SOME OBSERVATIONS ON DYSON’S NEW SYMMETRIES OF PARTITIONS

ALEXANDER BERKOVICH AND FRANK G. GARVAN

Abstract. We utilize Dyson’s concept of the adjoint of a partition to derive new polynomial analogues of Euler’s Pentagonal Number Theorem. We streamline Dyson’s bijection relating partitions with crank ≤ k and those with k in the Rank-Set of partitions. Also, we extend Dyson’s adjoint of a partition to MacMahon’s “modular” partitions with modulus 2. This way we find a new combinatorial proof of Gauss’s famous identity.

1. Introduction

Let p(n) denote the number of unrestricted partitions of n. Ramanujan discovered three beautiful arithmetic properties of p(n), namely:

(1.1)  p(5n + 4) ≡ 0 (mod 5),
(1.2)  p(7n + 5) ≡ 0 (mod 7),
(1.3)  p(11n + 6) ≡ 0 (mod 11).

The partition congruences modulo 5 and 7 were proved by Ramanujan in [14]. In [15] he proved (1.3) by a different method. The most elementary proof of (1.3) similar to the one in [14] is due to Winquist [16].

Dyson [7] discovered empirically remarkable combinatorial interpretations of (1.1) and (1.2). Defining the rank of a partition as the largest part minus the number of parts, he observed that

(1.4)  N(k, 5, 5n + 4) = \frac{p(5n + 4)}{5}, 0 ≤ k ≤ 4,
(1.5)  N(k, 7, 7n + 5) = \frac{p(7n + 5)}{7}, 0 ≤ k ≤ 6,

where N(k, m, n) denotes the number of partitions of n with rank congruent to k modulo m. Identities (1.4) and (1.5) were later proved by Atkin and Swinnerton-Dyer [6]. However, the rank failed to explain (1.3), and so Dyson conjectured the existence of some analogue of the rank that would explicate the Ramanujan congruence modulo 11. He named his hypothetical statistic the crank.

Date: March 12, 2002.

1991 Mathematics Subject Classification. Primary: 11P81, 11P83, 05A17, 33D15.

Key words and phrases. partitions, congruences, Dyson’s rank, cranks, Euler’s pentagonal number theorem, modular partitions, q-series, polynomial analogues.

Research of the second author supported in part by the NSF under grant number DMS-9870052.

Forty four years later, Andrews and Garvan [5], building on the work of Garvan [11], finally unveiled Dyson’s crank of a partition π:

(1.6)  crank(π) = \begin{cases}λ(π), & if μ(π) = 0, \\ ν^{∼}(π) − μ(π), & if μ(π) > 0, \end{cases}

where λ(π) denotes the largest part of π, μ(π) denotes the number of ones in π and ν^{∼}(π) denotes the number of parts of π larger than μ(π).

Remarkably, the crank provides combinatorial interpretations of all three Ramanujan congruences (1.1)–(1.3). Namely,

(1.7)  M(k, 5, 5n + 4) = \frac{p(5n + 4)}{5}, 0 ≤ k ≤ 4,
(1.8)  M(k, 7, 7n + 5) = \frac{p(7n + 5)}{7}, 0 ≤ k ≤ 6,
(1.9)  M(k, 11, 11n + 6) = \frac{p(11n + 6)}{11}, 0 ≤ k ≤ 10,

where M(k, m, n) denotes the number of partitions of n with crank congruent to k modulo m.

Let P_{m}(q) denote the generating function

(1.10)  P_{m}(q) = ∑_{n=1}^{∞} p_{m}(n)q^{n},

where p_{m}(n) is the number of partitions of n with rank m. Here we are using the convention that p_{m}(0) = 0. As a practical tool for his empirical calculations Dyson used the following formula for P_{m}(q):

(1.11)  P_{m}(q) = \frac{1}{(q)_{∞}} ∑_{j≥1}(−1)^{j−1}(1 − q^{j})q^{\frac{j(3j−1)}{2} +|m|j},

with

(1.12)  (q)_{∞} = ∏_{j≥1}(1 − q^{j}), for |q| < 1.

For later use we also define

(1.13)  (a; q)_{n} = (a)_{n} = ∏_{j=0}^{n−1}(1 − aq^{j}), n ≥ 0

and note that \frac{1}{(q)_{∞}} is the generating function for unrestricted partitions.

Dyson knew how to prove (1.11) in 1942 [8]. However the first published proof of (1.11) was given by Atkin and Swinnerton-Dyer [6] in 1954. In 1968, Dyson [9] found a simple combinatorial argument which not only explained (1.11) but also led to a new proof of Euler’s celebrated pentagonal number theorem:

(1.14)  1 = \frac{1}{(q)_{∞}} ∑_{j=−∞}^{∞}(−1)^{j}q^{\frac{j(3j−1)}{2}} .

To paraphrase Dyson’s argument in [9] we introduce the generating function

(1.15)  Q_{m}(q) = ∑_{n≥1} q^{∼}_{m}(n)q^{n},

where q^{∼}_{m}(n) is the number of partitions of n with rank ≥ m. We adopt the convention that

(1.16)  q^{∼}_{m}(0) = 0.

Clearly,

(1.17)  P_{m}(q) = Q_{m}(q) − Q_{m+1}(q).

Next, following the treatment in [9] we will show that

(1.18)  Q_{m}(q) + Q_{1−m}(q) + 1 = \frac{1}{(q)_{∞}} ,

and for m ≥ 0

(1.19)  Q_{m}(q) = q^{m+1} (Q_{−2−m}(q) + 1) .

To prove (1.18) we note that any given nonempty partition π counted by \frac{1}{(q)_{∞}} has either rank ≥ m or rank < m. If rank ≥ m, then π is counted by Q_{m}(q). If rank < m, then we conjugate π to get π^{∗} as illustrated in Fig. 1.

π : ν(π)  λ(π)  π^{∗} : λ(π)  ν(π)

Fig 1. Conjugation of a partition π with largest part λ(π) and the number of parts ν(π), λ(π) − ν(π) < m.

It is obvious in this case that rank(π^{∗}) ≥ 1 − m. Hence, π^{∗} is counted by Q_{1−m}(q). Finally, the empty partition is counted by 1 on the left side of (1.18) and on the right side by \frac{1}{(q)_{∞}}. The proof of (1.19) is more subtle. Here we will use a different conjugation transformation (Dyson’s adjoint) as follows. Consider some partition π with rank(π) ≥ m ≥ 0. This partition is counted by Q_{m}(q) in (1.19). Clearly,

(1.20)  λ(π) − ν(π) ≥ m ≥ 0.

Let us now remove the largest part of π to end up with π^{∼}. Next, we conjugate π^{∼} to get π^{∼}^{∗}. Finally, we attach to π^{∼}^{∗} a new largest part of size λ(π) − m − 1. These transformations are illustrated in Fig. 2.

π:  π^{∼}:  π^{∼}^{∗}:  π′:

Fig 2. Dyson’s adjoint of π = 4 + 3 + 1 with rank = m = 1.

Note that the map π → π′ is reversible. It is obvious that

(1.21)  λ(π′) = λ(π) − m − 1,
(1.22)  ν(π′) ≤ λ(π) + 1,
(1.23)  |π′| = |π| − m − 1,

where |π| denotes the sum of parts of π.

Since rank(π′) ≥ −2 − m, we see that π′ is counted by Q_{−2−m}(q) in (1.19), provided |π′| ≠ 0. If |π| = m + 1 and rank(π) ≥ m, then ν(π) = 1 and λ(π) = 1 + m. So in this case π′ represents the empty partition, which is counted by 1 in (1.19). This concludes the proof of (1.19).

Combining (1.18) and (1.19), we see that

(1.24)  Q_{m}(q) + q^{m+1}Q_{m+3} = \frac{q^{m+1}}{(q)_{∞}} .

Iterating (1.24) we obtain for m ≥ 0

Q_{m}(q) = \frac{q^{m+1}}{(q)_{∞}} − q^{m+1}Q_{m+3}

= \frac{q^{m+1} − q^{2m+5}}{(q)_{∞}} + q^{2m+5}Q_{m+6} = ⋯ (1.25)

= \frac{1}{(q)_{∞}} ∑_{j≥1}(−1)^{j−1}q^{\frac{j(3j−1)}{2} +mj}.

We observe that (1.25) together with (1.18) with m = 0 yields Euler’s theorem (1.14). On the other hand, (1.25) together with (1.17) yields (1.11) with m ≥ 0. To treat the m < 0 case in (1.17) we make use of

(1.26)  P_{m}(q) = P_{−m}(q),

which is a straightforward consequence of the conjugation transformation.

In [2] Andrews utilized Dyson’s adjoint to give a new proof of a partition theorem due to Fine. This seems to be the only known application of the Dyson transformation.

In the next section we will show that Dyson’s formulas (1.18), (1.19) can be generalized to yield a binary tree of polynomial analogues of (1.14). This tree contains Schur’s wellknown formula

(1.27)  1 = ∑_{j=−∞}^{∞}(−1)^{j}q^{\frac{j(3j−1)}{2}} [\begin{matrix}2L \\ L + ⌊\frac{3j}{2} ⌋ \end{matrix}]_{q} ,

as well as a new polynomial version of (1.14)

(1.28)  1 = ∑_{j=−∞}^{∞}(−1)^{j}q^{\frac{j(3j+1)}{2}} [\begin{matrix}2L − j \\ L + j \end{matrix}]_{q} ,

where ⌊x⌋ denotes the integer part of x and q-binomial coefficients are defined as

(1.29)  [\begin{matrix}n + m \\ n \end{matrix}]_{q} = \begin{cases}\frac{(q)_{n+m}}{(q)_{n}(q)_{m}} , & if n, m ≥ 0, \\ 0, & otherwise. \end{cases}

In §§3 and 4 we will streamline and generalize Dyson’s treatment of partitions with crank ≤ k. In §5 we will use modular representations with modulus 2 of partitions in which odd parts do not repeat, and an appropriate modification of Dyson’s adjoint transformation, to obtain a new proof of the Gauss formula

(1.30)  \frac{(q^{2}; q^{2})_{∞}}{(q; q^{2})_{∞}} = ∑_{j≥0} q^{\frac{j(j+1)}{2}} .

We conclude in §6 with a brief description of some open questions for future investigation.

2. Polynomial analogues of Euler’s pentagonal number theorem

We say that a partition π is in the box [L, M] if its largest part does not exceed L and the number of parts does not exceed M. In other words,

λ(π) ≤ L,
ν(π) ≤ M.

It is well known [3] that the generating functions for partitions in the box [L, M] is

[\begin{matrix}L + M \\ L \end{matrix}]_{q} .

Let us define Q_{m}^{L}(q) as

(2.1)  Q_{m}^{L}(q) = ∑_{n≥1} q^{∼}_{m}^{L}(n)q^{n},

where q^{∼}_{m}^{L}(n) is the number of partitions of n with rank ≥ m and largest part ≤ L. As before, we assume that q^{∼}_{m}^{L}(0) = 0. Clearly, Q_{m}^{L}(q) = 0 whenever L ≤ m. We now prove that

(2.2)  Q_{m}^{L}(q) − Q_{m}^{L−1}(q) = q^{L} [\begin{matrix}2L − m − 1 \\ L \end{matrix}]_{q} .

To this end we observe that the left side of (2.2) counts partitions with rank ≥ m and λ(π) = L. We note that these partitions are in the box [L, L − m]. If we remove the largest part L from one of those partitions we obtain a partition in the box [L, L−m−1], and this partition is counted by the q-binomial coefficient on the right side of (2.2), as desired.

We now move on to derive the bounded analogues of (1.18), (1.19), namely:

(2.3)  Q_{m}^{L}(q) + Q_{1−m}^{L−m}(q) + 1 = [\begin{matrix}2L − m \\ L \end{matrix}]_{q} ,  L > m,

and

(2.4)  Q_{m}^{L}(q) = q^{m+1} (Q_{−2−m}^{L−1−m}(q) + 1) ,  L > m ≥ 0.

To prove (2.3) we note that any given nonempty partition π, counted by [\begin{matrix}2L − m \\ m \end{matrix}]_{q}, has

either rank ≥ m or rank < m. Moreover, π is in the box [L, L − m]. Now if rank(π) ≥ m, then π is counted by Q_{m}^{L}(q). If rank(π) < m, then we conjugate π to get π^{∗}. Obviously, π^{∗} is counted by Q_{1−m}^{L−m}(q). Finally, the empty partition is counted by 1 and by the q-binomial coefficient on the left and right sides of (2.3), respectively.

Next, to prove (2.4) we observe that any partition π counted by Q_{m}^{L}(q) is in the box [L, L − m]. Performing Dyson’s transformation π → π′, as explained in the previous section, we see that λ(π′) ≤ L − 1 − m and rank(π′) ≥ −2 − m. Therefore, if |π| ≠ m + 1, then π′ is counted by Q_{−2−m}^{L−1−m}(q). If |π| = m + 1, then π′ is empty. In this case it is counted by 1 on the right side of (2.4).

Combining (2.3) and (2.4) yields

(2.5)  Q_{m}^{L}(q) + q^{m+1}Q_{m+3}^{L+2}(q) = q^{m+1} [\begin{matrix}2L − m + 1 \\ L + 2 \end{matrix}]_{q} , m ≥ 0.

We remark that when L ≤ m the above formula becomes 0 + 0 = 0, and when L tends to infinity (2.5) reduces to (1.24). Actually, it is possible to derive another bounded analogue of (1.24). To this end we employ (2.2) together with the well known recurrence

(2.6)  [\begin{matrix}n + m \\ n \end{matrix}]_{q} = q^{n} [\begin{matrix}n + m − 1 \\ n \end{matrix}]_{q} + [\begin{matrix}n + m − 1 \\ m − 1 \end{matrix}]_{q}

to transform (2.5) as

(2.7)  Q_{m}^{L}(q) + q^{m+1}Q_{m+3}^{L+1}(q) = q^{m+1} \{[\begin{matrix}2L − m + 1 \\ L + 2 \end{matrix}]_{q} − q^{L+2} [\begin{matrix}2L − m \\ L + 2 \end{matrix}]_{q}\}

= q^{m+1} [\begin{matrix}2L − m \\ L + 1 \end{matrix}]_{q} .

The power of (2.5) and (2.7) lies in the fact that these transformations can be employed to generate an infinite binary tree of representations for Q_{m}^{L}(q). Here we confine ourselves to four cases, namely:

(2.8)  Q_{m}^{L}(q) = ∑_{j≥1}(−1)^{j−1}q^{\frac{j(3j−1)}{2} +mj} [\begin{matrix}2L − m + j \\ L − m − j \end{matrix}]_{q} , m ≥ 0,

(2.9)  Q_{m}^{L}(q) = ∑_{j≥1}(−1)^{j−1}q^{\frac{j(3j−1)}{2} +mj} [\begin{matrix}2L − m − j + 1 \\ L + j \end{matrix}]_{q} , m ≥ 0,

(2.10)  Q_{m}^{L}(q) = ∑_{j≥1}(−1)^{j−1}q^{\frac{j(3j−1)}{2} +mj} [\begin{matrix}2L − m + 1 \\ L − ⌊−\frac{3j}{2} ⌋ \end{matrix}]_{q} , m ≥ 0,

(2.11)  Q_{m}^{L}(q) = ∑_{j≥1}(−1)^{j−1}q^{\frac{j(3j−1)}{2} +mj} [\begin{matrix}2L − m \\ L + ⌊\frac{3j}{2} ⌋ \end{matrix}]_{q} , m ≥ 0.

To derive (2.8)–(2.11) we use the iteration schemes which we denote symbolically as

(2.5) − (2.5) − (2.5) − (2.5) − (2.5) − (2.5) − ⋯, (2.12) (2.7) − (2.7) − (2.7) − (2.7) − (2.7) − (2.7) − ⋯, (2.13) (2.5) − (2.7) − (2.5) − (2.7) − (2.5) − (2.7) − ⋯, (2.14) (2.7) − (2.5) − (2.7) − (2.5) − (2.7) − (2.5) − ⋯, (2.15)

respectively. For example, the scheme (2.12) means each transformation uses only equation (2.5), and the scheme (2.14) means that we use both (2.5) and (2.7) in an alternating fashion with (2.5) being used first.

Now, (2.3) with m = 0 yields

(2.16)  Q_{0}^{L}(q) + Q_{1}^{L}(q) + 1 = [\begin{matrix}2L \\ L \end{matrix}]_{q} .

Equation (1.28) then follows by using (2.8) with m = 0 and (2.9) with m = 1.

Schur’s formula (1.27) follows in a similar fashion. We use (2.16), (2.10) with m = 1, (2.11) with m = 0 and the fact that

(2.17)  [\begin{matrix}2L \\ L + a \end{matrix}]_{q} = [\begin{matrix}2L \\ L − a \end{matrix}]_{q} .

We now move on to generalize (1.11). To this end we define P_{m}^{L}(q) as

(2.18)  P_{m}^{L}(q) = ∑_{n≥1} p_{m}^{L}(n)q^{n},

where p_{m}^{L}(n) is the number of partitions of n with largest part ≤ L and rank m. Obviously,

(2.19)  P_{m}^{L}(q) = Q_{m}^{L}(q) − Q_{m+1}^{L}(q).

So using (2.10), (2.11) and (2.17) we obtain (2.20)

P_{m}^{L}(q) = ∑_{j≥1}(−1)^{j−1}q^{\frac{j(3j−1)}{2} +mj} [\begin{matrix}2L − m \\ L + ⌊\frac{3j}{2} ⌋ \end{matrix}]_{q} − ∑_{j≥1}(−1)^{j−1}q^{\frac{j(3j+1)}{2} +mj} [\begin{matrix}2L − m \\ L − ⌊−\frac{3j}{2} ⌋ \end{matrix}]_{q} ,

provided m ≥ 0. Using the obvious conjugation symmetry

(2.21)  P_{−|m|}^{L}(q) = P_{|m|}^{L+|m|}(q)

it is straightforward to extend (2.20) to negative m. This way we obtain the following polynomial analogue of (1.11)

(2.22)  P_{m}^{L}(q) = ∑_{j≥1}(−1)^{j−1}q^{\frac{j(3j−1)}{2} +mj} [\begin{matrix}2L − m \\ L + sign(m)⌊\frac{3j}{2} ⌋ \end{matrix}]_{q}

− ∑_{j≥1}(−1)^{j−1}q^{\frac{j(3j+1)}{2} +mj} [\begin{matrix}2L − m \\ L − sign(m)⌊−\frac{3j}{2} ⌋ \end{matrix}]_{q} ,

where

sign(m) = \begin{cases}1, & if m ≥ 0, \\ −1, & otherwise. \end{cases}

3. Partitions with prescribed cranks

Dyson [7] conjectured that the generating function for the crank should have a form similar to (1.11), and it does as can be seen from the following formula

(3.1)  Ĉ_{k}(q) = \frac{1}{(q)_{∞}} ∑_{j≥1}(−1)^{j−1}q^{T_{j−1}+j|k|}(1 − q^{j}) + q(δ_{k,o} − δ_{k,1}),

where

(3.2)  T_{j} = \frac{j(j + 1)}{2},

and

(3.3)  Ĉ_{k}(q) = ∑_{n≥0} ĉ_{k}(n)q^{n},

with ĉ_{k}(n) denoting the number of partitions of n with crank k. Formula (3.1) is a consequence of Theorem (7.19) in [11] and Theorem 1 in [5].

To explain (3.1) in a combinatorial fashion Dyson [10] introduced the concept of the rank-set R(π) of a partition π = p_{1} + p_{2} + p_{3} + ⋯ with parts p_{1} ≥ p_{2} ≥ p_{3} ≥ · · ·. R(π) is defined as

(3.4)  R(π) = [j − p_{j+1}, j = 0, 1, 2, …].

To prove (3.1) Dyson first established that

(3.5)  C_{k}(q) = G_{k}(q) + qδ_{k,0},

where

C_{k}(q) = ∑_{n≥0} c_{k}(n)q^{n}, (3.6)

G_{k}(q) = ∑_{n≥0} g_{k}(n)q^{n}, (3.7)

with c_{k}(n), g_{k}(n) denoting the number of partitions of n with crank ≤ k and k in the rank-set of these partitions, respectively. In (3.6)–(3.7) we use the convention that c_{k}(0) = g_{k}(0) = 1 if k ≥ 0 and 0, otherwise. He then showed that

(3.8)  G_{−k}(q) + G_{k−1}(q) = \frac{1}{(q)_{∞}} ,

and

(3.9)  G_{k}(q) + q^{k+1}G_{k+1}(q) = \frac{1}{(q)_{∞}} , k ≥ −1.

Iteration of (3.9) yields

(3.10)  G_{k}(q) = \frac{1}{(q)_{∞}} ∑_{j≥0}(−1)^{j}q^{T_{j}+kj}, k ≥ −1.

Now (3.10), (3.5) and the obvious relation

(3.11)  Ĉ_{k}(q) = C_{k}(q) − C_{k−1}(q)

together imply that

(3.12)  Ĉ_{k}(q) = \frac{1}{(q)_{∞}} ∑_{j≥0}(−1)^{j−1} (q^{T_{j−1}+kj} − q^{T_{j}+kj}) + q(δ_{k,0} − δ_{k,1}), k ≥ 0,

which is (3.1) with k ≥ 0. To extend (3.12) to negative k, we observe that (3.8) implies that

(3.13)  Ĝ_{−k}(q) = Ĝ_{k}(q),

where

(3.14)  Ĝ(k) = G_{k}(q) − G_{k−1}(q).

From (3.5) we deduce that

(3.15)  Ĉ_{k}(q) = Ĝ_{k}(q) + q(δ_{k,0} − δ_{k,1}).

If we now replace k by −k in (3.15) with k > 0 and use (3.13) we obtain

(3.16)  Ĉ_{−k}(q) = Ĝ_{−k}(q) = Ĝ_{k}(q), k > 0.

The last equation together with (3.10), (3.14) gives (3.1) for k < 0.

To deal with (3.8), (3.9) Dyson introduced a simple graphical tool to determine whether or not k ∈ R(π). To explain it we follow Dyson [10] and define the boundary of the Ferrers graph of π as the infinite zig-zag line consisting of vertical and horizontal segments each of unit length (see Fig. 3).

Next, we draw two 45o lines, namely

y = k + x, y = 1 + k + x,

as shown in Fig. 4 and Fig. 5.

Let BS_{k}(π) denote the segment of B(π) lying in the strip

k + x ≤ y ≤ k + 1 + x

y  x

Fig 3. Graph of π = 5 + 2 + 1, the boundary B(π) is indicated by the thick line.

1 + kk y  x

Fig 4. Graph of π = 3 + 2 + 1 + 1 with k = 2 ∈ R(π).

1 + kk y  x

Fig 5. Graph of π = 2 + 1 with k = 1 ∉ R(π).

determined by these two lines. Now if BS_{k}(π) is vertical, then k ∈ R(π), otherwise k ∉ R(π). Using this criterion it is easy to verify that ν(π) ≠ 1 + k, whenever k ∈ R(π).

We are now ready to prove (3.8). First, it is obvious that any given partition π counted by \frac{1}{(q)_{∞}} has either −k ∈ R(π) or −k ∉ R(π). In the first case, π is counted by G_{−k}(q) in (3.8). In the second case, BS_{−k}(π) is a horizontal segment, and so if we conjugate π to get π^{∗}, then it is clear that BS_{k−1}(π^{∗}) is vertical. Therefore, k − 1 ∈ R(π^{∗}) and consequently π^{∗} is counted by G_{k−1}(q) in (3.8).

To prove (3.9), we remove the row containing the segment BS_{k}(π) from some given partition π counted by G_{k}(q). Next, we insert a vertical column of height j + k to the right of the rectangle [j, j + k], where j is the length of the row removed. This procedure is illustrated in Fig. 6.

2 j j + k 2 1 π: k 1 + k  j + k π′: k −1 + k 1 j

Fig 6. The transformation π → π′ used in the proof of (3.9) (k ≥ 0).

Let us call the resulting partition π′. It is easy to see that

|π′| = |π| + k,

and, because BS_{−1+k}(π′) is a horizontal segment,

k − 1 ∉ R(π′).

Since the map π → π′ is reversible, we immediately infer that

(3.17)  g_{k}(n) = p(n + k) − g_{k−1}(n + k),  k ≥ 0,

where n = |π|. The last equation can be easily transformed in (3.9).

In [10], Dyson proves (3.5) first by mapping partitions π with k ∈ R(π) onto certain vector partitions introduced in [11], and then mapping these vector partitions onto ordinary partitions with crank ≤ k. This approach involved ten separate cases. Here, we choose to prove (3.5) directly, without any reference to vector partitions. Our analysis requires consideration of only three separate cases, as we now explain.

\underline{Case 1.} Here we consider partitions π with k ∈ R(π) and ν(π) ≥ k + 2. This case is illustrated in Fig. 7.

We now remove the row bounded by the vertical segment BS_{k}(π) and then add a vertical column representing j ones to the resulting graph, where j > 0 is the length of the row removed. We call this last partition π′. It is easy to see that

ν(π′) ≥ k + 2, μ(π′) ≥ j > 0, and ν^{∼}(π′) ≤ j + k,

where μ and ν^{∼} were defined in (1.6). Clearly, crank(π′) = ν^{∼}(π′) − μ(π′) ≤ k. Perhaps, it is not immediately obvious that the map π → π′ is reversible. To see that it is, we consider partitions π′ with crank(π′) ≤ k, μ(π′) > 0 and ν(π′) ≥ k + 2. Next we define j to be the x-coordinate of the intersection point of the line y = x + k and the boundary B(π′). Since ν(π′) ≥ k + 2, j is positive. Moreover, j ≤ μ because otherwise crank(π′)

j + k 2 1 π: k 1 + k  j + k π′: k 1 j 45o  2 j j μ 45o

Fig 7. Map π → π′ from partitions π with k ∈ R(π), ν(π) ≥ k+2 to partitions π′ with crank(π′) ≤ k, μ(π′) > 0, ν(π′) ≥ k + 2.

would be > k. So we can remove from π′ a vertical column of length j representing ones and place it as a row of length j right underneath the [j, j + k] rectangle. This way we obtain π with k ∈ R(π), ν(π) ≥ k + 2.

\underline{Case 2.} Here we consider partitions π with ν(π) ≤ k and unique largest part λ(π). In this case the segment BS_{k}(π) is necessarily vertical, implying that k ∈ R(π). We now transform π into π′ as follows. If |π| > 1, then we add a part of size 1 to π and subtract 1 from λ(π), giving λ(π′) = λ(π) − 1, ν(π′) = ν(π) + 1, μ(π′) > 0. If |π| = 1, then we define π′ = π. It is obvious that the map π → π′ is reversible and that crank(π′) ≤ k − 1, μ(π′) > 0, ν(π′) ≤ k + 1.

\underline{Case 3.} Here we consider partitions π with k ≥ 2, ν(π) ≤ k and the largest part λ(π) is repeated. Once again, it is clear that k ∈ R(π). We now conjugate π to get π′ = π^{∗}. Since the smallest part of π′ is at least 2, we have μ(π′) = 0, and crank(π′) = λ(π′) = ν(π) ≤ k.

We now recall that ν(π) ≠ k + 1 whenever k ∈ R(π). Thus the three cases above are exhaustive. Hence, (3.5) holds for k > 0.

If k < 0 there is no need to consider cases 2 and 3, because there are no partitions with a negative number of parts. In addition, case 1 requires no modification. Hence, (3.5) is valid in this case k < 0, as well.

If k = 0, then there is no need to consider case 3. Once again, case 1 requires no modification. However, in case 2 the map π → π′ is not bijective. To see this, we note that the set of partitions π with ν(π) ≤ 0 is empty, but the set of partitions π′ with crank(π′) ≤ −1, μ(π′) > 0, ν(π′) ≤ 1 consists of the single partition π′ with |π′| = 1, ν(π′) = 1 and crank(π′) = −1. Thus,

(3.18)  c_{0}(n) = g_{0}(n) + δ_{n,1},  n ≥ 0.

The last equation can be easily transformed into (3.5) with k = 0.

4. Partitions with bounds on the largest part and the crank

Let C_{k}^{L}(q), Ĉ_{k}^{L}(q), G_{k}^{L}(q) denote the generating functions for partitions with crank ≤ k and largest part ≤ L, with crank k and largest part ≤ L, with k in the rank-set and largest part ≤ L, respectively. In this section we will establish the following bounded analogues of (3.5) and (3.9):

C_{k}^{L}(q) = G_{k}^{L}(q) + \frac{1 − q}{(q)_{k}} + (q − 1) [\begin{matrix}L + k \\ k \end{matrix}]_{q} + qδ_{k,0}, (4.1)

G_{k}^{L}(q) + q^{k+1}G_{k+1}^{L−1}(q) = \frac{1}{(q)_{L}} , (4.2)

where for the sake of simplicity here (and throughout this section) we assume that 0 ≤ k ≤ L, L ≠ 0, unless otherwise stated.

The proof of (4.2) is essentially the same as that of (3.9). Iterating (4.2) we derive

(4.3)  G_{k}^{L}(q) = ∑_{j=0}^{L}(−1)^{j} \frac{q^{T_{j}+kj}}{(q)_{L−j}} .

To prove (4.1) we need to follow the three separate cases of the map π → π′ we used to prove (3.5).

Case 1 requires no modification. In case 2 the map π → π′ produces partitions π′ with λ(π′) = λ(π) − 1 ≤ L − 1 and, therefore, misses partitions π′ counted by C_{k}^{L}(q) in (4.1) such that λ(π′) = L, μ(π′) > 0 and ν(π′) ≤ k + 1, and when k = 0 this map also misses the partition π′ = 1, as discussed earlier. In other words, the correction term needed in this case is

(4.4)  CT_{2} = q^{1+L} [\begin{matrix}L + k − 1 \\ k − 1 \end{matrix}]_{q} + qδ_{k,0}.

In case 3, the map π → π′ fails to account for partitions π′ counted by C_{k}^{L}(q) such that λ(π′) ≤ k, ν(π′) > L, μ(π′) = 0. The correction term needed in this case is

(4.5)  CT_{3} = \{\frac{1 − q}{(q)_{k}} − ([\begin{matrix}L + k \\ k \end{matrix}]_{q} − q [\begin{matrix}L − 1 + k \\ k \end{matrix}]_{q})\} θ(k > 1),

where

(4.6)  θ(statement) = \begin{cases}1, & if statement is true, \\ 0, & otherwise. \end{cases}

To understand (4.5) we observe that \frac{1−q}{(q)_{k}} is the generating function for partitions without ones and largest part not exceeding k, and

([\begin{matrix}L + k \\ k \end{matrix}]_{q} − q [\begin{matrix}L − 1 + k \\ k \end{matrix}]_{q})

is the generating function for partitions π^{∼} with λ(π^{∼}) ≤ k, ν(π^{∼}) ≤ L, μ(π^{∼}) = 0. Combining (4.4) and (4.5) and using the q-binomial recurrence (2.6) we get the total correction term

(4.7)  T = CT_{2} + CT_{3} = \frac{1 − q}{(q)_{k}} + (q − 1) [\begin{matrix}L + k \\ k \end{matrix}]_{q} + qδ_{k,0},

as desired. Since

(4.8)  Ĉ_{k}^{L}(q) = C_{k}^{L}(q) − C_{k−1}^{L}(q)

we have for L ≥ k > 0

(4.9)  Ĉ_{k}^{L}(q) = ∑_{j=1}^{L}(−1)^{j−1}q^{T_{j−1}+kj} \frac{(1 − q^{j})}{(q)_{L−j}}

+ θ(k > 1)\frac{1 − q}{(q)_{k}} q^{k} + (q − 1)q^{k} [\begin{matrix}L − 1 + k \\ k \end{matrix}]_{q} ,

by using (4.1) and (4.3).

We now derive a very different representation for Ĉ_{k}^{L}(q) using (1.6). Because the crank is defined in (1.6) in a piece-wise fashion we have to treat two separate cases.

\underline{Case A.} Here we consider partitions \overline{π} with crank(\overline{π}) = k > 0, λ(\overline{π}) ≤ L, and μ(\overline{π}) > 0. We decompose the graph of some given \overline{π} as shown in Fig. 8 below.

μ + 1  μ + k μ > 0  ≤ μ  ≤ L 1  2
Fig 8. Decomposition of partition \overline{π} with crank(\overline{π}) = k > 0, L ≥ λ(\overline{π}), μ(\overline{π}) > 0.

From this decomposition it is clear that the generating function for these partitions is

(4.10)  A(q) = ∑_{μ=1}^{L−1} q^{μ}q^{(μ+1)(μ+k)} \frac{1}{(q^{2}; q)_{μ−1}} [\begin{matrix}L − 1 + k \\ μ + k \end{matrix}]_{q} .

\underline{Case B.} Here we consider partitions \overline{π} without ones with crank(\overline{π}) = λ(\overline{π}) = k, 2 ≤ k ≤ L. Clearly, the generating function for these partitions is

(4.11)  B(q) = \frac{q^{k}}{(q^{2}; q)_{k−1}} θ(k > 1).

Combining (4.10), (4.11) we find that

(4.12) Ĉ_{k}^{L}(q) = A(q) + B(q)

= \frac{q^{k}(1 − q)}{(q)_{k}}θ(k > 1) + ∑_{μ=1}^{L−1} \frac{q^{(μ+1)(μ+k)+μ}}{(q^{2}; q)_{μ−1}} [\begin{matrix}L − 1 + k \\ μ + k \end{matrix}]_{q} , 0 < k ≤ L.

Comparing (4.9) and (4.12) we arrive at the following identity

(4.13)  ∑_{j=1}^{L}(−1)^{j−1}q^{T_{j−1}+kj} \frac{(1 − q^{j})}{(q)_{L−j}} = (1 − q) ∑_{μ=0}^{L−1} \frac{q^{(μ+1)(μ+k)+μ}}{(q)_{μ}} [\begin{matrix}L − 1 + k \\ μ + k \end{matrix}]_{q} .

Remarkably, this identity is nothing else but a limiting case of Heine’s second transformation of a _{2}ϕ_{1}-series [13]:

(4.14)  _{2}ϕ_{1} ( \begin{matrix}a, b \\ c \end{matrix} ; q, z) = \frac{(\frac{c}{b})_{∞}(bz)_{∞}}{(c)_{∞}(z)_{∞}}_{2}ϕ_{1} ( \begin{matrix}\frac{abz}{c}, b \\ bz \end{matrix} ; q, \frac{c}{b}) ,

where

(4.15)  _{2}ϕ_{1} ( \begin{matrix}a, b \\ c \end{matrix} ; q, z) = ∑_{n=1}^{∞} \frac{(a)_{n}(b)_{n}}{(c)_{n}(q)_{n}} z^{n}.

To see this we rewrite the left side of (4.13) in q-hypergeometric form as

(4.16)  ∑_{j=1}^{L}(−1)^{j−1}q^{T_{j−1}+kj} \frac{(1 − q^{j})}{(q)_{L−j}} = \frac{q^{k}(1 − q)}{(q)_{L−1}} lim_{c→0}_{2}ϕ_{1} ( \begin{matrix}q^{2}, q^{1−L} \\ c \end{matrix} ; q, q^{L+k}) ,

Here we have used

(4.17)  (q)_{L−j} = \frac{(q)_{L−1}}{(q^{1−L})_{j−1}} (−1)^{j−1}q^{T_{j−2}−(L−1)(j−1)},

and

(4.18)  \frac{1 − q^{1+j}}{1 − q} = \frac{(q^{2})_{j}}{(q)_{j}} ,

along with the trivial relation

(4.19)  lim_{c→0}(c)_{n} = 1.

Next we employ (4.14) with a = q^{2}, b = q^{1−L}, z = q^{L+k} together with

(4.20)  \frac{(q^{1+k})_{∞}}{(q^{L+k})_{∞}} = (q^{1+k})_{L−1},

and

(4.21)  lim_{ρ→∞}(ρ)_{i}ρ^{−i} = (−1)^{i}q^{T_{i−1}}

to derive

∑_{j=1}^{L}(−1)^{j−1}q^{T_{j−1}+kj} \frac{(1 − q^{j})}{(q)_{L−j}} = \frac{q^{k}(1 − q)}{(q)_{L−1}}(q^{1+k})_{L−1} lim_{c→0}_{2}ϕ_{1} ( \begin{matrix}\frac{q^{3+k}}{c} , q^{1−L} \\ q^{1+k} \end{matrix} ; q, cq^{L−1}) (4.22)

= \frac{q^{k}(1 − q)}{(q)_{L−1}}(q^{1+k})_{L−1} ∑_{i=0}^{L−1}(−1)^{i} \frac{(q^{1−L})_{i}}{(q)_{i}(q^{1+k})_{i}} q^{T_{i−1}+(L+k+2)i}.

Finally, verifying that

(4.23)  (−1)^{i} \frac{(q^{1+k})_{L−1}}{(q)_{L−1}} \frac{(q^{1−L})_{i}}{(q^{1+k})_{i}} = q^{T_{i}−Li} [\begin{matrix}L − 1 + k \\ i + k \end{matrix}]_{q}

we see that

(4.24)  ∑_{j=1}^{L}(−1)^{j−1}q^{T_{j−1}+kj} \frac{(1 − q^{j})}{(q)_{L−j}} = (1 − q) ∑_{i=0}^{L−1} \frac{q^{i^{2}+(2+k)i+k}}{(q)_{i}} [\begin{matrix}L − 1 + k \\ i + k \end{matrix}]_{q} .

This last equation is essentially (4.13), as desired.

The q-hypergeometric proof of (4.13) clearly suggests that our analysis can be extended further to treat partitions π with crank(π) = k, λ(π) ≤ L and ν(π) ≤ M. However, we will not pursue this here.

5. A variant of Dyson’s transformation and a new proof of Gauss’s formula

Let e(n) denote the number of partitions of n into distinct odd parts with all other parts being even. The generating function E(q) for these partitions can be written in the form of a product as

(5.1)  E(q) = ∑_{n≥0} e(n)q^{n} = \frac{(−q; q^{2})_{∞}}{(q^{2}; q^{2})_{∞}} .

We will use MacMahon’s graphs with modulus 2 to depict these partitions. For example, the mod 2 graph of the partition π = 7 + 6 + 6 + 5 + 2 is given in Fig. 9.

A nice thing about mod 2 representations of the partitions counted by E(q) is that these representations have certain invariance properties under conjugation. Namely, if we conjugate the mod 2 graph of some given partition counted by E(q) we obtain a partition that is also counted by E(q). For instance if we conjugate the mod 2 graph of the partition depicted in Fig. 9 we get π^{∗} = 10 + 8 + 7 + 1 whose mod 2 graph is given in Fig. 10.

22  12 22 2 22 2 2 2 2 1

Fig 9. mod 2 and regular mod 1 representations of π = 7 + 6 + 6 + 5 + 2

12 2 2 12 2 2 22 2 2 2 2

Fig 10. mod 2 representation of π^{∗} = 10 + 8 + 7 + 1

Note that the ordinary Ferrers graph representations do not possess this invariance property. For example, if we conjugate the mod 1 graph in Fig. 9 we get the partition 5 + 5 + 4 + 4 + 4 + 3 + 1, which has repeated odd part 5.

Next, we define the M_{2}-rank of a partition as the largest row minus the number of rows of its mod 2 graph. It is easy to check that the M_{2}-rank of the partition, 7 + 6 + 6 + 5 + 2, depicted in Fig. 9, is equal to 4 − 5 = −1, while its rank is 7 − 5 = 2. Also, it is straightforward to verify that under conjugation the M_{2}-rank changes its sign, as does the ordinary rank.

Let us define E^{∼}_{r}(q) as

(5.2)  E^{∼}_{r}(q) = ∑_{n≥1} e^{∼}_{r}(q)q^{n},

where e^{∼}_{r}(n) denotes the number of partitions of n into distinct odd parts and unrestricted even parts such that the M_{2}-rank ≥ r. We assume that e^{∼}_{r}(0) = 0. We now show that

(5.3)  E^{∼}_{r}(q) + E^{∼}_{1−r}(q) + 1 = E(q),

and

(5.4)  E^{∼}_{r}(q) = q^{2r+1} (E^{∼}_{−1−r}(q) + 1) , r ≥ 0.

To prove (5.3) we will follow a well-trodden path and observe that any nonempty partition counted by E(q) whose M_{2}-rank ≥ r is also counted by E^{∼}_{r}(q). Any nonempty partition counted by E(q) whose M_{2}-rank < r gives rise to a partition with M_{2}-rank ≥ −1−r, after conjugation. Thus, this conjugated partition is counted by E^{∼}_{−1−r}(q) in (5.3). Finally, the empty partition is counted by 1 and E(q) on the left and right sides of (5.3), respectively.

The proof of (5.4) requires modification of Dyson’s transformation, which we now proceed to describe. Let π denote the mod 2 graph of some partition counted by E^{∼}_{r}(q) in (5.4). Let r + ℓ(π) denote the length of the largest row of π, and h(π) denote the number of rows of π. Clearly,

h(π) ≤ ℓ(π)

for π to have M_{2}-rank ≥ r. Next, we remove the largest row from π to get a mod 2 graph π^{∼}. Conjugating π^{∼} we obtain π^{∼}^{∗}. Now, if the removed row represented the odd part 2ℓ + 2r − 1, then we add to π^{∼}^{∗} a new largest row of length ℓ − 1, representing the even part 2ℓ − 2. On the other hand, if the removed row represented the even part 2ℓ + 2r, then we add to π^{∼}^{∗} a new largest row of length ℓ, representing the odd part 2ℓ − 1. These operations are illustrated in Fig. 11, 12 where the resulting partition is denoted by π′.

2 ^{2} ^{2} ^{1} π: ^{2}  ^{1}^{2}^{2}  π^{∼}: ^{2}  ^{1}^{2}^{2}  π^{∼}^{∗}: ^{2}^{2}^{1}^{2}  ^{2}^{2}^{1}^{2}^{2} ^{2} ^{2} π′:

Fig 11. Modification of Dyson’s adjoint for π = 7 + 5 + 2 with M_{2}-rank = 1 > r = 0.

2 ^{2} ^{2} ^{2} π: ^{2}  ^{1}^{2}^{2}  π^{∼}: ^{2}  ^{1}^{2}^{2}  π^{∼}^{∗}: ^{2}^{2}^{1}^{2}  ^{2}^{2}^{1}^{2}^{2} ^{2} ^{2} ^{1} π′:

Fig 12. Modification of Dyson’s adjoint for π = 8 + 5 + 2 with M_{2}-rank = 1 > r = 0.

Remarkably, regardless of whether the largest part of π is even of odd we have

(5.5)  |π′| = |π| − 2r − 1,

and

(5.6)  M_{2}-rank(π′) ≥ −1 − r.

It is easy to check that the map π → π′ is reversible, except when |π| = 2r + 1. In the last case π′ is empty. This concludes the proof of (5.4).

Combining (5.3), (5.4) we obtain

(5.7)  E^{∼}_{r}(q) + q^{2r+1}E^{∼}_{2+r}(q) = q^{2r+1} \frac{(−q; q^{2})_{∞}}{(q^{2}; q^{2})_{∞}} , r ≥ 0.

Iteration of (5.7) yields

(5.8)  E^{∼}_{r}(q) = \frac{(−q; q^{2})_{∞}}{(q^{2}; q^{2})_{∞}} ∑_{j≥1}(−1)^{j−1}q^{2rj+j(2j−1)}, r ≥ 0.

Now (5.3) with r = 0 states that

(5.9)  E^{∼}_{0}(q) + E^{∼}_{1}(q) + 1 = \frac{(−q; q^{2})_{∞}}{(q^{2}; q^{2})_{∞}} .

Thanks to (5.8) we may cast (5.9) in the form

(5.10)  1 = \frac{(−q; q^{2})_{∞}}{(q^{2}; q^{2})_{∞}} ∑_{j=−∞}^{∞}(−1)^{j}q^{j(2j+1)},

and so

(5.11)  \frac{(q^{2}; q^{2})_{∞}}{(−q; q^{2})_{∞}} = ∑_{j=−∞}^{∞}(−1)^{j}q^{j(2j+1)}.

Finally, replacing q by −q in (5.11) we obtain the Gauss identity

(5.12)  \frac{(q^{2}; q^{2})_{∞}}{(q; q^{2})_{∞}} = ∑_{j=−∞}^{∞} q^{j(2j+1)} = ∑_{j≥0} q^{T_{j}}.

Formula (5.8) implies that

(5.13)  Ê_{r}(q) = E(q) − E^{∼}_{r+1}(q)

= \frac{(−q; q^{2})_{∞}}{(q^{2}; q^{2})_{∞}} ∑_{j≥0}(−1)^{j}q^{2rj+j(2j+1)}, r ≥ 0,

where Ê_{r}(q) denotes the generating function for partitions into distinct odd, and unrestricted even parts with M_{2}-rank ≤ r. We now develop very different representations for Ê_{r}(q). To this end we decompose partitions counted by Ê_{r}(q) into even and odd parts. Let’s assume that this decomposition gives π_{1} with j distinct odd parts and π_{2} with i even parts. Clearly, λ(π_{1}) ≤ 2(i + j + r) − 1 and λ(π_{2}) ≤ 2(i + j + r), and so, for r ≥ 0 we have

(5.14)  Ê_{r}(q) = ∑_{i,j≥0} q^{j+2T_{j−1}} [\begin{matrix}i + j + r \\ j \end{matrix}]_{q^{2}} q^{2i} [\begin{matrix}i + j + r − 1 + i \\ i \end{matrix}]_{q^{2}} .

Comparing (5.13) and (5.14), we see that (5.15)

∑_{i,j≥0} q^{j^{2}+2i} [\begin{matrix}i + j + r \\ j \end{matrix}]_{q^{2}} [\begin{matrix}2i + j + r − 1 \\ i \end{matrix}]_{q^{2}} = \frac{(−q; q^{2})_{∞}}{(q^{2}; q^{2})_{∞}} ∑_{j≥0}(−1)^{j}q^{j(2j+1)} (q^{2r})^{j} , r ≥ 0.

Next, since

(5.16)  [\begin{matrix}n + m \\ n \end{matrix}]_{q} = \frac{(q^{1+m}; q)_{n}}{(q; q)_{n}} ,

we can rewrite (5.13) as

(5.17)  ∑_{i,j≥0} q^{j^{2}+2i} \frac{(aq^{2i+2}; q^{2})_{j}(aq^{2i+2j}; q^{2})_{i}}{(q^{2}; q^{2})_{j}(q^{2}; q^{2})_{i}} = \frac{(−q; q^{2})_{∞}}{(q^{2}; q^{2})_{∞}} ∑_{j≥0}(−1)^{j}q^{2j^{2}+j}a^{j},

where a = q^{2r}, r ≥ 0. Since the limit of the sequence {q^{2r}} is equal to zero, we may treat a in (5.17) as a free parameter.

In the past, fundamental as they are, modular representations have not received the attention they deserve. Recently, Alladi [1] used 2-modular representations to provide an elegant combinatorial bijection for a variant of Göllnitz’s partition theorem. However, in [1] partitions into only distinct odd parts are considered, whereas here we allow even parts to appear with possible repetition.

In this regard, Alladi pointed out to us that Andrews [4, ex.6, p.13] used mod 2 representations on the set of partitions treated here, subject to the extra condition that no part = 1, in order to establish a partition theorem, which is equivalent to Cauchy’s identity in the form:

(5.18)  ∑_{n≥0} \frac{(−aq; q^{2})_{n}}{(q^{2}; q^{2})_{n}} t^{n}q^{2n} = \frac{(−atq^{3}; q^{2})_{∞}}{(tq^{2}; q^{2})_{∞}} .

6. Open questions

In [4] Andrews proposed a dissection of a partition π into successive Durfee squares with sizes n_{1}(π) ≥ n_{2}(π) ≥ n_{3}(π) ≥ · · ·. For example, the partition π, depicted in Fig. 13, has two Durfee squares with sizes n_{1}(π) = 3, n_{2}(π) = 2.

Fig 13. Ferrers graph of π = 6 + 5 + 4 + 2 + 2 + 1. This graph can be dissected into two Durfee squares of sizes 3 and 2. 3-rank(π) = 2 − 1 = 1.

Garvan [12] introduced a generalization of Dyson’s rank for partitions with at least k − 1 successive Durfee squares. He called this generalization the k-rank of a partition π. The k-rank is defined as

(6.1)  k-rank(π) = the number of columns in the Ferrers graph of π which lie to the right of the first Durfee square and whose length ≤ n_{k−1}(π) minus the number of parts of π that lie below the (k − 1)-th Durfee square.

For instance, the partition π depicted in Fig. 13 has 3-rank(π) = 2 − 1 = 1. Since any nonempty partition π has at least one Durfee square we can easily infer that the 2-rank is the same as Dyson’s rank.

Formula (1.10) in [12] implies that for m ≥ 0

(6.2)  F G_{k,m}(q) = \frac{1}{(q)_{∞}} ∑_{j=1}^{∞}(−1)^{j−1}q^{\frac{j((2k−1)j−1)}{2} +mj},

where F G_{k,m}(q) denotes the generating function for partitions π with at least k − 1 successive Durfee squares and with k-rank(π) ≥ m ≥ 0. Using (6.2) it is easy to verify that

(6.3)  F G_{k,m}(q) + q^{k+m−1}F G_{k,2k−1+m}(q) = \frac{q^{k+m−1}}{(q)_{∞}} .

We note that (6.3) with k = 2 becomes (1.24). Despite its speciously simple appearance the functional equation (6.3) with k > 2 turned out to be very difficult to prove in a combinatorial fashion. Perhaps the appropriate generalization of Dyson’s notion of rankset may provide a key to a combinatorial proof of (6.3).

We feel that it would be worthwhile to determine the precise q-hypergeometric status of the new polynomial analogue of Euler’s pentagonal number theorem (1.28). Finally, we would like to pose the problem of finding a natural bounded extension of formulas (1.4), (1.5), and (1.7)–(1.9).

Acknowledgements. We are grateful to Krishnaswami Alladi for interesting discussion and for his comments and suggestions, and to George E. Andrews for his help and encouragement.

References

[1] K. Alladi, A variation on a theme of Sylvester — a smoother road to Göllnitz’s (Big) theorem, Discrete Mat. 196 (1999), 1–11.

[2] G. E. Andrews, On a partition theorem of N. J. Fine, J. Nat. Acad. Math. India 1 (1983), 105–107.

[3] G. E. Andrews, The Theory of Partitions, Encyclopedia of Mathematics and Its Applications, Vol. 2 (G. - C. Rota, ed.), Addison-Wesley, Reading, Mass., 1976. (Reissued: Cambridge Univ. Press, London and New York, 1985)

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[8] F. J. Dyson, A walk through Ramanujan’s garden, in “Ramanujan revisited” (Urbana-Champaign, Ill., 1987), Academic Press, Boston, MA, 1988.

[9] F. J. Dyson, A new symmetry of partitions, J. Combinatorial Theory 7 (1969), 56–61.

[10] F. J. Dyson, Mappings and symmetries of partitions, J. Combin. Theory Ser. A 51 (1989), 169–180.

[11] F. G. Garvan, New combinatorial interpretations of Ramanujan’s partition congruences mod 5, 7 and 11, Trans. Amer. Math. Soc. 305 (1988), 47–77.

[12] F. G. Garvan, Generalizations of Dyson’s rank and non-Rogers-Ramanujan partitions, Manuscripta Math. 84 (1994), 343–359.

[13] G. Gasper and M. Rahman, Basic hypergeometric series, Cambridge Univ. Press, Cambridge, 1990.

[14] S. Ramanujan, Some properties of p(n), the number of partitions of n, Proc. Cambridge Philos. Soc. 19 (1919), 207–210.

[15] S. Ramanujan, Congruence properties of partitions, Math. Zeitschr. 9 (1921), 147–153.

[16] L. Winquist An elementary proof of p(11m + 6) ≡ 0 (mod 11), J. Combin. Theory 6 (1969), 56–59.

E-mail address: alexb@math.ufl.edu

Department of Mathematics, University of Florida, Gainesville, FL 32611

E-mail address: frank@math.ufl.edu

22 A. Berkovich and F.G. Garvan

Department of Mathematics, University of Florida, Gainesville, FL 32611