Two Families of Buffered Frobenius Representations of Overpartitions
Thomas Morrill
TL;DR
This paper introduces two new families of buffered Frobenius representations that generalize existing overpartition rank generating series, providing combinatorial interpretations for these extended structures.
Contribution
It extends the generating series of overpartition ranks to k-fold variants and offers combinatorial interpretations for these new representations.
Findings
Generalization of Dyson and M_2 ranks to k-fold variants
Introduction of two families of buffered Frobenius representations
Combinatorial interpretations for the new representations
Abstract
We generalize the generating series of the Dyson ranks and -ranks of overpartitions to obtain -fold variants, and give a combinatorial interpretation of each. The -fold generating series correspond to the full ranks of two families of buffered Frobenius representations, which generalize Lovejoy's first and second Frobenius representations of overpartitions, respectively.
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Two Families of Buffered Frobenius Representations of Overpartitions
Thomas Morrill
Kidder Hall 368
Oregon State University
Corvallis, OR, 97331
United States
Abstract.
We generalize the generating series of the Dyson ranks and -ranks of overpartitions to obtain -fold variants, and give a combinatorial interpretation of each. The -fold generating series correspond to the full ranks of two families of buffered Frobenius representations, which generalize Lovejoy’s first and second Frobenius representations of overpartitions, respectively.
Key words and phrases:
basic hypergeometric series, overpartitions, rank, conjugation, Frobenius symbols
1991 Mathematics Subject Classification:
Primary 11P81; Secondary 05A17
1. Introduction and Statement of Results
A partition of is a nonincreasing sequence of integers such that the sum of the equals . Each of the is called a part of . We use the term partition statistic loosely to refer to any integer valued function on the set of partitions. For example, the weight of an arbitrary partition is the sum of its parts,
[TABLE]
We use to denote the largest part of , and to denote the number of parts of .
Historically, the theory of partition ranks was developed to give combinatorial evidence for the Ramanujan congruences, which state that for all ,
[TABLE]
where denotes the number of partitions of . Given a partition , Dyson [9] defined the rank of to be
[TABLE]
that is, the largest part of minus the number of parts of . For example, the partitions of are given with their ranks in Table 1. Note that , which agrees with (1.1).
Moreover, each equivalence class of appears exactly once in the second row of Table 1. Atkin and Swinnerton-Dyer [5] proved that for all and all ,
[TABLE]
where denotes the number of partitions of with rank modulo . Consequently, the set of partitions of can be separated into five classes of equal size by their ranks, which proves (1.1) via a counting argument. Atkin and Swinnerton-Dyer also proved that
[TABLE]
which treats (1.2) similarly. However, it is easy to confirm that
[TABLE]
does not even hold for . A counting argument for (1.3) was later found by using the partition crank function, which was predicted by Dyson [9] and later defined by Andrews and Garvan [4].
We now generalize. An overpartition is a nonincreasing sequence of positive integers , where the first occurrence of each part may be overlined. For example, the fourteen overpartitions of are given by
[TABLE]
Since every partition is an overpartition, we retain the notation , , and for the weight, largest part, and number of parts of an overpartition , respectively.
It is useful to represent partitions or overpartitions graphically as arrays of boxes. The Young tableau of a partition or overpartition is a left aligned array where the th row of the array consists of boxes. For overpartitions, if the first occurrence of the integer is overlined in , then we mark the last row of boxes with a dot111This convention ensures that mirroring the diagram across its main diagonal will produce the Young tableau of another overpartition, more commonly known as conjugating the overpartition.. An example is given in Figure 1.
Because these objects generalize partitions, it is natural to ask if partition statistics can be extended to overpartitions in a meaningful way. We begin by recapping some results for overpartition ranks. The full proofs are given in work of Lovejoy [12] [13].
The Dyson rank of an overpartition is defined to be
[TABLE]
an extension of Dyson’s rank function for ordinary partitions. For example, if , then . We see the generating series for the Dyson ranks of overpartitions in the following theorem.
Theorem 1.1** (Lovejoy [12]).**
The coefficient of in the series
[TABLE]
is equal to the number of overpartitions with and .
Lovejoy also developed an -rank for overpartitions [13], which expands on Berkovich and Garvan’s -rank for ordinary partitions whose odd parts cannot repeat [6]. Given an overpartition , the -rank of is defined to be
[TABLE]
where is the subpartition of consisting of all non-overlined odd parts of , and is defined to be
[TABLE]
For example, let . Then , and we see that . We see the generating series for the -ranks of overpartitions in the following theorem.
Theorem 1.2** (Lovejoy [13]).**
The coefficient of in the series
[TABLE]
is equal to the number of overpartitions with and .
The proofs of these theorems are based on Lovejoy’s first and second Frobenius representations for overpartitions [12] [13], which we summarize in Section 2. Note the similarity in the summands in (1.5) and (1.6); they are identical apart from the exponents of in the summation.
We now continue this pattern. For , define the series
[TABLE]
It is natural to ask is if can be interpreted as the generating series of an overpartition rank. In this paper we give a partial answer in terms of Frobenius representations. We may think of a Frobenius representation as an array
[TABLE]
where and are partitions or overpartitions. As we will see in Section 2, certain Frobenius representations correspond bijectively to overpartitions.
In Section 3, we introduce buffered Frobenius representations, which are arrays of the form
[TABLE]
where each of the entries and are partitions or overpartitions. A buffered Frobenius representation can be interpreted as an exploded Young tableau for an ordinary Frobenius representation . Thus, every overpartition admits multiple buffered Frobenius representations.
We now present our first main result, which interprets in terms of buffered Frobenius representations.
Theorem 1.3**.**
Let be a primitive th root of unity. The coefficient of in is equal to the weighted count of buffered Frobenius representations of the first kind with at most columns, , and full rank , where the count is weighted by
[TABLE]
In particular, the count vanishes for buffered Frobenius representations whose full rank is not a multiple of .
Following Lovejoy’s work on the -rank and the second Frobenius representation of an overpartition [13], our second main result interprets in terms of a second family of buffered Frobenius representations.
Theorem 1.4**.**
Let be a primitive th root of unity. The coefficient of in is equal to the weighted count of buffered Frobenius representations of the second kind with at most columns, , and full rank , where the count is weighted by
[TABLE]
In particular, the count vanishes for buffered Frobenius representations whose full rank is not a multiple of .
Each of these families is equipped with rank functions, and , respectively, and rank-reversing conjugation maps, which are developed in Sections 4 and 5. The observant reader will note that and are generating series for the ranks of buffered Frobenius representations, rather than for the ranks of overpartitions. We discuss this gap and the potential for improvement in Section 6.
The organization of this paper is as follows. In Section 2, we outline our -series techniques and summarize the motivating results for the Dyson rank and -rank. In Section 3, we define a generic buffered Frobenius representation and give a combinatorial map from buffered Frobenius representations to generalized Frobenius representations. This allows us to construct our first family of buffered Frobenius representations and prove Theorem 1.3 in Section 4. Then, in Section 5, we construct our second family of buffered Frobenius representations and prove Theorem 1.4. Finally, we give our closing remarks in Section 6.
2. Preliminaries
2.1. The -Pochhammer Symbol and -Hypergeometric Series.
We begin with the definition of the -Pochhammer symbol and its conventional shorthand notations. For , define
[TABLE]
Manipulating -Pochhammer symbols typically entails expanding the product and canceling individual factors, as seen in the following lemma.
Lemma 2.1**.**
For all nonnegative integers and ,
[TABLE]
Proof.
The case ,
[TABLE]
is trivial.
Next, consider . By expanding the -Pochhammer symbol and canceling like terms, we have
[TABLE]
∎
The -Pochhammer symbol is necessary for the definition of the -hypergeometric series,
[TABLE]
These series admit many beautiful transformation formulas; see Gasper and Rahman [10] for examples. In this paper, we only require Andrews’ -fold generalization of the Watson-Whipple transformation.
Theorem 2.2** (Andrews [1]).**
Let be complex numbers, and let and . Then,
[TABLE]
where we write and for all .
Observe that the left hand side of (2.6) is a symmetric function in the variables . Thus, we may permute the indices of and on the right hand side while leaving the corresponding indices fixed on the left hand side. We map
[TABLE]
which gives the following corollary to Theorem 2.2.
Corollary 2.3**.**
Let be complex numbers, and let and . Then,
[TABLE]
where we write and for all .
We now summarize Lovejoy’s work on the Dyson rank and -rank.
2.2. Summary of Lovejoy’s Work
In this context, it is convenient to allow partitions and overpartitions to contain [math] as a part, such as . We call these partitions into nonnegative parts and overpartitions into nonnegative parts, respectively222When unspecified, the terms partition and overpartition should be taken to mean partitions and overpartitions into positive parts.. The reader may consider this approach as a way for shorter partitions and overpartitions to attain a longer length requirement. For example, we can admit in contexts where a partition with exactly five parts is required. This is a common technique when working with generalized Frobenius representations, which we now define.
Definition 2.4** (Andrews [3]).**
Let and be sets of partitions or overpartitions, possibly into nonnegative parts. A generalized Frobenius representation is a two rowed array
[TABLE]
where , and .
We define the weight of a generalized Frobenius representation to be the sum of its entries333 Note that Lovejoy uses Andrews’ convention in his earlier work [12]. Statements of these results have been adjusted for consistency. ,
[TABLE]
For example,
[TABLE]
is a generalized Frobenius representation with weight 28. The top row is an ordinary partition, and the bottom row is an overpartition into nonnegative parts. With the correct choice of sets and , the corresponding Frobenius representations are equivalent to overpartitions, as seen in the following theorem.
Theorem 2.5** (Corteel, Lovejoy [8]).**
There is a bijection between overpartitions and generalized Frobenius representations where is a partition into distinct parts and is an overpartition into nonnegative parts such that .
Using this bijection, we can define the Dyson rank of to be . We see a generating series for the Dyson ranks of Frobenius representations in the following lemma.
Lemma 2.6** (Lovejoy [12]).**
The coefficient of in the series
[TABLE]
is equal to the number of generalized Frobenius representations with , where is a partition into distinct parts and is an overpartition into nonnegative parts, and .
Thus, Theorem 1.1 reduces to the following -series transformation.
Lemma 2.7** (Lovejoy [12]).**
For ,
[TABLE]
The proof of Lemma 2.7 involves a limiting case of the -Watson-Whipple transformation, or equivalently, the case in Theorem 2.2. Full details of the transformation may be seen as the case in Section 4. We now state the algorithm which produces the bijection in Theorem 2.5.
Algorithm 2.1** (Corteel, Lovejoy [8]).**
Input: A Frobenius representation
[TABLE]
as described in Proposition 2.5.
Output: An overpartition such that .
- (1)
Initialize . 2. (2)
We treat as a partition into nonnegative parts. Delete from and add 1 to each part of . 3. (3)
Delete from . If was overlined, append as a part of . Otherwise, if was not overlined, append as a part of . 4. (4)
Repeat Steps (2) and (3) until all parts of are exhausted. 5. (5)
Because was a partition into distinct parts, is also a partition into distinct parts. We define the output to be the overpartition with non-overlined parts given by and overlined parts given by .
An example of Algorithm 2.1 is shown in Table 2. Further details may be found in work of Lovejoy [12].
The generating series for the -rank involves a second family of Frobenius representations, which appear in the following theorem.
Theorem 2.8** (Lovejoy[13]).**
There is a bijection between overpartitions and generalized Frobenius partitions where is an overpartition into odd parts and is a partition into nonnegative parts where odd parts may not repeat such that .
As was the case with the Dyson rank, we can define the -rank of to be . We see a generating series for the -ranks of Frobenius representations in the following lemma.
Lemma 2.9** (Lovejoy [13]).**
The coefficient of in the series
[TABLE]
is equal to the number of Frobenius representations with , where is an overpartition into odd parts and is a partition into nonnegative parts, and .
Then Theorem 1.2 reduces to the following -series transformation.
Lemma 2.10** (Lovejoy [13]).**
For ,
[TABLE]
As before, the proof utilizes a limiting case of the -Watson-Whipple transformation. Full details may be seen as the case in Section 5. We now state the algorithm which gives the bijection in Theorem 2.8.
Algorithm 2.2** (Lovejoy [13]).**
Input: A Frobenius representation
[TABLE]
as described in Theorem 2.8.
Output: An overpartition such that .
- (1)
Initialize . 2. (2)
For each odd integer which does not appear overlined in , we insert in its correct position in . We also append as a part of . 3. (3)
Reindex the parts of so that from left to right, odd integers appear in increasing order, followed by even integers in decreasing order. 4. (4)
For each pair , let . If is even, append as a part of with the same overline marking as . If is odd, append as a part of with the opposite overline marking as . Reindex the in non-increasing order, with the convention that .
An example of Algorithm 2 is demonstrated in Table 3. The reverse algorithm is a modification of Corteel and Lovejoy’s work on vector partitions [7]. We present it below for completeness. For this algorithm, we let denote the smallest part of the overpartition .
Algorithm 2.3** **(Corteel, Lovejoy
Input: An overpartition .
Output: A second Frobenius representation such that .
- (1)
Initialize and . Dissect into four partitions , , , and as follows. Let be the subpartition consisting of all even overlined parts of . Let be the subpartition consisting of all even non-overlined parts of . We define and analogously for the odd parts of . 2. (2)
If , or if , then append as a part of , append as a part of , and delete the smallest part of . 3. (3)
Otherwise, append as a part of , append as a part of , delete the smallest part of , and set . 4. (4)
Repeat Steps (2) and (3) until both and are exhausted. 5. (5)
If , or if , then append as a part of , append as a part of , and delete the smallest part of . 6. (6)
Otherwise, append as a part of , append as a part of , delete the smallest part of , and set . 7. (7)
Repeat Steps (5) and (6) until both and are exhausted. 8. (8)
If a part occurs in , delete both from and from .
An example of Algorithm 2.3 is given in Table 4.
This ends our presentation of previous results. We now introduce the notion of buffered Frobenius representations.
3. Buffered Frobenius Representations
We use the following abbreviated notation for the rest of the paper. If , , …, and , , …, are sets, we write
[TABLE]
to mean that and for all .
Definition 3.1**.**
Let denote the set of overpartitions into nonnegative parts, and let denote the set of partitions into nonnegative parts. A buffered Frobenius representation is a two rowed array
[TABLE]
where for all , we have and . Additionally, we may mark either of or with a hat if .
The weight of a buffered Frobenius representation is defined to be
[TABLE]
We see that every generalized Frobenius representation as in Section 2
[TABLE]
can be interpreted as a buffered Frobenius representation
[TABLE]
although this only produces simple examples. The hat notation serves to enrich the combinatorics of buffered Frobenius representations, similar to the purpose of overlining the parts of an overpartition. For example,
[TABLE]
is a buffered Frobenius representation. Note that ; only the sequences and must be nonincreasing.
3.1. Buffered Young Tableaux
Given a buffered Frobenius representation
[TABLE]
we construct buffered Young tableaux to represent the entries of by using colors as follows.
First, we draw the Young tableau for in the first color. Next, we draw the Young tableau for in the second color. However, we align the boxes for to the right edge of the tableau for . If is marked with a hat, we shift the tableau for to the right by one unit and leave a buffer between and . For example, if and , then we produce the tableaux in Figure 2.
We then continue by drawing the tableau for each in the th color, aligned to the right edge of the preceding tableau, and shifted to the right by one unit if is marked with a hat. We draw the tableaux for the in the same manner. For example, Figure 3 shows the buffered Young tableaux for the buffered Frobenius representation in (3.1).
Note that entries marked with a hat increase the width of the tableaux without increasing the number of boxes. There are no tableaux which could indicate a buffer to the right of or , which corresponds to the restriction that neither or can be marked with a hat.
3.2. The Jigsaw Map
Visualizing buffered Frobenius representations by their tableaux suggests that we should interpret buffered Frobenius representations as the exploded Young tableaux of generalized Frobenius representations. To reassemble the generalized Frobenius representation, we use the jigsaw map.
Let be a buffered Frobenius representation
[TABLE]
where for all ,
[TABLE]
We seek to construct a generalized Frobenius representation
[TABLE]
where and are partitions or overpartitions into nonnegative parts.
First, discard any hats from the entries of . We then rewrite each and as a partition into nonnegative parts,
[TABLE]
For all , we define the integers to be
[TABLE]
Finally, we overline or if and only if the th part of or is overlined, respectively444 This is why only and may be overpartitions. . Graphically, this is equivalent to removing the colors from the buffered Young tableaux and aligning the boxes to the left, with careful attention paid to the 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 , we consider the series
[TABLE]
bearing in mind that
[TABLE]
We see a transformation of in the theorem below.
Theorem 4.1**.**
Let be a positive integer. Then we have
[TABLE]
where we write and for all .
Proof.
We begin by substituting into Corollary 2.3. Letting turns the transformation of terminating series into a transformation of infinite series. The left side becomes
[TABLE]
When , the -Pochhammer symbols take their trivial value, and the summand is equal to . For , we may simplify the summand using the relation
[TABLE]
Thus the left hand side is equal to
[TABLE]
On the right hand side, we use the relation
[TABLE]
to obtain
[TABLE]
Setting , the equation becomes
[TABLE]
We set , for , and . This cancels the term
[TABLE]
On the left hand side, we use the identity
[TABLE]
and obtain
[TABLE]
The right hand side becomes
[TABLE]
We now let . On the left hand side, we use the simple identities
[TABLE]
to obtain
[TABLE]
On the right hand side, applying (4.5) and (4.6) produces
[TABLE]
Applying Lemma 2.1 to the left hand side of the equation and multiplying by gives us
[TABLE]
On the right hand side of the equation, we use the fact that for all with Lemma 2.1 to write
[TABLE]
Multiplying the right hand side of the equation by gives
[TABLE]
Finally, as , we may rewrite the right hand side using
[TABLE]
which gives us the desired equation,
[TABLE]
∎
4.1. Overpartition Statistics
In order to interpret (4.7) as a generating series, we must introduce some partition and overpartition statistics. The first statistic we 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 name it the bracket of a partition.
Given a partition , the bracket of is defined to be the length of the longest sequence of the form , where for all , we have . We retain Andrews’ notation of to denote the bracket of .
For example, if , then we consider the sequences
[TABLE]
the longest of which has length three. Therefore, .
We see how the partition rank and the partition bracket relate to (4.7) in the following lemma.
Lemma 4.2**.**
Fix nonnegative integers . The coefficient of in
[TABLE]
is equal to the number of partitions of into distinct parts with and .
Proof.
The term
[TABLE]
generates the columns of a Young tableau, where tracks the number of columns generated. The length of these columns is bounded between and . Then we may consider as a partition into exactly nonnegative parts, . Note that has at least occurrences of its largest part, that is, .
To account for , we add a staircase to . That is, we add to the first part, to the second part, and so on, adding 1 to the last part. At this stage, contains the sequence , which implies that . Finally, since and , we see that . ∎
We also need an overpartition statistic introduced by Corteel and Lovejoy [8] [12]. Given an overpartition , the overpartition rank of is defined to be
[TABLE]
where is the suboverpartition whose parts are all the overlined parts of smaller than . Here we have chosen the notation in order to avoid confusion in the ranks.
For example, if , then , and . Note that if every part of is overlined, then .
We introduce a variant of the bracket for overpartitions. If is an overpartition, then the overpartition bracket of is defined to be the length of the longest sequence of the form , where for all , we have one of the following:
- •
- •
and at least one of and is overlined.
We denote the overpartition bracket of by .
For example, if , then we consider the sequences
[TABLE]
the longest of which has length four. Therefore, .
We see how the overpartition rank and the overpartition bracket relate to (4.7) in the following lemma.
Lemma 4.3**.**
Fix nonnegative integers . The coefficient of in
[TABLE]
is equal to the number of overpartitions of into nonnegative parts with and .
The proof of Lemma 4.3 relies on an an algorithm originally due to Joichi and Stanton [11].
Algorithm 4.1** (Joichi, Stanton [11]).**
Input: a partition into parts, and a partition into distinct nonnegative parts, each less than .
Output: An overpartition into parts.
- (1)
Delete from , and add 1 to the first parts of . This operation is well defined, as all parts of are strictly less than the number of parts of . Because is a partition into nonnegative parts, 0 may occur as a part of . If , then the parts of are unchanged. 2. (2)
Overline the -st part of . If , then we overline . 3. (3)
Relabel the parts of , if any exist, so that is the largest part of . Repeat Steps (1) to (3) until the parts of are exhausted.
Because the parts of are distinct, we see that is an overpartition into parts. An example of the Joichi Stanton map shown in Table 5. Algorithm 4.1 is not difficult to reverse; additional details may be found in work of Lovejoy [12]. We now prove Lemma 4.3.
Proof of Lemma 4.3.
As in the proof of Lemma 4.2, the term
[TABLE]
generates a partition into exactly nonnegative parts, with at least occurrences of its largest part, and with its largest part equal to . The term generates a partition into distinct nonnegative parts less than . We now apply Algorithm 4.1 to produce an overpartition . We claim that the overpartition bracket of is equal to the number of occurrences of the largest part of .
We induct on the number of parts of . If , then has no overlined parts, and is equal to the number of occurrences of the largest part of , which is at least .
Suppose that and let be overpartition corresponding to the pair . Let be the sequence which determines the overpartition bracket of . It is sufficient to show that Algorithm 4.1 leaves the length of unchanged. If , then all parts of are increased by 1. Thus is eligible for determining , but neither of the sequences or are eligible. Therefore, the length of is unchanged.
Otherwise, if , then the sequence
[TABLE]
is eligible for determining , but
[TABLE]
is not. Therefore, the length of is unchanged. That is, is invariant under iterations of Algorithm 4.1.
Recall that . Each iteration of Algorithm 4.1 increases the largest part of by 1, except for the case . Thus, the largest part of is equal to plus the number of overlined parts less than . Then
[TABLE]
as desired. ∎
We can now give a combinatorial interpretation of (4.7) in terms of buffered Frobenius representations.
Definition 4.4**.**
A buffered Frobenius representation of the first kind, or a -representation for short, is a buffered Frobenius representation
[TABLE]
in which
- (1)
is the set of nonempty partitions into distinct parts. 2. (2)
is the set of nonempty partitions with . 3. (3)
For all , the set is the set of nonempty partitions with less than or equal to the number of occurrences of the largest part of . 4. (4)
is the set of overpartitions into nonnegative parts with . 5. (5)
For all , the set is the set of partitions into nonnegative parts with at least occurrences of its largest part. 6. (6)
is the set of partitions into nonnegative parts.
We also define the empty array to be a -representation with .
For example, consider the array:
[TABLE]
On the top row, is a partition into distinct parts, which satisfies (1). Next, is a partition into three parts with two occurrences of its largest part. Because this satisfies (2). Finally, is a nonempty partition with one part. Because has two occurrences of its largest part, this satisfies (3).
On the bottom row, is an overpartition into three parts with , which satisfies (4). Next, is a partition into three nonnegative parts, with one occurrence of its largest part, which satisfies (5). Finally, is a partition into one nonnegative part, which satisfies (6). Additionally, both and are marked with hats.
As in Section 3, we see that Lovejoy’s first Frobenius representations of overpartitions correspond to the case above. For , we can collapse -representations using the jigsaw map.
Proposition 4.5**.**
Let denote the set of -representations, and let denote the set of first Frobenius representations of overpartitions. Then is a surjective map.
Taken with Theorem 2.5, we see that every -representation corresponds to an overpartition , although this correspondence is many-to-one. Thus the ranks we will establish to study do not immediately carry over to the set of overpartitions.
4.2. Ranks of -representations
If
[TABLE]
then admits different rank functions, corresponding to the variables in . We first define the indicator function to be
[TABLE]
We see that detects buffers in the tableaux of . The first rank of is defined to be
[TABLE]
We also define .
For , the th rank of is defined to be
[TABLE]
We also define whenever has fewer than columns.
For example, let
[TABLE]
Then
[TABLE]
and for .
We now establish as the generating series for the ranks of -representations.
4.3. Generating Series
Let denote the set of -representations with at most columns,
[TABLE]
Theorem 4.6**.**
The coefficient of in
[TABLE]
is equal to the number of -representations such that and , where the count is weighted by .
Proof.
Consider an arbitrary summand of the form
[TABLE]
If , then the summand reduces to , which corresponds to the empty -representation . Otherwise, for some . Let be the smallest index so that . Then the summand reduces to
[TABLE]
We claim that the coefficient of in (4.10) is equal to the number of -representations
[TABLE]
where , such that and , where the count is weighted by . Note that
[TABLE]
The parts of and are generated by the multiplicand, which we write as
[TABLE]
We use the fact that to apply Lemmas 4.2 and 4.3 with and . Then we see that is a partition into distinct parts with and is an overpartition into nonnegative parts with . Here, the exponents of and track and , respectively.
Given an arbitrary , the coefficient of in is equal to the weighted count of ways to mark or with hats, where and the count is weighted by . Therefore, the coefficient of in (4.11) is equal to the weighted count of possible columns of a -representation such that , , , and
[TABLE]
where the count is weighted by .
For , the parts of and are generated by the multiplicand, which we write as
[TABLE]
As in Lemma 4.2,
[TABLE]
generates the Young tableau of , whose columns’ lengths are bounded between and . We add to each part of to account for . Thus, is a nonempty partition with positive parts and at least occurrences of its largest part, and is a partition into nonnegative parts with at least occurrences of its largest part. Here, the exponents of and track and , respectively.
Because is not the rightmost column of , either entry may be marked with a hat. As with the previous column, entries marked by a hat are tracked by the term . Therefore, the coefficient of in (4.12) is equal to the weighted count of possible columns of such that , , , and
[TABLE]
where the count is weighted by .
Finally, the parts of and are generated by the multiplicand,
[TABLE]
By minimality of , we see that . Thus, , and the multiplicand reduces to
[TABLE]
This reflects the fact that neither or can be marked with a hat. As with the previous column, we see that the coefficient of in (4.13) is equal to the weighted count of possible columns of such that , , and , where the count is weighted by .
By combining these terms, we have counted all possible with , , , and entries marked with a hat, where the count is weighted by . By summing over all values of , we generate all possible -representations in .
∎
4.4. Full Rank and Proof of Theorem 1.3
We have one final statistic in this section. We define the full rank of a -representation to be the sum of the th ranks of ,
[TABLE]
This sum converges for any -representation , as all but finitely many of the summands vanish. We may now prove Theorem 1.3.
Proof of Theorem 1.3.
Let be a primitive th root of unity. The desired generating series,
[TABLE]
is given by
[TABLE]
∎
We now have our combinatorial interpretation of . Observe that one of the series in (4.14) is a series in with coefficients in , and the other is a series in with integer coefficients. Thus, the weighted count must vanish for -representations whose full rank is not a multiple of .
We close this section by discussing conjugation maps on .
4.5. Conjugation
Given a buffered Frobenius representation of the first kind
[TABLE]
we define different conjugation maps corresponding to the columns of . To perform the first conjugation, delete a staircase from by removing from the first part, from the second part and so on until removing from the smallest part. We next reverse Algorithm 4.1 on . Let and be the partition and partition into distinct parts produced this way, respectively. Both and are partitions into nonnegative parts with at least occurrences of their largest parts. Add a staircase to to produce , and perform Algorithm 4.1 on and to produce . We mark with a hat if and only if was marked with a hat, and vice versa. We call
[TABLE]
the first conjugate of .
For example, let
[TABLE]
Then removing the staircase from produces
[TABLE]
while reversing Algorithm 4.1 on produces
[TABLE]
Next, we add a staircase to , and perform Algorithm 4.1 on and , producing
[TABLE]
Because was marked with a hat, and was not marked with a hat, we see that
[TABLE]
For , the th conjugation map is performed as follows. First, subtract 1 from each part of to produce , and add 1 to each part of to produce . We mark with a hat if and only if was marked with a hat, and vice versa. We call
[TABLE]
the th conjugate of . We also define if has fewer than columns.
For example, we see that
[TABLE]
Each of the th conjugation maps exchange the roles of
[TABLE]
in (4.7). This fact immediately implies two propositions.
Proposition 4.7**.**
For all , we have .
Proposition 4.8**.**
For all nonnegative integers and , .
Finally, if we define the full conjugation to be
[TABLE]
then is defined for all , and .
We now consider a second family of buffered Frobenius representations.
5. Buffered Frobenius Representations of the Second Kind
Recall that is the generating series for the -rank of overpartitions. We consider the series
[TABLE]
bearing in mind that
[TABLE]
The thoughtful reader may be concerned that we are reproducing the work of Section 4. We will see that buffered Frobenius representations of the second kind directly generalize Lovejoy’s second Frobenius representation of overpartitions, as opposed to the multi-to-one correspondence that -representations require. We hope that studying both of these families will allow us to define an infinite family of overpartition ranks, as we discuss in Section 6.
We see a transformation of in the theorem below.
Theorem 5.1**.**
Let be a positive integer. Then we have
[TABLE]
where we write and for all , we write .
Proof.
We begin by substituting and in Corollary 2.3. Then we have
[TABLE]
Next, we take the limit as and set . As in the proof of Theorem 4.1, we use (4.2) and (4.3) to simplify the -Pochhammer symbols. The equation becomes
[TABLE]
Continue, setting and for . We now diverge from the proof of Theorem 4.1 by setting and . The term
[TABLE]
in the left hand side of the equation reduces to , and we obtain
[TABLE]
The right hand side of the equation becomes
[TABLE]
On the left hand side of the equation, we use Lemma 2.1 to obtain
[TABLE]
On the right hand side of the equation, we use Lemma 2.1 and the relations
[TABLE]
to obtain
[TABLE]
Since , the right side becomes
[TABLE]
Here we have rewritten as in order to simplify the product notation. Multiplying both sides by gives us the desired equation,
[TABLE]
∎
5.1. Overpartition Statistics
In order to interpret (5.1) as a generating series, we must introduce additional partition and overpartition statistics. The first is a variation of Berkovich and Garvan’s -rank for partitions [6] implied by work of Lovejoy [13]. Given a partition into nonnegative parts where odd parts may not repeat, the second partition rank of is defined to be
[TABLE]
where is the subpartition of consisting of all odd parts of which are less than . For example, if , then .
We introduce another variation of the partition bracket. Let be a partition into nonnegative parts where odd parts may not repeat. The second bracket of is the length of the longest substring of of the form , where for all , we have . We denote the second bracket of by .
For example, if , then we consider the substrings
[TABLE]
the longest of which has length . Therefore, . We see how the second rank and the second bracket relate to (5.1) in the following lemma.
Lemma 5.2**.**
Fix nonnegative integers . The coefficient of in
[TABLE]
is equal to the number of partitions of into nonnegative parts where odd parts may not repeat with and .
The proof rests on Lovejoy’s modification of Algorithm 4.1.
Algorithm 5.1** (Lovejoy [13]).**
Input: A partition into nonnegative even parts , and a partition into distinct odd parts less than .
Output: A partition into nonnegative parts with distinct odd parts.
- (1)
Delete the largest part of , which we may write as . 2. (2)
Add 2 to the first parts of , then add 1 to . Note that is now odd. If , then we instead add to . This operation is well defined, as has exactly parts and , which implies . 3. (3)
Relabel the parts of , if any exist, so that the largest part of is . We now repeat Steps (1) and (2) until the parts of are exhausted.
Because the parts of are distinct, we see that is a partition into nonnegative parts with distinct odd parts.
Proof of Lemma 5.2.
The term
[TABLE]
generates pairs of columns in the Young tableau of a partition . Therefore, has even nonnegative parts with at least occurrences of the largest part, and the coefficient of tracks one half of the largest part of . The term generates a partition into distinct odd parts less than . We use Algorithm 5.1 to produce a partition into even nonnegative parts where odd parts may not repeat. We claim that the second bracket of is equal to the number of occurrences of the largest part of , which is at least .
To show that , we induct on the number of parts of . If is empty, then only consists of even parts. In this case, is equal to the number of occurrences of the largest part of , which is , and the second rank is equal to .
Suppose that with parts, and let be the partition corresponding to . By assumption, . Write and . Because , the first parts of must have the same parity.
If , then adding 2 to the first parts of will leave the second bracket unchanged. Otherwise, . In this case, adding 2 to the first parts of and adding to also leaves the second bracket unchanged. In either case, we have shown that the result holds for a with parts. Therefore, is a partition of into nonnegative parts with .
Each step in Algorithm 5.1 adds an odd part to and increases the largest part by either or . Let denote the subpartition whose parts are the odd parts of which are less than . Then if is even, and if is odd. In either case, we see that . ∎
We need a variation of the overpartition rank implied by the work of Lovejoy [13]. Given an overpartition into odd parts, the second overpartition rank of is defined to be
[TABLE]
where we recall is the sub-overpartition of consisting of all overlined parts of less than . For example, if , then the second overpartition rank of is given by .
We also introduce a variation of the overpartition bracket corresponding to . Given an overpartition into odd parts, the second overpartition bracket of is the length of the longest substring of of the form , where for all , one of the following holds:
- •
- •
and at least one of or is overlined.
We denote the second overpartition bracket of by .
For example, if , the substrings we consider are
[TABLE]
the longest of which has length . Therefore, . We see how the second overpartition rank and the second overpartition bracket relate to (5.1) in the following lemma.
Lemma 5.3**.**
Fix nonnegative integers . The coefficient of in
[TABLE]
is equal to the number of overpartitions of into odd parts with and .
The proof of Lemma 5.3 is almost identical to that of Lemma 4.3. We can now give a combinatorial interpretation of (5.1) in terms of a second family of buffered Frobenius representations.
5.2. Buffered Frobenius Representations of the Second Kind
Definition 5.4**.**
A buffered Frobenius representation of the second kind, or a -representation, is a buffered Frobenius representation
[TABLE]
where
- (1)
is the set of nonempty overpartitions into odd parts. 2. (2)
is the set of nonempty partitions into even parts, with . 3. (3)
For all , is the set of nonempty partitions into even parts with less than or equal to the number of occurrences of the largest part of 4. (4)
is the set of partitions into nonnegative parts where odd parts may not repeat, with . 5. (5)
For all , is the set of partitions into nonnegative even parts and at most occurrences of their largest part. 6. (6)
is the set of partitions into nonnegative even parts.
We also define the empty array to be a -representation with .
For example, consider the array
[TABLE]
On the top row, is an overpartition into odd parts, which satisfies (1). Next, is a partition into two even parts, with two occurrences of its largest part. Because , this satisfies (2). Finally, is an partition into a single even part. Because has two occurrences of its largest part, this satisfies (3).
On the bottom row, is a partition into two parts with no repeating odd parts, and , which satisfies (4). Next, is a partition into two nonnegative even parts with a single occurrence of its largest part, which satisfies (5). Finally, is a partition into one nonnegative part, which satisfies (6). Additionally, is marked with a hat.
As in Section 3, we see that Lovejoy’s second Frobenius representations of overpartitions correspond to the case above. For , we can collapse -representations using the jigsaw map.
Proposition 5.5**.**
Let denote the set of -representations, and let denote the set of second Frobenius representations of overpartitions. Then is a surjective map.
Taken with Theorem 2.8, we see that every -representation corresponds to an overpartition , although this correspondence is many-to-one. Thus the ranks we will establish to study do not immediately carry over to the set of overpartitions.
5.3. Ranks of -representations
Recall the definition of from Section 4. If
[TABLE]
then we define the first rank of to be
[TABLE]
that is, the second overpartition rank of minus the second partition rank of plus . We also define .
For , we define the th rank of to be
[TABLE]
which is an integer since and have even parts. We also define whenever has fewer than columns.
For example, let
[TABLE]
Then
[TABLE]
and for .
We now establish as the generating series for the ranks of -representations.
5.4. Generating Series
Let denote the set of -representations with at most columns,
[TABLE]
We see the generating series for the th ranks of -representations in in the following theorem.
Theorem 5.6**.**
The coefficient of in
[TABLE]
is equal to the number of -representations such that and , where the count is weighted by .
Proof.
Consider an arbitrary summand of the form
[TABLE]
If , then the summand reduces to , which corresponds to the empty -representation . Otherwise, for some . Let be the smallest index so that . Then the summand reduces to
[TABLE]
We claim that the coefficient of in (5.3) is equal to the number of -representations
[TABLE]
where , such that and , where the count is weighted by . Note that
[TABLE]
The parts of and are generated by the multiplicand, which we write as
[TABLE]
We use the fact that to apply Lemmas 5.2 and 5.3 with and . Then we see that is an overpartition into odd parts with , and is a partition into nonnegative parts where odd parts may not repeat with . Here, the exponents of and track and , respectively.
As in the proof of Theorem 4.6, the term tracks whether or not and are marked with a hat. Thus, the coefficient of in (5.4) is equal to the weighted count of of possible columns in a -representation such that , , , and , where the count is weighted by .
For , the parts of and are generated by the multiplicand, which we write as
[TABLE]
As in the proof of Lemma 5.2, (5.5) generates pairs of columns in the tableau for and . We see that is a nonempty partition into even parts with at least occurrences of its largest part, and is a nonempty partition into nonnegative even parts with at least occurrences of its largest part. Here, the exponents of and track and , respectively. As with the previous column, entries marked with a hat are tracked by . Thus, the coefficient of in (5.5) is equal to the weighted count of of possible columns in a -representation such that , , , and
[TABLE]
where the count is weighted by .
The parts of and are generated by the multiplicand
[TABLE]
By minimality of , we see that . Thus, , and the multiplicand reduces to
[TABLE]
This reflects the fact that neither or can be marked with a hat. As with the previous column, we see that the coefficient of in (5.6) is equal to the weighted count of possible columns of such that , , and , where the count is weighted by .
By combining these terms, we have counted all possible with , , , and entries marked with a hat, where the count is weighted by . By summing over all values of , we count all possible -representations in .
∎
5.5. Full Rank and Proof of Theorem 1.4
As in Section 4, we define the full rank of a -representation to be the sum of the th ranks of ,
[TABLE]
This sum converges for any -representation , as all but finitely many of the summands vanish. We may now prove Theorem 1.4.
Proof of Theorem 1.4.
Let be a primitive th root of unity. The desired generating series,
[TABLE]
is given by
[TABLE]
∎
We now have our combinatorial interpretation of . As in Section 4, the weighted count in (5.7) must vanish for -representations whose full rank is not a multiple of .
We close this section by discussing conjugation maps on .
5.6. Conjugation
Given a -representation
[TABLE]
we define different conjugation maps corresponding to the columns of . To perform the first conjugation, we subtract from each part of and reverse Algorithm 4.1 to obtain a partition into nonnegative even parts and a partition into distinct even parts . We reverse Algorithm 5.1 on and obtain a partition into nonnegative even parts and a partition into distinct odd parts . Note that by construction.
We then perform Algorithm 4.1 on and to produce and perform Algorithm 5.1 on and to produce . Next, add to each part of . Finally, mark with a hat if and only if was marked with a hat, and vice versa. We call
[TABLE]
the first conjugate of .
For example, if
[TABLE]
then we see that
[TABLE]
Performing Algorithms 4.1 and 5.1, produces
[TABLE]
and adding to each part of yields
[TABLE]
For , the th conjugation map is performed as follows. First, subtract 2 from each part of to produce , and add 2 to each part of to produce . Mark with a hat if and only if was marked with a hat, and vice versa. We call
[TABLE]
the th conjugate of . Keeping as above, we have
[TABLE]
Each of the th conjugation maps exchange the roles of
[TABLE]
in (5.1). We find the same relations between conjugation maps as in Section 4.
Proposition 5.7**.**
For all , we have .
Proposition 5.8**.**
For all nonnegative integers and , .
Finally, if we define the full conjugation to be
[TABLE]
then is defined for all , and .
This concludes our results.
6. Conclusion
We began with the series and , which arose from observing a pattern between the generating series of the Dyson ranks and -ranks of overpartitions, and asked whether these new series related to the ranks of overpartitions. By generalizing the notion of Frobenius representations of overpartitions, we found that and are weighted generating series for the full ranks of buffered Frobenius representations, which lie over the set of overpartitions and generalize the first and second Frobenius representations of overpartitions. It is somewhat disappointing then that the full rank functions are not well defined on the set of overpartitions – compare for example
[TABLE]
Note that the full conjugation maps are well-defined. That is, whenever , for . Additionally, it not immediately clear why a sum weighted by roots of unity should produce a meaningful count.
One would hope that there exists a family of “-ranks” of overpartitions, whose generating series are given by
[TABLE]
By setting in (6.1), we at least have that
[TABLE]
as expected. It seems likely that the coefficients are nonnegative integers, which remains open.
It is sufficient that an -rank candidate satisfy
[TABLE]
which is a generalization of Proposition 3.2 [12] and Corollary 1.3 [13]. We see an avenue for this work via the two interpretations of as both the generating series of the -ranks of overpartitions, and as the weighted generating series of the full ranks of -representations in . One might wonder if the parity of determines behavior in . Perhaps understanding how to map will shed light on how to treat the rest of the and . Alternatively, there may be a “th Frobenius representation” of overpartitions closer in spirit to Lovejoy’s work.
Of course, we should be interested in determining the congruences arising from any rank-like function. We may be able to use (6.2) to move from congruences of buffered Frobenius representations back to congruencies of overpartitions.
There is also the question of analytics to consider. Since the series and are related to overpartition ranks, and can be obtained from the -hypergeometric series, it is natural to ask if these series exhibit any modular properties. This could be investigated separately of establishing a higher -rank.
Acknowledgments
The author is very grateful to the referee for uncovering multiple 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.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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- 2[2] George E. Andrews. The theory of partitions . Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. Encyclopedia of Mathematics and its Applications, Vol. 2.
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