpartition into nonnegative parts. With thecorrect choice of sets A and B , the corresponding Frobenius representations areequivalent to overpartitions, as seen in the following theorem. Theorem 2.5 (Corteel, Lovejoy [8]) . There is a bijection between overpartitions λ and generalized Frobenius representations ν = ( α, β ) T where α is a partition intodistinct parts and β is an overpartition into nonnegative parts such that | λ | = | ν | . Using this bijection, we can define the Dyson rank of ν to be r D ( λ ). We see agenerating series for the Dyson ranks of Frobenius representations in the followinglemma. Lemma 2.6 (Lovejoy [12]) . The coefficient of z m q n in the series ∞ (cid:88) n =0 ( − q ) n q n + n ( zq, z − q ; q ) n is equal to the number of generalized Frobenius representations ν = ( α, β ) T with | ν | = n , where α is a partition into distinct parts and β is an overpartition intononnegative parts, and r D ( ν ) = m . Thus, Theorem 1.1 reduces to the following q -series transformation. Lemma 2.7 (Lovejoy [12]) . For z (cid:54) = 0 , (2.7) ( − q ; q ) ∞ ( q ; q ) ∞ (cid:18) ∞ (cid:88) n =1 (1 − z )(1 − z − )( − n q n + n (1 − zq n )(1 − z − q n ) (cid:19) = ∞ (cid:88) n =0 ( − q ) n q n + n ( zq, z − q ; q ) n . Note that Lovejoy uses Andrews’ convention | ν | = k + (cid:80) ( a i + b i ) in his earlier work [12].Statements of these results have been adjusted for consistency. MORRILL
The proof of Lemma 2.7 involves a limiting case of the q -Watson-Whipple trans-formation, or equivalently, the case k = 2 in Theorem 2.2. Full details of thetransformation may be seen as the case k = 1 in Section 4. We now state thealgorithm which produces the bijection in Theorem 2.5. Algorithm 2.1 (Corteel, Lovejoy [8]) . Input: A Frobenius representation ν = (cid:18) a a . . . a k b b . . . b k (cid:19) as described in Proposition 2.5.Output: An overpartition λ such that | λ | = | ν | . (1) Initialize λ = λ = ∅ . (2) We treat λ as a partition into b k nonnegative parts. Delete b k from ν andadd 1 to each part of λ . (3) Delete a k from ν . If b k was overlined, append a k as a part of λ . Otherwise,if b k was not overlined, append a k as a part of λ . (4) Repeat Steps (2) and (3) until all parts of ν are exhausted. (5) Because ( a , a , . . . , a k ) was a partition into distinct parts, λ is also apartition into distinct parts. We define the output λ to be the overpartitionwith non-overlined parts given by λ and overlined parts given by λ . An example of Algorithm 2.1 is shown in Table 2. Further details may be foundin work of Lovejoy [12].Iteration α β λ λ , ,
1) (4 , , ∅ ∅ ,
2) (4 ,
4) (1 , , , ∅ , , ,
2) (2)3 ∅ ∅ (3 , , , ,
3) (2)
Table 2.
A demonstration of Algorithm 2.1. This produces theoverpartition λ = (3 , , , , , M -rank involves a second family of Frobeniusrepresentations, which appear in the following theorem. Theorem 2.8 (Lovejoy[13]) . There is a bijection between overpartitions λ andgeneralized Frobenius partitions ν = ( α, β ) T where α is an overpartition into oddparts and β is a partition into nonnegative parts where odd parts may not repeatsuch that | λ | = | ν | . As was the case with the Dyson rank, we can define the M -rank of ν to be r M ( λ ). We see a generating series for the M -ranks of Frobenius representationsin the following lemma. Lemma 2.9 (Lovejoy [13]) . The coefficient of z m q n in the series (cid:88) n ≥ ( − q ) n q n ( zq , z − q ; q ) n UFFERED FROBENIUS REPRESENTATIONS 9 is equal to the number of Frobenius representations ν = ( α, β ) T with | ν | = n , where α is an overpartition into odd parts and β is a partition into nonnegative parts, and r M ( ν ) = m . Then Theorem 1.2 reduces to the following q -series transformation. Lemma 2.10 (Lovejoy [13]) . For z (cid:54) = 0 , ( − q ; q ) ∞ ( q ; q ) ∞ (cid:88) n ≥ (1 − z )(1 − z − )( − n q n +2 n (1 − zq n )(1 − z − q n ) = (cid:88) n ≥ ( − q ) n q n ( zq , z − q ; q ) n . As before, the proof utilizes a limiting case of the q -Watson-Whipple transfor-mation. Full details may be seen as the case k = 1 in Section 5. We now state thealgorithm which gives the bijection in Theorem 2.8. Algorithm 2.2 (Lovejoy [13]) . Input: A Frobenius representation ν = (cid:18) αβ (cid:19) = (cid:18) a a . . . a k b b . . . b k (cid:19) as described in Theorem 2.8.Output: An overpartition λ such that | λ | = | ν | . (1) Initialize λ = ∅ . (2) For each odd integer n < a which does not appear overlined in α , we insert n in its correct position in α . We also append − n as a part of β . (3) Reindex the parts of β so that from left to right, odd integers appear inincreasing order, followed by even integers in decreasing order. (4) For each pair ( a i , b i ) , let (cid:96) i = a i + b i . If b i is even, append (cid:96) i as a part of λ with the same overline marking as a i . If b i is odd, append (cid:96) i as a part of λ with the opposite overline marking as a i . Reindex the (cid:96) i in non-increasingorder, with the convention that n > n . Step α β λ ,
1) (6 , ∅ , ,
1) (6 , , − ∅ , ,
1) ( − , , ∅ ∅ ∅ (8 , , Table 3.
Demonstration of Algorithm 2.2.An example of Algorithm 2 is demonstrated in Table 3. The reverse algorithm is amodification of Corteel and Lovejoy’s work on vector partitions [7]. We present itbelow for completeness. For this algorithm, we let s ( λ ) denote the smallest part ofthe overpartition λ . Algorithm 2.3 (Corteel, Lovejoy [7] [13]) . Input: An overpartition λ .Output: A second Frobenius representation ν = ( α, β ) T such that | ν | = | λ | . (1) Initialize α = β := ∅ and a := 1 . Dissect λ into four partitions π e , π e , π o ,and π o as follows. Let π e be the subpartition consisting of all even overlinedparts of λ . Let π e be the subpartition consisting of all even non-overlinedparts of λ . We define π o and π o analogously for the odd parts of λ . (2) If π o = ∅ , or if s ( π o ) ≤ s ( π o ) , then append a as a part of α , append s ( π o ) − a as a part of β , and delete the smallest part of π o . (3) Otherwise, append a as a part of α , append s ( π o ) − a as a part of β , deletethe smallest part of π o , and set a := a + 2 . (4) Repeat Steps (2) and (3) until both π o and π o are exhausted. (5) If π e = ∅ , or if s ( π e ) < s ( π o ) , then append a as a part of α , append s ( π e ) − a as a part of β , and delete the smallest part of π e . (6) Otherwise, append a as a part of α , append s ( π e ) − a as a part of β , deletethe smallest part of π e , and set a := a + 2 . (7) Repeat Steps (5) and (6) until both π o and π o are exhausted. (8) If a part − n occurs in β , delete both − n from β and n from α . An example of Algorithm 2.3 is given in Table 4.This ends our presentation of previous results. We now introduce the notion ofbuffered Frobenius representations.Iteration π e π e π o π o a α β ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ,
1) (6 , ∅ ∅ ∅ ∅ , ,
1) (6 , , − ∅ ∅ ∅ ∅ ,
1) (6 , Table 4.
Demonstration of Algorithm 2.3.3.
Buffered Frobenius Representations
We use the following abbreviated notation for the rest of the paper. If A , A ,. . . , A k and B , B , . . . , B k are sets, we write (cid:18) α α . . . α k β β . . . β k (cid:19) ∈ (cid:18) A A . . . A k B B . . . B k (cid:19) to mean that α i ∈ A i and β i ∈ B i for all 1 ≤ i ≤ k . Definition 3.1.
Let P denote the set of overpartitions into nonnegative parts, andlet P denote the set of partitions into nonnegative parts. A buffered Frobeniusrepresentation is a two rowed array ν = (cid:18) α α . . . α k β β . . . β k (cid:19) ∈ (cid:18) P P . . . P P P . . . P (cid:19) , where for all i , we have α i ) ≥ α i +1 ) and β i ) = α i ). Additionally, wemay mark either of α i or β i with a hat if i < k .The weight of a buffered Frobenius representation is defined to be | ν | := (cid:88) ≤ i ≤ k | α i | + | β i | . UFFERED FROBENIUS REPRESENTATIONS 11
We see that every generalized Frobenius representation as in Section 2 (cid:18) a a . . . a k b b . . . b k (cid:19) can be interpreted as a buffered Frobenius representation (cid:18) α β (cid:19) = (cid:18) ( a , a , . . . , a k )( b , b , . . . , b k ) (cid:19) , although this only produces simple examples. The hat notation serves to enrichthe combinatorics of buffered Frobenius representations, similar to the purpose ofoverlining the parts of an overpartition. For example, (cid:18) α α β β (cid:19) = (cid:32) (cid:92) (3 , , ,
1) (1 , , , , ,
2) (4 , , (cid:33) (3.1)is a buffered Frobenius representation. Note that (cid:96) ( β ) > (cid:96) ( β ); only the sequences { α i ) } and { β i ) } must be nonincreasing.3.1. Buffered Young Tableaux.
Given a buffered Frobenius representation ν = (cid:18) α α . . . α k β β . . . β k (cid:19) we construct buffered Young tableaux to represent the entries of ν by using k colorsas follows.First, we draw the Young tableau for α in the first color. Next, we draw theYoung tableau for α in the second color. However, we align the boxes for α to theright edge of the tableau for α . If α is marked with a hat, we shift the tableaufor α to the right by one unit and leave a buffer between α and α . For example,if α = (cid:92) (3 , ,
1) and α = (2 , , Figure 2.
The buffered Young tableaux for α = (cid:92) (3 , ,
1) and α = (2 , , α i in the i th color, aligned tothe right edge of the preceding tableau, and shifted to the right by one unit if α i is marked with a hat. We draw the tableaux for the β i in the same manner. Forexample, Figure 3 shows the buffered Young tableaux for the buffered Frobeniusrepresentation in (3.1).Note that entries marked with a hat increase the width of the tableaux withoutincreasing the number of boxes. There are no tableaux which could indicate a bufferto the right of α k or β k , which corresponds to the restriction that neither α k or β k can be marked with a hat. Figure 3.
The buffered Young Tableaux for the buffered Frobe-nius representations in (3.1).3.2.
The Jigsaw Map.
Visualizing buffered Frobenius representations by theirtableaux suggests that we should interpret buffered Frobenius representations as theexploded Young tableaux of generalized Frobenius representations. To reassemblethe generalized Frobenius representation, we use the jigsaw map .Let ν be a buffered Frobenius representation ν = (cid:18) α α . . . α k β β . . . β k (cid:19) , where for all i , α i = ( a ( i, , a ( i, , . . . , a ( i,k i ) ) β i = ( b ( i, , b ( i, , . . . , b ( i,k i ) ) . We seek to construct a generalized Frobenius representation j ( ν ) = (cid:18) a a . . . a k b b . . . b k (cid:19) , where ( a , a , . . . , a k ) and ( b , b , . . . , b k ) are partitions or overpartitions into non-negative parts.First, discard any hats from the entries of ν . We then rewrite each α i and β i asa partition into k nonnegative parts, α i = ( k (cid:122) (cid:125)(cid:124) (cid:123) a ( i, , a ( i, , . . . , a ( i,k i ) , , . . . , ,β i = ( k (cid:122) (cid:125)(cid:124) (cid:123) b ( i, , b ( i, , . . . , b ( i,k i ) , , . . . , . For all 1 ≤ j ≤ k , we define the integers a j to be a j = k (cid:88) i =1 a ( i,j ) ,b j = k (cid:88) i =1 b ( i,j ) . UFFERED FROBENIUS REPRESENTATIONS 13
Finally, we overline a j or b j if and only if the j th part of α or β is overlined, re-spectively . Graphically, this is equivalent to removing the colors from the bufferedYoung tableaux and aligning the boxes to the left, with careful attention paid tothe convention for overlined parts.We now move away from the generic treatment in order to present Theorem 1.3.4. Buffered Frobenius Representations of the First Kind
In order to apply Corollary 2.3 to R [ k ]( z, q ), we consider the series(4.1) R k ( x , x , . . . , x k ; q ):= ( − q ; q ) ∞ ( q ; q ) ∞ (cid:18) ∞ (cid:88) n =1 ( − n q n + kn k (cid:89) i =1 (1 − x i )(1 − x − i )(1 − x i q n )(1 − x − i q n ) (cid:19) , bearing in mind that R k ( k √ z, ζ k k √ z, . . . , ζ k − k k √ z ; q ) = R [ k ]( z, q ) . We see a transformation of R k ( x , x , . . . , x k ; q ) in the theorem below. Theorem 4.1.
Let k ≥ be a positive integer. Then we have ( − q ; q ) ∞ ( q ; q ) ∞ (cid:18) ∞ (cid:88) n =1 ( − n q n + kn k (cid:89) i =1 (1 − x i )(1 − x − i )(1 − x i q n )(1 − x − i q n ) (cid:19) = (cid:88) n ,...,n k ≥ ( − q ) N k q N k − N k k (cid:89) i =1 (1 − x k − i +1 )(1 − x − k − i +1 ) q N i ( x k − i +1 q N i − , x − k − i +1 q N i − ) n i +1 , where we write N = 0 and N i = n + n + · · · + n i for all i ≥ .Proof. We begin by substituting k (cid:55)→ k +1 into Corollary 2.3. Letting N → ∞ turnsthe transformation of terminating series into a transformation of infinite series. Theleft side becomes ∞ (cid:88) n =0 ( a, qa , − qa , b , c , . . . , b k +1 , c k +1 ; q ) n ( − n q n − n ( q, a , − a , aqb , aqc , . . . , aqb k +1 , aqc k +1 ; q ) n × (cid:18) a k +1 q k +1 b c · · · b k +1 c k +1 (cid:19) n . When n = 0, the q -Pochhammer symbols take their trivial value, and the summandis equal to 1. For n >
0, we may simplify the summand using the relation( a, qa , − qa ; q ) n ( a , − a ; q ) n = (1 − aq n )( aq ; q ) n − . (4.2) This is why only α and β may be overpartitions. Thus the left hand side is equal to1 + ∞ (cid:88) n =1 (1 − aq n )( aq ; q ) n − ( b , c , . . . , b k +1 , c k +1 ; q ) n ( − n q n − n ( q, aqb , aqc , . . . , aqb k +1 , aqc k +1 ; q ) n × (cid:18) a k +1 q k +1 b c · · · b k +1 c k +1 (cid:19) n . On the right hand side, we use the relationlim N →∞ ( q − N ; q ) N k ( a − b k +1 c k +1 q − N ; q ) N k = lim N →∞ N k − (cid:89) i =0 ( q N − q i )( q N − a − b k +1 c k +1 q i )(4.3) = N k − (cid:89) i =0 − q i − a − b k +1 c k +1 q i = (cid:18) ab k +1 c k +1 (cid:19) N k (4.4)to obtain( aq, aqb k +1 c k +1 ; q ) ∞ ( aqb k +1 , aqc k +1 ; q ) ∞ (cid:88) n ,...,n k ≥ ( aqb k c k ; q ) n ( q ; q ) n · · · ( aqb c ; q ) n k ( q ; q ) n k × ( b k − , c k − ; q ) N ( aqb k , aqc k ; q ) N ( b k − , c k − ; q ) N ( aqb k − , aqc k − ; q ) N · · · ( b , c ; q ) N k − ( aqb , aqc ; q ) N k − × ( b k +1 , c k +1 ; q ) N k ( aqb , aqc ; q ) N k ( aq ) N + N + ··· + N k ( b k c k ) N · · · ( b c ) N k − ( b k +1 c k +1 ) N k . Setting a = 1, the equation becomes1 + ∞ (cid:88) n =1 (1 + q n ) ( b , c , . . . , b k +1 , c k +1 ; q ) n ( − n q n − n ( qb , qc , . . . , qb k +1 , qc k +1 ; q ) n (cid:18) q k +1 b c · · · b k +1 c k +1 (cid:19) n = ( q, qb k +1 c k +1 ; q ) ∞ ( qb k +1 , qc k +1 ; q ) ∞ (cid:88) n ,...,n k − ≥ ( qb k c k ; q ) n ( q ; q ) n · · · ( qb c ; q ) n k ( q ; q ) n k × ( b k − , c k − ; q ) N ( qb k , qc k ; q ) N ( b k − , c k − ; q ) N ( qb k − , qc k − ; q ) N · · · ( b , c ; q ) N k − ( qb , qc ; q ) N k − × ( b k +1 , c k +1 ; q ) N k ( aqb , qc ; q ) N k q N + N + ··· + N k ( b k c k ) N · · · ( b c ) N k − ( b k +1 c k +1 ) N k . We set b i = x i , c i = x − i for 1 ≤ i ≤ k , and b k +1 = −
1. This cancels the term( − n b nk +1 . On the left hand side, we use the identity(1 + q n ) ( − q ) n ( − q ; q ) n = 2 , and obtain 1 + 2 ∞ (cid:88) n =1 ( x , x − , . . . , x k , x − k , c k +1 ; q ) n ( x q, x − q, . . . , x k q, x − k q, c − k +1 q ; q ) n q n − n k +1) n c nk +1 . UFFERED FROBENIUS REPRESENTATIONS 15
The right hand side becomes( q, − qc k +1 ; q ) ∞ ( − q, qc k +1 ; q ) ∞ (cid:88) n ,...,n k ≥ ( x k − , x − k − ; q ) N ( x k q, x − k q ; q ) N × ( x k − , x − k − ; q ) N ( x k − q, x − k − q ; q ) N · · · ( x , x − ; q ) N k − ( x q, x − q ; q ) N k − × ( − , c k +1 ; q ) N k ( x q, x − q ; q ) N k q N + N + ··· + N k ( − c k +1 ) N k . We now let c k +1 → ∞ . On the left hand side, we use the simple identitieslim c k +1 →∞ ( c k +1 ; q ) n c nk +1 = ( − n q n − n (4.5) lim c k +1 →∞ ( c − k +1 q ; q ) n = 1(4.6)to obtain 1 + 2 ∞ (cid:88) n =1 ( x , x − , . . . , x k , x − k ; q ) n ( − n q n + kn ( x q, x − q, . . . , x k q, x − k q ; q ) n . On the right hand side, applying (4.5) and (4.6) produces( q ; q ) ∞ ( − q ; q ) ∞ (cid:88) n ,...,n k ≥ ( x k − , x − k − ; q ) N ( x k q, x − k q ; q ) N · · · ( x , x − ; q ) N k − ( x q, x − q ; q ) N k − × ( − q ) N k ( x q, x − q ; q ) N k q N + N + ··· + N k − + N k + N k . Applying Lemma 2.1 to the left hand side of the equation and multiplying by ( − q ; q ) ∞ ( q ; q ) ∞ gives us( − q ; q ) ∞ ( q ; q ) ∞ (cid:18) ∞ (cid:88) n =1 ( − n q n + kn k (cid:89) i =1 (1 − x i )(1 − x − i )(1 − x i q n )(1 − x − i q n ) (cid:19) . On the right hand side of the equation, we use the fact that N i = N i − + n i for all1 ≤ i ≤ k with Lemma 2.1 to write( x ; q ) N i − ( xq ; q ) N i = (1 − x )( xq N i − ; q ) n i +1 . Multiplying the right hand side of the equation by ( − q ; q ) ∞ ( q ; q ) ∞ gives (cid:88) n ,...,n k ≥ ( − q ) N k q N k − N k N ( x k q, x − k q ) n (cid:32) k (cid:89) i =2 (1 − x k − i +1 )(1 − x − k − i +1 ) q N i ( x k − i +1 q N i − , x − k − i +1 q N i − ) n i +1 (cid:33) . Finally, as N := 0, we may rewrite the right hand side using1( x k q, x k − q ; q ) n = (1 − x k )(1 − x − k )( x k q N , x − k q N ; q ) n +1 , which gives us the desired equation,(4.7) ( − q ; q ) ∞ ( q ; q ) ∞ (cid:18) ∞ (cid:88) n =1 ( − n q n + kn k (cid:89) i =1 (1 − x i )(1 − x − i )(1 − x i q n )(1 − q n x − i ) (cid:19) = (cid:88) n ,...,n k ≥ ( − q ) N k q N k − N k k (cid:89) i =1 (1 − x k − i +1 )(1 − x − k − i +1 ) q N i ( x k − i +1 q N i − , x − k − i +1 q N i − ) n i +1 . (cid:3) Overpartition Statistics.
In order to interpret (4.7) as a generating series,we must introduce some partition and overpartition statistics. The first statisticwe consider appears in Franklin’s proof of Euler’s pentagonal number theorem [2].We will use several variations of this statistic, so we take the opportunity to nameit the bracket of a partition.Given a partition λ = ( (cid:96) , (cid:96) , . . . , (cid:96) n ), the bracket of λ is defined to be the lengthof the longest sequence of the form ( (cid:96) , (cid:96) , . . . , (cid:96) k ), where for all 1 ≤ i < k , we have (cid:96) i = (cid:96) i +1 + 1. We retain Andrews’ notation of σ ( λ ) to denote the bracket of λ .For example, if λ = (7 , , , , , (7 , , (7 , , , the longest of which has length three. Therefore, σ ( λ ) = 3.We see how the partition rank and the partition bracket relate to (4.7) in thefollowing lemma. Lemma 4.2.
Fix nonnegative integers ≤ s ≤ t . The coefficient of z m q n in q t + t ( zq s ; q ) t − s +1 is equal to the number of partitions λ of n into t distinct parts with σ ( λ ) ≥ s and r ( λ ) = m .Proof. The term 1( zq s ; q ) t − s +1 = 1(1 − zq s ) 1(1 − zq s +1 ) · · · − zq t )generates the columns of a Young tableau, where m tracks the number of columnsgenerated. The length of these columns is bounded between s and t . Then we mayconsider λ as a partition into exactly t nonnegative parts, λ = ( (cid:96) , (cid:96) , . . . , (cid:96) t ). Notethat λ has at least s occurrences of its largest part, that is, (cid:96) = (cid:96) = · · · = (cid:96) s .To account for q t + t , we add a staircase to λ . That is, we add t to the firstpart, t − λ contains the sequence ( (cid:96) + t, (cid:96) + ( t − , . . . , (cid:96) + ( t − s + 1)), which implies that σ ( λ ) ≥ s . Finally, since (cid:96) ( λ ) = m + t and λ ) = t , we see that r ( λ ) = m . (cid:3) We also need an overpartition statistic introduced by Corteel and Lovejoy [8][12]. Given an overpartition λ , the overpartition rank of λ is defined to be r CL ( λ ) := (cid:96) ( λ ) − − λ < ) , UFFERED FROBENIUS REPRESENTATIONS 17 where λ < is the suboverpartition whose parts are all the overlined parts of λ smallerthan (cid:96) ( λ ). Here we have chosen the notation r CL ( λ ) in order to avoid confusion inthe ranks.For example, if λ = (5 , , , λ < = (3 , r CL ( λ ) = 5 − − λ is overlined, then r D ( λ ) = r CL ( λ ).We introduce a variant of the bracket for overpartitions. If λ = ( (cid:96) , (cid:96) , . . . , (cid:96) n )is an overpartition, then the overpartition bracket of λ is defined to be the lengthof the longest sequence of the form ( (cid:96) , (cid:96) , . . . , (cid:96) k ), where for all 1 ≤ i < k , we haveone of the following: • (cid:96) i = (cid:96) i +1 • (cid:96) i = (cid:96) i +1 + 1 and at least one of (cid:96) i and (cid:96) i +1 is overlined.We denote the overpartition bracket of λ by σ ( λ ).For example, if λ = (7 , , , , , (7 , , (7 , , , (7 , , , , the longest of which has length four. Therefore, σ ( λ ) = 4.We see how the overpartition rank and the overpartition bracket relate to (4.7)in the following lemma. Lemma 4.3.
Fix nonnegative integers ≤ s ≤ t . The coefficient of z m q n in ( − q ) t ( zq s ; q ) t − s +1 is equal to the number of overpartitions λ of n into t nonnegative parts with σ ( λ ) ≥ s and m = r CL ( λ ) + 1 . The proof of Lemma 4.3 relies on an an algorithm originally due to Joichi andStanton [11].
Algorithm 4.1 (Joichi, Stanton [11]) . Input: a partition λ = ( (cid:96) , (cid:96) , . . . , (cid:96) n ) into n parts, and a partition µ = ( m , m , . . . , m k ) into k distinct nonnegative parts,each less than n .Output: An overpartition λ (cid:48) = ( (cid:96) (cid:48) , (cid:96) (cid:48) , . . . , (cid:96) (cid:48) n ) into n parts. (1) Delete m from µ , and add 1 to the first m parts of λ . This operation iswell defined, as all parts of µ are strictly less than the number of parts of λ . Because µ is a partition into nonnegative parts, 0 may occur as a partof µ . If m = 0 , then the parts of λ are unchanged. (2) Overline the ( m + 1) -st part of λ . If m = 0 , then we overline (cid:96) . (3) Relabel the parts of µ , if any exist, so that m is the largest part of µ .Repeat Steps (1) to (3) until the parts of µ are exhausted. Because the parts of µ are distinct, we see that λ (cid:48) is an overpartition into n parts. An example of the Joichi Stanton map shown in Table 5. Algorithm 4.1 isnot difficult to reverse; additional details may be found in work of Lovejoy [12]. Wenow prove Lemma 4.3. Proof of Lemma 4.3.
As in the proof of Lemma 4.2, the term1( zq s ; q ) t − s +1 generates a partition λ into exactly t nonnegative parts, with at least s occurrencesof its largest part, and with its largest part equal to m . The term ( − q ) t generates a partition µ into distinct nonnegative parts less than t . We now apply Algorithm4.1 to produce an overpartition λ (cid:48) . We claim that the overpartition bracket of λ (cid:48) is equal to the number of occurrences of the largest part of λ .We induct on the number of parts of µ . If µ = ∅ , then λ (cid:48) has no overlined parts,and σ ( λ (cid:48) ) is equal to the number of occurrences of the largest part of λ (cid:48) , which isat least s .Suppose that µ = ( m , m , . . . , m k +1 ) and let λ (cid:48) be overpartition correspondingto the pair ( λ, ( m , m , . . . , m k )). Let α = ( (cid:96) (cid:48) , (cid:96) (cid:48) , . . . , (cid:96) (cid:48) j ) be the sequence whichdetermines the overpartition bracket of λ . It is sufficient to show that Algorithm4.1 leaves the length of α unchanged. If m k +1 < j , then all parts of α are increasedby 1. Thus ( (cid:96) (cid:48) + 1 , (cid:96) (cid:48) + 1 , . . . , (cid:96) (cid:48) j + 1) is eligible for determining σ ( λ ), but neither ofthe sequences ( (cid:96) (cid:48) + 1 , (cid:96) (cid:48) + 1 , . . . , (cid:96) (cid:48) j + 1 , (cid:96) (cid:48) j +1 + 1) or ( (cid:96) (cid:48) + 1 , (cid:96) (cid:48) + 1 , . . . , (cid:96) (cid:48) j + 1 , (cid:96) (cid:48) j +1 )are eligible. Therefore, the length of α is unchanged.Otherwise, if m k +1 ≤ j , then the sequence( (cid:96) (cid:48) + 1 , . . . , (cid:96) (cid:48) m k +1 − + 1 , (cid:96) m k +1 , (cid:96) m k +2 , . . . , (cid:96) (cid:48) j )is eligible for determining σ ( λ ), but( (cid:96) (cid:48) + 1 , . . . , (cid:96) (cid:48) m k +1 − + 1 , (cid:96) m k +1 , (cid:96) m k +2 , . . . , (cid:96) (cid:48) j , (cid:96) (cid:48) j +1 )is not. Therefore, the length of α is unchanged. That is, σ ( λ (cid:48) ) is invariant underiterations of Algorithm 4.1.Recall that (cid:96) ( λ ) = m . Each iteration of Algorithm 4.1 increases the largest partof λ (cid:48) by 1, except for the case m k = 0. Thus, the largest part of λ (cid:48) is equal to m plus the number of overlined parts less than (cid:96) ( λ (cid:48) ). Then r CL ( λ (cid:48) ) = [ (cid:96) ( λ ) + λ (cid:48) < )] − − λ (cid:48) < ) = (cid:96) ( λ ) − m − , as desired. (cid:3) Iteration λ µ σ ( λ ) r CL ( λ )0 (4 , , ,
2) (3 , ,
0) 1 31 (5 , , ,
2) (1 ,
0) 1 32 (6 , , ,
2) (0) 1 33 (6 , , , ∅ Table 5.
An example of Algorithm 4.1.We can now give a combinatorial interpretation of (4.7) in terms of bufferedFrobenius representations.
Definition 4.4.
A buffered Frobenius representation of the first kind, or a B -representation for short, is a buffered Frobenius representation ν ∈ (cid:18) A A . . . A k B B . . . B k (cid:19) , in which(1) A is the set of nonempty partitions α into distinct parts.(2) A is the set of nonempty partitions α with α ) ≤ σ ( α ).(3) For all i ≥
3, the set A i is the set of nonempty partitions α i with α i )less than or equal to the number of occurrences of the largest part of α i − . UFFERED FROBENIUS REPRESENTATIONS 19 (4) B is the set of overpartitions β into α ) nonnegative parts with σ ( β ) ≥ α ).(5) For all 2 ≤ i < k , the set B i is the set of partitions into α i ) nonnegativeparts with at least α i +1 ) occurrences of its largest part.(6) B k is the set of partitions into α k ) nonnegative parts.We also define the empty array to be a B -representation with k = 0.For example, consider the array: ν = (cid:32) (cid:92) (3 , ,
1) (2 , ,
1) (3)(4 , , (cid:92) (1 , ,
0) (0) (cid:33) (4.8)On the top row, α is a partition into distinct parts, which satisfies (1). Next, α is a partition into three parts with two occurrences of its largest part. Because σ ( α ) = 3 , this satisfies (2). Finally, α is a nonempty partition with one part.Because α has two occurrences of its largest part, this satisfies (3).On the bottom row, β is an overpartition into three parts with σ ( β ) = 3,which satisfies (4). Next, β is a partition into three nonnegative parts, with oneoccurrence of its largest part, which satisfies (5). Finally, β is a partition into onenonnegative part, which satisfies (6). Additionally, both α and β are marked withhats.As in Section 3, we see that Lovejoy’s first Frobenius representations of over-partitions correspond to the case k = 1 above. For k >
1, we can collapse B -representations using the jigsaw map. Proposition 4.5.
Let B denote the set of B -representations, and let F denotethe set of first Frobenius representations of overpartitions. Then j : B → F is asurjective map. Taken with Theorem 2.5, we see that every B -representation ν corresponds toan overpartition λ , although this correspondence is many-to-one. Thus the rankswe will establish to study R k ( x , x , . . . , x k ; q ) do not immediately carry over to theset of overpartitions.4.2. Ranks of B -representations. If ν = (cid:18) α α . . . α k β β . . . β k (cid:19) , then ν admits k different rank functions, corresponding to the x i variables in R k ( x , x , . . . , x k ; q ). We first define the indicator function χ i to be χ i ( ν ) := α i is marked with a hat, and β i is not marked with a hat − β i is marked with a hat, and α i is not marked with a hat0 : otherwise.We see that χ i detects buffers in the tableaux of ν . The first rank of ν is definedto be ρ ( ν ) := r ( α ) − ( r CL ( β ) + 1) + χ ( ν ) . (4.9)We also define ρ ( ∅ ) := 0. For 1 < i ≤ k , the i th rank of ν is defined to be ρ i ( ν ) = ( (cid:96) ( α i ) − − (cid:96) ( β i ) + χ i ( ν ) . We also define ρ i ( ν ) := 0 whenever ν has fewer than i columns.For example, let ν = (cid:32) (cid:92) (3 , ,
1) (2 , ,
1) (3)(4 , , (cid:92) (1 , ,
0) (0) (cid:33)
Then ρ ( ν ) = (3 − − ((3 −
1) + 1) + 1 = − ρ ( ν ) = (2 − − − − ρ ( ν ) = (3 − − − , and ρ i ( ν ) = 0 for i > R k ( x , x , . . . , x k ; q ) as the generating series for the ranks of B -representations.4.3. Generating Series.
Let B k denote the set of B -representations with at most k columns, B k := (cid:26)(cid:18) α α . . . α j β β . . . β j (cid:19) ∈ B (cid:12)(cid:12)(cid:12)(cid:12) j ≤ k (cid:27) . Theorem 4.6.
The coefficient of x m x m · · · x m k k q n in (cid:88) n ,...,n k ≥ ( − q ) N k q N k − N k k (cid:89) i =1 (1 − x k − i +1 )(1 − x − k − i +1 ) q N i ( x k − i +1 q N i − , x − k − i +1 q N i − ) n i +1 is equal to the number of B -representations ν ∈ B k such that | ν | = n and ρ i ( ν ) = m i , where the count is weighted by ( − h ( ν ) .Proof. Consider an arbitrary summand of the form( − q ) N k q N k − N k k (cid:89) i =1 (1 − x k − i +1 )(1 − x − k − i +1 ) q N i ( x k − i +1 q N i − , x − k − i +1 q N i − ) n i +1 . If n = · · · = n k = 0, then the summand reduces to 1, which corresponds to theempty B -representation ν = ∅ . Otherwise, n i > i . Let j be the smallestindex so that n j >
0. Then the summand reduces to( − q ) N k q N k − N k k (cid:89) i = j (1 − x k − i +1 )(1 − x − k − i +1 ) q N i ( x k − i +1 q N i − , x − k − i +1 q N i − ) n i +1 . (4.10)We claim that the coefficient of x m x m · · · x m k k q n in (4.10) is equal to the numberof B -representations ν = (cid:18) α α . . . α k − j +1 β β . . . β k − j +1 (cid:19) where α i ) = N k − i +1 , such that | ν | = n and ρ i ( ν ) = m i , where the count isweighted by ( − h ( ν ) . Note that( − h ( ν ) = ( − (cid:80) χ i ( ν ) . UFFERED FROBENIUS REPRESENTATIONS 21
The parts of α and β are generated by the i = k multiplicand, which we writeas (cid:18) (1 − x ) q N k + N k ( x q N k − ; q ) n k +1 (cid:19)(cid:18) (1 − x − )( − q ) N k ( x − q N k − ; q ) n k +1 (cid:19) . (4.11)We use the fact that N k = N k − + n k to apply Lemmas 4.2 and 4.3 with t = N k and s = N k − . Then we see that α is a partition into distinct parts with σ ( α ) ≥ N k − and β is an overpartition into N k nonnegative parts with σ ( β ) ≥ N k − .Here, the exponents of x and x − track r ( α ) and r CL ( β ) + 1, respectively.Given an arbitrary ( α , β ), the coefficient of x m in (1 − x )(1 − x − ) is equalto the weighted count of ways to mark α or β with hats, where m = χ ( ν ) andthe count is weighted by ( − χ ( ν ) . Therefore, the coefficient of x m q n in (4.11) isequal to the weighted count of possible columns ( α , β ) T of a B -representation ν such that α ) = N k , α ) = N k − , n = | α | + | β | , and m = r ( α ) − ( r CL ( β ) + 1) + χ ( ν ) = ρ ( ν ) , where the count is weighted by ( − χ ( ν ) .For 1 < i < k − j + 1, the parts of α i and β i are generated by the k − i + 1multiplicand, which we write as (cid:18) (1 − x i ) q N k − i +1 ( x i q N k − i ; q ) n k − i +1 +1 (cid:19)(cid:18) − x − i ( x − i q N k − i ; q ) n k − i +1 +1 (cid:19) . (4.12)As in Lemma 4.2, q N k − i +1 ( x i q N k − i ; q ) n k − i +1 +1 generates the Young tableau of α i , whose columns’ lengths are bounded between N k − i and N k − i +1 . We add 1 to each part of α i to account for q N k − i +1 . Thus, α i is anonempty partition with N k − i +1 positive parts and at least N k − i occurrences of itslargest part, and β i is a partition into N k − i +1 nonnegative parts with at least N k − i occurrences of its largest part. Here, the exponents of x i and x − i track (cid:96) ( α ) − (cid:96) ( β ), respectively.Because ( α i , β i ) T is not the rightmost column of ν , either entry may be markedwith a hat. As with the previous column, entries marked by a hat are tracked bythe term (1 − x i )(1 − x − i ). Therefore, the coefficient of x m i i q n in (4.12) is equal tothe weighted count of possible columns ( α i , β i ) T of ν such that α i ) = N k − i +1 , α i +1 ) = N k − i , n = | α i | + | β i | , and m i = ( (cid:96) ( α i ) − − (cid:96) ( β i ) + χ i ( ν ) = ρ i ( ν ) , where the count is weighted by ( − χ i ( ν ) .Finally, the parts of α k − j +1 and β k − j +1 are generated by the i = k − j + 1multiplicand, (cid:18) (1 − x k − j +1 ) q N j ( x k − j +1 q N j − ; q ) n j +1 (cid:19)(cid:18) (1 − x − j )( x − k − j +1 q N j − ; q ) n j +1 (cid:19) . By minimality of j , we see that n = · · · = n j − = 0. Thus, N j − = 0, and themultiplicand reduces to (cid:18) q N j ( x k − j +1 q ; q ) n j (cid:19)(cid:18) x − k − j +1 ; q ) n j (cid:19) . (4.13)This reflects the fact that neither α k − j +1 or β k − j +1 can be marked with a hat.As with the previous column, we see that the coefficient of x m k − j +1 k − j +1 q n in (4.13) isequal to the weighted count of possible columns ( α k − j +1 , β k − j +1 ) T of ν such that α k − j +1 ) = N k − i +1 , n = | α k − j +1 | + | β k − j +1 | , and m k − j +1 = ρ k − j +11 ( ν ), wherethe count is weighted by ( − χ k − j +1 ( ν ) .By combining these terms, we have counted all possible ν ∈ B k with | α i | = N k − i +1 , | ν | = n , ρ i ( ν ) = m i , and h ( ν ) entries marked with a hat, where the countis weighted by ( − h ( ν ) . By summing over all values of n , n , . . . , n k , we generateall possible B -representations in B k . (cid:3) Full Rank and Proof of Theorem 1.3.
We have one final statistic in thissection. We define the full rank of a B -representation ν to be the sum of the i thranks of ν , ρ ( ν ) := (cid:88) i ≥ ρ i ( ν ) . This sum converges for any B -representation ν , as all but finitely many of thesummands vanish. We may now prove Theorem 1.3. Proof of Theorem 1.3.
Let ζ k be a primitive k th root of unity. The desired gener-ating series, (cid:88) ν ∈B k ( − h ( ν ) k (cid:89) i =1 ζ ( i − ρ i ( ν ) k z ρ ( ν ) k q | ν | , is given by R k ( k √ z, ζ k k √ z, . . . , ζ k − k k √ z ; q ) = R [ k ]( z, q ) . (4.14) (cid:3) We now have our combinatorial interpretation of R [ k ]( z, q ). Observe that oneof the series in (4.14) is a series in k √ z with coefficients in Z [ ζ k ], and the other isa series in z with integer coefficients. Thus, the weighted count must vanish for B -representations whose full rank is not a multiple of k .We close this section by discussing conjugation maps on B k .4.5. Conjugation.
Given a buffered Frobenius representation of the first kind ν = (cid:18) α α . . . α k β β . . . β k (cid:19) , we define k different conjugation maps corresponding to the columns of ν . Toperform the first conjugation , delete a staircase from α by removing α ) fromthe first part, α ) − β . Let λ and µ be the partitionand partition into distinct parts produced this way, respectively. Both α and λ are UFFERED FROBENIUS REPRESENTATIONS 23 partitions into α ) nonnegative parts with at least α ) occurrences of theirlargest parts. Add a staircase to λ to produce α (cid:48) , and perform Algorithm 4.1 on α and µ to produce β (cid:48) . We mark α (cid:48) with a hat if and only if β was marked witha hat, and vice versa. We call φ ( ν ) := (cid:18) α (cid:48) α . . . α k β (cid:48) β . . . β k (cid:19) the first conjugate of ν .For example, let ν = (cid:32) (cid:92) (3 , ,
1) (2 , ,
1) (3)(4 , , (cid:92) (1 , ,
0) (0) (cid:33) . Then removing the staircase from α produces α = (0 , , , while reversing Algorithm 4.1 on β produces λ = (3 , , µ = (2 , . Next, we add a staircase to λ , and perform Algorithm 4.1 on α and µ , producing α (cid:48) = (6 , , β (cid:48) = (1 , , . Because α was marked with a hat, and β was not marked with a hat, we see that φ ( ν ) = (cid:32) (6 , ,
4) (2 , ,
1) (3) (cid:92) (1 , , (cid:92) (1 , ,
0) (0) (cid:33) . For i >
1, the i th conjugation map is performed as follows. First, subtract 1from each part of α i to produce β (cid:48) i , and add 1 to each part of β i to produce α (cid:48) i . Wemark α (cid:48) i with a hat if and only if β i was marked with a hat, and vice versa. We call φ i ( ν ) := (cid:18) α . . . α (cid:48) i . . . α k β . . . β (cid:48) i . . . β k (cid:19) the i th conjugate of α . We also define φ i ( ν ) := ν if ν has fewer than i columns.For example, we see that φ ( ν ) = (cid:32) (cid:92) (3 , , (cid:92) (2 , ,
1) (3)(4 , ,
3) (1 , ,
0) (0) (cid:33) φ ( ν ) = (cid:32) (cid:92) (3 , ,
1) (2 , ,
1) (1)(4 , , (cid:92) (1 , ,
0) (2) (cid:33) . Each of the i th conjugation maps exchange the roles of1 − x i ( x i q N k − i ; q ) n k − i +1 +1 and 1 − x − i ( x − i q N k − i ; q ) n k − i +1 +1 in (4.7). This fact immediately implies two propositions. Proposition 4.7.
For all i ≥ , we have ρ i ( φ i ( ν )) = − ρ i ( ν ) . Proposition 4.8.
For all nonnegative integers i and j , φ i φ j = φ j φ i . Finally, if we define the full conjugation to be φ := (cid:89) i ≥ φ i , then φ is defined for all ν ∈ B , and ρ ( φ ( ν )) = − ρ ( ν ).We now consider a second family of buffered Frobenius representations.5. Buffered Frobenius Representations of the Second Kind
Recall that R [2]( z ; q ) is the generating series for the M -rank of overpartitions.We consider the series R k ( x , x , . . . , x k ; q ) := ( − q ; q ) ∞ ( q ; q ) ∞ × (cid:32) ∞ (cid:88) n =1 ( − n q n +2 kn k (cid:89) i =1 (1 − x i )(1 − x − i )(1 − x i q n )(1 − x − i q n ) (cid:33) , bearing in mind that R k ( k √ z, ζ k k √ z, . . . , ζ k − k k √ z ; q ) = R [2 k ]( z, q ) . The thoughtful reader may be concerned that we are reproducing the work ofSection 4. We will see that buffered Frobenius representations of the second kinddirectly generalize Lovejoy’s second Frobenius representation of overpartitions, asopposed to the multi-to-one correspondence that B -representations require. Wehope that studying both of these families will allow us to define an infinite familyof overpartition ranks, as we discuss in Section 6.We see a transformation of R k ( x , x , . . . , x k ; q ) in the theorem below. Theorem 5.1.
Let k ≥ be a positive integer. Then we have ( − q ; q ) ∞ ( q ; q ) ∞ (cid:32) ∞ (cid:88) n =1 ( − n q n +2 kn k (cid:89) i =1 (1 − x i )(1 − x − i )(1 − x i q n )(1 − x − i q n ) (cid:33) = (cid:88) n ,...,n k ≥ ( − q ) N k q N k k (cid:89) i =1 (1 − x k − i +1 )(1 − x − k − i +1 ) q N i ( x k − i +1 q N i − , x − k − i +1 q N i − ; q ) n i +1 , where we write N := 0 and for all ≤ i ≤ k , we write N i = n + n + · · · + n i . UFFERED FROBENIUS REPRESENTATIONS 25
Proof.
We begin by substituting k (cid:55)→ ( k + 1) and q (cid:55)→ q in Corollary 2.3. Thenwe have k +6 Φ k +5 a, q a , − q a , b , c , . . . , b k +1 , c k +1 , q − N a , − a , aqb , aqc , . . . , aqb k +1 , aqc k +1 , aq N +2 ; q ; a k q k +2 N (cid:81) k +1 i =1 b i c i = ( aq , aq b k +1 c k +1 ; q ) N ( aq b k +1 , aq c k +1 ; q ) N (cid:88) n ,...,n k ≥ ( aq b k c k ; q ) n ( q ; q ) n ( aq b k − c k − ; q ) n ( q ; q ) n · · · ( aq b c ; q ) n k ( q ; q ) n k × ( b k − , c k − ; q ) N ( aq b k , aq c k ; q ) N ( b k − , c k − ; q ) N ( aq b k − , aq c k − ; q ) N · · · ( b , c ; q ) N k − ( aq b , aq c ; q ) N k − × ( b k +1 , c k +1 ; q ) N k ( aq b , aq c ; q ) N k ( q − N ; q ) N k ( a − b k +1 c k +1 q − N ; q ) N k × ( aq ) N + N + ··· + N k − q N k ( b k c k ) n ( b k − c k − ) N · · · ( b c ) N k − . Next, we take the limit as N → ∞ and set a = 1. As in the proof of Theorem4.1, we use (4.2) and (4.3) to simplify the q -Pochhammer symbols. The equationbecomes1 + ∞ (cid:88) n =1 (1 + q n ) ( b , c , . . . , b k +1 , c k +1 ; q ) n ( − n q n − n ( q b , q c , . . . , q b k +1 , q c k +1 ; q ) n (cid:32) q k +2 (cid:81) k +1 i =1 b i c i (cid:33) n = ( q , q b k +1 c k +1 ; q ) ∞ ( q b k +1 , q c k +1 ; q ) ∞ (cid:88) n ,...,n k ≥ ( q b k c k ; q ) n ( q ; q ) n ( q b k − c k − ; q ) n ( q ; q ) n · · · ( q b c ; q ) n k ( q ; q ) n k × ( b k − , c k − ; q ) N ( q b k , q c k ; q ) N ( b k − , c k − ; q ) N ( q b k − , q c k − ; q ) N · · · ( b , c ; q ) N k − ( q b , q c ; q ) N k − ( b k +1 , c k +1 ; q ) N k ( q b , q c ; q ) N k × q N +2 N + ··· +2 N k ( b k +1 c k +1 ) N k ( b k c k ) N ( b k − c k − ) N · · · ( b c ) N k − . Continue, setting b i = x i and c i = x − i for 1 ≤ i ≤ k . We now diverge from theproof of Theorem 4.1 by setting b k +1 = − c k +1 = − q . The term( − n q n − n ( c k +1 ; q ) n ( q c k +1 ; q ) n (cid:18) q k +2 b k +1 c k +1 (cid:19) n in the left hand side of the equation reduces to ( − n q n +2 kn , and we obtain1 + ∞ (cid:88) n =1 (1 + q n ) ( x , x − , x , x − , . . . , x k , x − k , − q ) n ( − n q n +2 kn ( x q , x − q , x q , x − q , . . . , x k q , x − k q , − q ; q ) n . The right hand side of the equation becomes( q , q ; q ) ∞ ( − q , − q ; q ) ∞ (cid:88) n ,...,n k ≥ ( x k − , x − k − ; q ) N ( x k q , x − k q ; q ) N × ( x k − , x − k − ; q ) N ( x k − q , x − k − q ; q ) N · · · ( x , x − ; q ) N k − ( x q , x − q ; q ) N k − × ( − , − q ; q ) N k q N +2 N + ··· +2 N k − + N k ( x q , x − q ; q ) N k . On the left hand side of the equation, we use Lemma 2.1 to obtain1 + 2 ∞ (cid:88) n =1 ( − n q n +2 kn k (cid:89) i =1 (1 − x i )(1 − x − i )(1 − x i q n )(1 − x − i q n ) . On the right hand side of the equation, we use Lemma 2.1 and the relations( q , q ; q ) ∞ ( − q , − q ; q ) ∞ = ( q ; q ) ∞ ( − q ; q ) ∞ , ( − , − q ; q ) n = ( − q ) n to obtain( q ; q ) ∞ ( − q ; q ) ∞ (cid:88) n ,...,n k ≥ ( − q ) N k q N +2 N + ··· +2 N k − + N k ( x k q , x − k q ; q ) N × (1 − x k − )(1 − x − k − )( x k − q N , x − k − q N ; q ) n +1 × (1 − x k − )(1 − x − k − )( x k − q N , x − k − q N ; q ) n +1 · · · (1 − x )(1 − x − )( x q N k − , x − q N k − ; q ) n k +1 . Since N := 0, the right side becomes( q ; q ) ∞ ( − q ; q ) ∞ (cid:88) n ,...,n k ≥ ( − q ) N k q N k k (cid:89) i =1 (1 − x k − i +1 )(1 − x − k − i +1 ) q N i ( x k − i +1 q N i − , x − k − i +1 q N i − ; q ) n i +1 . Here we have rewritten q N k as q N k /q N k in order to simplify the product notation.Multiplying both sides by ( − q ; q ) ∞ ( q ; q ) ∞ gives us the desired equation,(5.1) ( − q ; q ) ∞ ( q ; q ) ∞ (cid:32) ∞ (cid:88) n =1 ( − n q n +2 kn k (cid:89) i =1 (1 − x i )(1 − x − i )(1 − x i q n )(1 − x − i q n ) (cid:33) = (cid:88) n ,...,n k ≥ ( − q ) N k q N k k (cid:89) i =1 (1 − x k − i +1 )(1 − x − k − i +1 ) q N i ( x k − i +1 q N i − , x − k − i +1 q N i − ; q ) n i +1 . (cid:3) Overpartition Statistics.
In order to interpret (5.1) as a generating series,we must introduce additional partition and overpartition statistics. The first is avariation of Berkovich and Garvan’s M -rank for partitions [6] implied by work of UFFERED FROBENIUS REPRESENTATIONS 27
Lovejoy [13]. Given a partition λ into nonnegative parts where odd parts may notrepeat, the second partition rank of λ is defined to be r ( λ ) := (cid:98) (cid:96) ( λ )2 (cid:99) − λ o,< ) , where λ o,< is the subpartition of λ consisting of all odd parts of λ which are lessthan (cid:96) ( λ ). For example, if λ = (6 , r ( λ ) = 3 − λ = ( (cid:96) , (cid:96) , . . . , (cid:96) n )be a partition into nonnegative parts where odd parts may not repeat. The secondbracket of λ is the length of the longest substring of λ of the form ( (cid:96) , (cid:96) , . . . , (cid:96) k ),where for all 1 ≤ i < k , we have | (cid:96) i +1 − (cid:96) i | <
2. We denote the second bracket of λ by σ ( λ ).For example, if λ = (8 , , , , , , , (8 , , (8 , , , (8 , , , , the longest of which has length 4. Therefore, σ ( λ ) = 4. We see how the secondrank and the second bracket relate to (5.1) in the following lemma. Lemma 5.2.
Fix nonnegative integers ≤ s ≤ t . The coefficient of z m q n in ( − q ; q ) t ( zq s ; q ) t − s +1 is equal to the number of partitions λ of n into t nonnegative parts where odd partsmay not repeat with r ( λ ) = m and σ ( λ ) ≥ s . The proof rests on Lovejoy’s modification of Algorithm 4.1.
Algorithm 5.1 (Lovejoy [13]) . Input: A partition into n nonnegative even parts λ = ( (cid:96) , (cid:96) , . . . , (cid:96) n ) , and a partition µ = ( m , m , . . . , m k ) into k distinct odd partsless than n .Output: A partition λ (cid:48) = ( (cid:96) (cid:48) , (cid:96) (cid:48) , . . . , (cid:96) (cid:48) n ) into n nonnegative parts with k distinctodd parts. (1) Delete the largest part of µ , which we may write as m = 2 s + 1 . (2) Add 2 to the first s parts of λ , then add 1 to (cid:96) s +1 . Note that λ s +1 is nowodd. If s = 0 , then we instead add to λ . This operation is well defined,as λ has exactly n parts and m = 2 s + 1 < n , which implies s + 1 ≤ n . (3) Relabel the parts of µ , if any exist, so that the largest part of µ is m . Wenow repeat Steps (1) and (2) until the parts of µ are exhausted. Because the parts of µ are distinct, we see that λ is a partition into n nonnegativeparts with k distinct odd parts. Proof of Lemma 5.2.
The term 1( zq s ; q ) t − s +1 generates pairs of columns in the Young tableau of a partition λ . Therefore, λ has t even nonnegative parts with at least s occurrences of the largest part, and thecoefficient of z tracks one half of the largest part of λ . The term ( − q ; q ) t generatesa partition µ into distinct odd parts less than 2 t . We use Algorithm 5.1 to producea partition λ (cid:48) into t even nonnegative parts where odd parts may not repeat. Weclaim that the second bracket of λ (cid:48) is equal to the number of occurrences of thelargest part of λ , which is at least s . To show that σ ( λ (cid:48) ) ≥ s , we induct on the number of parts of µ . If µ is empty,then λ (cid:48) only consists of even parts. In this case, σ ( λ (cid:48) ) is equal to the number ofoccurrences of the largest part of λ (cid:48) , which is s , and the second rank is equal to m .Suppose that µ = ( m , m , . . . , m k +1 ) with k +1 parts, and let λ (cid:48) be the partitioncorresponding to ( λ, ( m , m , . . . , m k )). By assumption, σ ( λ (cid:48) ) ≥ s . Write m k +1 =2 s k +1 + 1 and m k = 2 s k + 1. Because m k +1 < m k , the first s k +1 parts of λ (cid:48) musthave the same parity.If σ ( λ (cid:48) ) ≤ s k +1 , then adding 2 to the first s k +1 parts of λ (cid:48) will leave the secondbracket unchanged. Otherwise, σ ( λ (cid:48) ) > s k +1 . In this case, adding 2 to the first s k +1 parts of λ (cid:48) and adding 1 to (cid:96) s k +1 +1 also leaves the second bracket unchanged.In either case, we have shown that the result holds for a µ with k + 1 parts.Therefore, λ is a partition of n into t nonnegative parts with σ ( λ ) ≥ s .Each step in Algorithm 5.1 adds an odd part to λ and increases the largestpart by either 1 or 2. Let λ (cid:48) o denote the subpartition whose parts are the oddparts of λ (cid:48) which are less than (cid:96) ( λ ). Then (cid:96) ( λ (cid:48) ) = 2 m + 2( λ o ) if (cid:96) ( λ (cid:48) ) is even,and (cid:96) ( λ (cid:48) ) = 2 m + 2( λ o ) + 1 if (cid:96) ( λ (cid:48) ) is odd. In either case, we see that m = (cid:98) (cid:96) ( λ )2 (cid:99) − λ (cid:48) o ) = r ( λ (cid:48) ). (cid:3) We need a variation of the overpartition rank implied by the work of Lovejoy[13]. Given an overpartition λ into odd parts, the second overpartition rank of λ isdefined to be r ( λ ) := (cid:96) ( λ ) − − λ < ) , where we recall λ < is the sub-overpartition of λ consisting of all overlined parts of λ less than (cid:96) ( λ ). For example, if λ = (3 , λ is given by 1 − r ( λ ).Given an overpartition λ into odd parts, the second overpartition bracket of λ isthe length of the longest substring of λ of the form ( (cid:96) , (cid:96) , . . . , (cid:96) k ), where for all1 ≤ i < k , one of the following holds: • (cid:96) i = (cid:96) i +1 • (cid:96) i = (cid:96) i +1 + 2 and at least one of (cid:96) i or (cid:96) i +1 is overlined.We denote the second overpartition bracket of λ by σ ( λ ).For example, if λ = (5 , , , , (5 , , (5 , , , the longest of which has length 3. Therefore, σ ( λ ) = 3. We see how the sec-ond overpartition rank and the second overpartition bracket relate to (5.1) in thefollowing lemma. Lemma 5.3.
Fix nonnegative integers ≤ s ≤ t . The coefficient of z m q n in ( − q ) t q t ( zq s ; q ) t − s +1 is equal to the number of overpartitions λ of n into t odd parts with r ( λ ) = m and σ ( λ ) ≥ s . The proof of Lemma 5.3 is almost identical to that of Lemma 4.3. We can nowgive a combinatorial interpretation of (5.1) in terms of a second family of bufferedFrobenius representations.
UFFERED FROBENIUS REPRESENTATIONS 29
Buffered Frobenius Representations of the Second Kind.Definition 5.4.
A buffered Frobenius representation of the second kind, or a B -representation, is a buffered Frobenius representation ν ∈ (cid:18) A A . . . A k B B . . . B k (cid:19) where(1) A is the set of nonempty overpartitions α into odd parts.(2) A is the set of nonempty partitions α into even parts, with α ) ≤ σ ( α ).(3) For all 3 < i ≤ k , A i is the set of nonempty partitions α i into even partswith α i ) less than or equal to the number of occurrences of the largestpart of α i − (4) B is the set of partitions β into α ) nonnegative parts where odd partsmay not repeat, with σ ( β ) ≥ α ).(5) For all 2 ≤ i < k , B i is the set of partitions β i into α i ) nonnegative evenparts and at most α i +1 ) occurrences of their largest part.(6) B k is the set of partitions β i into α i ) nonnegative even parts.We also define the empty array to be a B -representation with k = 0.For example, consider the array ν = (cid:32) (cid:91) (3 ,
1) (2 ,
2) (4)(6 ,
5) (2 ,
0) (2) (cid:33) . (5.2)On the top row, α is an overpartition into odd parts, which satisfies (1). Next, α is a partition into two even parts, with two occurrences of its largest part. Because σ ( α ) = 2, this satisfies (2). Finally, α is an partition into a single even part.Because α has two occurrences of its largest part, this satisfies (3).On the bottom row, β is a partition into two parts with no repeating odd parts,and σ ( β ) = 2, which satisfies (4). Next, β is a partition into two nonnegativeeven parts with a single occurrence of its largest part, which satisfies (5). Finally, β is a partition into one nonnegative part, which satisfies (6). Additionally, α ismarked with a hat.As in Section 3, we see that Lovejoy’s second Frobenius representations of over-partitions correspond to the case k = 1 above. For k >
1, we can collapse B -representations using the jigsaw map. Proposition 5.5.
Let B denote the set of B -representations, and let F denotethe set of second Frobenius representations of overpartitions. Then j : B → F isa surjective map. Taken with Theorem 2.8, we see that every B -representation ν corresponds toan overpartition λ , although this correspondence is many-to-one. Thus the rankswe will establish to study R k ( x , x , . . . , x k ; q ) do not immediately carry over tothe set of overpartitions.5.3. Ranks of B -representations. Recall the definition of χ i from Section 4. If ν = (cid:18) α α . . . α k β β . . . β k (cid:19) , then we define the first rank of ν to be ρ ( ν ) := r ( α ) − r ( β ) + χ ( ν ) , that is, the second overpartition rank of α minus the second partition rank of β plus χ ( ν ). We also define ρ ( ∅ ) := 0.For 2 ≤ i ≤ k , we define the i th rank of ν to be ρ i ( ν ) = (cid:18) (cid:96) ( α i )2 − (cid:19) − (cid:96) ( β i )2 + χ i ( ν ) , which is an integer since α i and β i have even parts. We also define ρ i ( ν ) := 0whenever ν has fewer than i columns.For example, let ν = (cid:32) (cid:91) (3 ,
1) (2 ,
2) (4)(6 ,
5) (2 ,
0) (2) (cid:33) . Then ρ ( ν ) = (1 − − (3 −
1) + 1 = − ρ ( ν ) = (1 − − − ρ ( ν ) = (2 − − , and ρ i ( ν ) = 0 for i > R k ( x , x , . . . , x k ; q ) as the generating series for the ranks of B -representations.5.4. Generating Series.
Let B k denote the set of B -representations with at most k columns, B k := (cid:26)(cid:18) α α . . . α j β β . . . β j (cid:19) ∈ B (cid:12)(cid:12)(cid:12)(cid:12) j ≤ k (cid:27) . We see the generating series for the i th ranks of B -representations in B k in thefollowing theorem. Theorem 5.6.
The coefficient of x m x m · · · x m k k q n in (cid:88) n ,...,n k ≥ ( − q ) N k q N k k (cid:89) i =1 (1 − x k − i +1 )(1 − x − k − i +1 ) q N i ( x k − i +1 q N i − , x − k − i +1 q N i − ; q ) n i +1 is equal to the number of B -representations ν ∈ B k such that | ν | = n and ρ i ( ν ) = m i , where the count is weighted by ( − h ( ν ) .Proof. Consider an arbitrary summand of the form( − q ) N k q N k k (cid:89) i =1 (1 − x k − i +1 )(1 − x − k − i +1 ) q N i ( x k − i +1 q N i − , x − k − i +1 q N i − ; q ) n i +1 . If n = · · · = n k = 0, then the summand reduces to 1, which corresponds to theempty B -representation ν = ∅ . Otherwise, n i > i . Let j be the smallestindex so that n j >
0. Then the summand reduces to( − q ) N k q N k k (cid:89) i = j (1 − x k − i +1 )(1 − x − k − i +1 ) q N i ( x k − i +1 q N i − , x − k − i +1 q N i − ; q ) n i +1 . (5.3) UFFERED FROBENIUS REPRESENTATIONS 31
We claim that the coefficient of x m x m · · · x m k k q n in (5.3) is equal to the numberof B -representations ν = (cid:18) α α . . . α k − j +1 β β . . . β k − j +1 (cid:19) where α i ) = N k − i +1 , such that | ν | = n and ρ i ( ν ) = m i , where the count isweighted by ( − h ( ν ) . Note that( − h ( ν ) = ( − (cid:80) χ i ( ν ) . The parts of α and β are generated by the i = k multiplicand, which we writeas (cid:18) (1 − x )( − q ) N k q N k ( x q N k − ; q ) n k +1 (cid:19)(cid:18) (1 − x − )( − q ; q ) N k ( x − q N k − ; q ) n k +1 (cid:19) . (5.4)We use the fact that N k = N k − + n k to apply Lemmas 5.2 and 5.3 with t = N k and s = N k − . Then we see that α is an overpartition into N k odd parts with σ ( α ) ≥ N k − , and β is a partition into N k nonnegative parts where odd partsmay not repeat with σ ( β ) ≥ N k − . Here, the exponents of x and x − track r ( α ) and r ( β ), respectively.As in the proof of Theorem 4.6, the term (1 − x )(1 − x − ) tracks whether or not α and β are marked with a hat. Thus, the coefficient of x m q n in (5.4) is equal tothe weighted count of of possible columns ( α , β ) T in a B -representation ν suchthat α ) = N k , α ) = N k − , n = | α | + | β | , and m = r ( α ) − r ( β ) + χ ( ν ) = ρ ( ν ), where the count is weighted by ( − χ ( ν ) .For j < i < k , the parts of α i and β i are generated by the k − i + 1 multiplicand,which we write as (cid:18) (1 − x i ) q N k − i +1 ( x i q N k − i ; q ) n k − i +1 +1 (cid:19)(cid:18) (1 − x − i )( x − i q N k − i ; q ) n k − i +1 +1 (cid:19) . (5.5)As in the proof of Lemma 5.2, (5.5) generates pairs of columns in the tableau for α i and β i . We see that α i is a nonempty partition into N k − i +1 even parts with at least N k − i occurrences of its largest part, and β i is a nonempty partition into N k − i +1 nonnegative even parts with at least N k − i occurrences of its largest part. Here, theexponents of x i and x − i track (cid:96) ( α i )2 − (cid:96) ( β i )2 , respectively. As with the previouscolumn, entries marked with a hat are tracked by (1 − x i )(1 − x − i ). Thus, thecoefficient of x m i i q n in (5.5) is equal to the weighted count of of possible columns( α i , β i ) T in a B -representation ν such that α i ) = N k − i +1 , α i +1 ) = N k − i , n = | α i | + | β i | , and m i = (cid:18) (cid:96) ( α i )2 − (cid:19) − (cid:96) ( β i )2 + χ ( ν ) = ρ i ( ν ) , where the count is weighted by ( − χ i ( ν ) .The parts of α k − j +1 and β k − j +1 are generated by the i = j multiplicand (cid:18) (1 − x k − j +1 ) q N j ( x k − j +1 q N j − ; q ) n j +1 (cid:19)(cid:18) (1 − x − k − j +1 )( x − k − j +1 q N j − ; q ) n j +1 (cid:19) . By minimality of j , we see that n = · · · = n j − = 0. Thus, N j − = 0, and themultiplicand reduces to (cid:18) q N j ( x k − j +1 q ; q ) n j (cid:19)(cid:18) x − k − j +1 q ; q ) n j (cid:19) . (5.6)This reflects the fact that neither α k − j +1 or β k − j +1 can be marked with a hat.As with the previous column, we see that the coefficient of x m k − j +1 k − j +1 q n in (5.6) isequal to the weighted count of possible columns ( α k − j +1 , β k − j +1 ) T of ν such that α k − j +1 ) = N k − i +1 , n = | α k − j +1 | + | β k − j +1 | , and m k − j +1 = ρ k − j +12 ( ν ), wherethe count is weighted by ( − χ k − j +1 ( ν ) .By combining these terms, we have counted all possible ν ∈ B k with | α i | = N k − i +1 , | ν | = n , ρ i ( ν ) = m i , and h ( ν ) entries marked with a hat, where the countis weighted by ( − h ( ν ) . By summing over all values of n , n , . . . , n k , we count allpossible B -representations in B k . (cid:3) Full Rank and Proof of Theorem 1.4.
As in Section 4, we define the fullrank of a B -representation ν to be the sum of the i th ranks of ν , ρ ( ν ) := (cid:88) i ≥ ρ i ( ν ) . This sum converges for any B -representation ν , as all but finitely many of thesummands vanish. We may now prove Theorem 1.4. Proof of Theorem 1.4.
Let ζ k be a primitive k th root of unity. The desired gener-ating series, (cid:88) ν ∈B k ( − h ( ν ) k (cid:89) i =1 ζ ( i − ρ i ( ν ) k z ρ ( ν ) k q | ν | , is given by R k ( k √ z, ζ k k √ z, . . . , ζ k − k k √ z ; q ) = R [2 k ]( z, q ) . (5.7) (cid:3) We now have our combinatorial interpretation of R [2 k ]( z, q ). As in Section 4,the weighted count in (5.7) must vanish for B -representations whose full rank isnot a multiple of k .We close this section by discussing conjugation maps on B k .5.6. Conjugation.
Given a B -representation ν = (cid:18) α α . . . α k β β . . . β k (cid:19) , we define k different conjugation maps corresponding to the columns of ν . To per-form the first conjugation , we subtract 1 from each part of α and reverse Algorithm4.1 to obtain a partition into nonnegative even parts λ and a partition into distincteven parts µ . We reverse Algorithm 5.1 on β and obtain a partition into nonnega-tive even parts γ and a partition into distinct odd parts δ . Note that λ ) = γ )by construction. UFFERED FROBENIUS REPRESENTATIONS 33
We then perform Algorithm 4.1 on γ and µ to produce α (cid:48) and perform Algorithm5.1 on λ and δ to produce β (cid:48) . Next, add 1 to each part of α (cid:48) . Finally, mark α (cid:48) with a hat if and only if β was marked with a hat, and vice versa. We call φ ( ν ) := (cid:18) α (cid:48) α . . . α k β (cid:48) β . . . β k (cid:19) the first conjugate of ν .For example, if ν = (cid:32) (cid:91) (3 ,
1) (2 ,
2) (4)(6 ,
5) (2 ,
0) (2) (cid:33) , then we see that λ = (0 , µ = (2) γ = (4 , δ = (3) . Performing Algorithms 4.1 and 5.1, produces λ (cid:48) = (6 , µ (cid:48) = (2 , , and adding 1 to each part of λ (cid:48) yields φ ( ν ) = (cid:32) (7 ,
5) (2 ,
2) (4) (cid:91) (2 ,
1) (2 ,
0) (2) (cid:33) . For 1 < i ≤ k , the i th conjugation map is performed as follows. First, subtract2 from each part of α i to produce α (cid:48) i , and add 2 to each part of β i to produce β (cid:48) i .Mark α (cid:48) i with a hat if and only if β i was marked with a hat, and vice versa. Wecall φ i ( ν ) := (cid:18) α . . . α i − α (cid:48) i α i +1 . . . α k β . . . β i − β (cid:48) i β i +1 . . . β k (cid:19) the i th conjugate of ν . Keeping ν as above, we have φ ( ν ) = (cid:32) (cid:91) (3 ,
1) (4 ,
2) (4)(6 ,
5) (0 ,
0) (2) (cid:33) ,φ ( ν ) = (cid:32) (cid:91) (3 ,
1) (2 ,
2) (4)(6 ,
5) (2 ,
0) (2) (cid:33) . Each of the i th conjugation maps exchange the roles of1 − x i ( x i q N k − i ; q ) n k − i +1 +1 and 1 − x − i ( x − i q N k − i ; q ) n k − i +1 +1 in (5.1). We find the same relations between conjugation maps as in Section 4. Proposition 5.7.
For all i ≥ , we have ρ i ( φ i ( ν )) = − ρ i ( ν ) . Proposition 5.8.
For all nonnegative integers i and j , φ i φ j = φ j φ i . Finally, if we define the full conjugation to be φ := (cid:89) i ≥ φ i , then φ is defined for all ν ∈ B , and ρ ( φ ( ν )) = − ρ ( ν ).This concludes our results. 6. Conclusion
We began with the series R [ k ]( z, q ) and R [2 k ]( z, q ), which arose from observ-ing a pattern between the generating series of the Dyson ranks and M -ranks ofoverpartitions, and asked whether these new series related to the ranks of overpar-titions. By generalizing the notion of Frobenius representations of overpartitions,we found that R [ k ]( z, q ) and R [2 k ]( z, q ) are weighted generating series for the fullranks of buffered Frobenius representations, which lie over the set of overpartitionsand generalize the first and second Frobenius representations of overpartitions. Itis somewhat disappointing then that the full rank functions are not well defined onthe set of overpartitions – compare for example ρ (cid:32)(cid:32) (cid:92) (3 , , ,
1) (1 , , , , ,
2) (4 , , (cid:33)(cid:33) and ρ (cid:18)(cid:18) (3 , , ,
1) (1 , , , , ,
2) (4 , , (cid:19)(cid:19) . Note that the full conjugation maps are well-defined. That is, j ( φ α ( ν )) = j ( φ α ( ν (cid:48) ))whenever j ( ν ) = j ( ν (cid:48) ), for α = 1 ,
2. Additionally, it not immediately clear why asum weighted by roots of unity should produce a meaningful count.One would hope that there exists a family of “ M k -ranks” of overpartitions, whosegenerating series are given by(6.1) (cid:88) n ≥ (cid:88) m ∈ Z N [ k ]( m, n ) z m q n = ( − q ; q ) ∞ ( q ; q ) ∞ (cid:18) ∞ (cid:88) n =1 (1 − z )(1 − z − )( − n q n + kn (1 − zq kn )(1 − z − q kn ) (cid:19) . By setting z = 1 in (6.1), we at least have that (cid:88) m ∈ Z N [ k ]( m, n ) = p ( n ) , (6.2)as expected. It seems likely that the coefficients N [ k ]( m, n ) are nonnegative inte-gers, which remains open.It is sufficient that an M k -rank candidate satisfy (cid:88) n ≥ N [ k ]( m, n ) q n = 2 ( − q ; q ) ∞ ( q ; q ) ∞ (cid:88) n ≥ ( − n +1 q n + k | m | n (1 − q kn )(1 + q kn ) , which is a generalization of Proposition 3.2 [12] and Corollary 1.3 [13]. We see anavenue for this work via the two interpretations of R [2]( z ; q ) as both the generatingseries of the M -ranks of overpartitions, and as the weighted generating series ofthe full ranks of B -representations in B . One might wonder if the parity of k determines behavior in R [ k ]( z, q ). Perhaps understanding how to map B → F will shed light on how to treat the rest of the B k and B k . Alternatively, there may UFFERED FROBENIUS REPRESENTATIONS 35 be a “ k th Frobenius representation” of overpartitions closer in spirit to Lovejoy’swork.Of course, we should be interested in determining the congruences arising fromany rank-like function. We may be able to use (6.2) to move from congruences ofbuffered Frobenius representations back to congruencies of overpartitions.There is also the question of analytics to consider. Since the series R [ k ]( z, q )and R [2 k ]( z, q ) are related to overpartition ranks, and can be obtained from the q -hypergeometric series, it is natural to ask if these series exhibit any modularproperties. This could be investigated separately of establishing a higher M k -rank. Acknowledgments.
The author is very grateful to the referee for uncoveringmultiple errors and suggesting improvements in the presentation of the results,to Jeremy Lovejoy for careful reading of earlier drafts and many helpful comments,and to Thomas Schmidt for a useful observation for future work.
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