MORE CRANKS AND t-CORES

F. G. GARVAN

Abstract. In 1990, new statistics on partitions (called cranks) were found which combinatorially prove Ramanujan’s congruences for the partition function modulo 5, 7, 11 and 25. The methods are extended to find cranks for Ramanujan’s partition congruence modulo 49. A more explicit form of the crank is given for the modulo 25 congruence.

Dedicated to George Szekeres on the occasion of his 90th Birthday

1. Introduction

Let p(n) be the number of partitions of n [1]. If α ≥ 1, and δ_{α}, λ_{α}, μ_{α} are the reciprocals of 24 modulo 5^{α}, 7^{α}, 11^{α} respectively, then

(1.1)  p(5^{α}n + δ_{α}) ≡ 0 (mod 5^{α}),
(1.2)  p(7^{2α−1}n + λ_{2α−1}) ≡ 0 (mod 7^{α}),
(1.3)  p(7^{2α}n + λ_{2α}) ≡ 0 (mod 7^{α+1}),
(1.4)  p(11^{α}n + μ_{α}) ≡ 0 (mod 11^{α}).

These are Ramanujan’s partition congruences. Watson [9] proved (1.1), (1.2), (1.3) and Atkin [3] proved (1.4). Dyson [5] was the first to consider explaining these congruences combinatorially. Dyson defined an integral statistic on partitions, called the rank, whose value mod 5 he conjectured split the partitions of 5n + 4 into 5 equal classes, thus giving a combinatorial refinement for the α = 1 case of (1.1). He further conjectured that the analogous result for the rank mod 7 gave the α = 1 case of (1.2), and that there was a statistic, called the crank, which would similarly give the α = 1 case of (1.4). Atkin and Swinnerton-Dyer [4] proved Dyson’s conjecture for 5 and 7. Andrews and Garvan [2] were able to find a crank which not only solved Dyson’s crank conjecture for 11 but gave new interpretations for 5 and 7. Later, Garvan, Kim and Stanton [6] found new cranks which gave new interpretations of Ramanujan’s congruences mod 5, 7, 11, and 25. Their approach was combinatorial and in terms of the t-core of a partition. They gave explicit bijections between the equinumerous classes. In the present paper we extend the methods of [6] and give a crank which is a combinatorial refinement of the α = 1 case of (1.3), namely

(1.5)  p(49n + 47) ≡ 0 (mod 49).

Date: July 22, 2001.

1991 Mathematics Subject Classification. Primary: 11P83; Secondary: 05A17.

Key words and phrases. partitions, congruences, t-cores, Dyson’s rank, cranks.

Research supported in part by the NSF under grant numbers DMS-9208813 and DMS-9870052.

In §2 we reexamine two bijections from [6]. A crank for the partitions of 25n+24 mod 25 was given in [6]. A more explicit form of this crank is given in Theorem 3.4. A new and explicit crank for the 7-cores of 49n + 47 is given in Theorem 3.5. This leads to a crank for the partitions of 49n + 47 (Corollary 3.1).

2. Two bijections for t-cores

We need to examine in detail the two bijections relating partitions and t-cores which were given in [6]. Following [6] we let P be the set of all partitions. For any λ ∈ P, let |λ| denote the number that λ partitions. Fix a positive integer t. Let P_{t-core} be the set of partitions which are t-cores. Recall, that a partition is a t-core if it has no hook numbers that are multiples of t or equivalently no rim hooks that are multiples of t. See [7] for background on t-cores, hook numbers and rim hooks. We let a_{t}(n) denote the number of partitions of n which are t-cores.

Bijection 1. ([7, 2.7.17], [6, p.2]) There is a bijection ϕ_{1} : P → P_{t-core}×P ×⋯×P,

ϕ_{1}(λ) = (λ̃, λ̂_{0}, λ̂_{1}, …, λ̂_{t−1}),

such that

|λ| = |λ̃| + t ∑_{i=0}^{t−1} |λ̂_{i}|.

Corollary 2.1.

∑_{n≥0} a_{t}(n)q^{n} = ∏_{n=1}^{∞} \frac{(1 − q^{tn})^{t}}{(1 − q^{n})} .

Given a partition λ we label a cell in the i-th row and j-th column by j − i (mod t). The resulting diagram is called a t-residue diagram [7, p.84]. We form the extended t-residue diagram by adding an infinite column 0 labelled in the same way. A region r of the extended diagram is the set of cells (i, j) with t(r −1) ≤ j −i < tr. A cell is exposed if it is at the end of a row. The partition λ is a t-core if and only if for each exposed cell labeled i in region r there is an exposed cell labeled i in each region < r. Now we construct t bi-infinite words W_{0}, W_{1}, …, W_{t−1} of words of two letters N (not exposed) and E (exposed):

The j-th element of W_{i} = \{ \begin{matrix}N & if i is not exposed in region j, \\ E & if i is exposed in region j. \end{matrix}

We now give the bijection. For each i we do the following steps: Step 1. Find the right most E. Step 2. Find the right most N to the left of this E. If no such N exists then END. Step 3. Remove the rim hook whose head is at E and whose tail is one cell to the

right of the N. Place a part of size (rim hook removed)/t in λ_{i}. Step 4. Go to Step 1. The operation in Step 3 above changes a substring of W_{i} of the form NEE … EEN to EEE … ENN, i.e. the N is pushed right.The other words W_{j} are left unchanged by removing this rim hook, and we can process the i’s in any order. Steps 1–4 create a partition λ_{i} starting from the smallest part to the largest part and the process is easily reversible. At the end when all the W_{i} have been processed we are left with the required t-core λ̃.

Bijection 2. ([6, p.3]) There is a bijection ϕ_{2} : P_{t-core} → {n⃗ = (n_{0}, n_{1}, …, n_{t−1}) : n_{i} ∈ ℤ, n_{0} + ⋯ + n_{t−1}}, where

|λ̃| = t||n⃗||^{2}/2 + b⃗ · n⃗, b⃗ = (0, 1, …, t − 1).

For a partition λ, we let r_{k}(λ) denote the number of cells in the t-residue diagram labeled k (mod t), and call

r⃗ = (r_{0}, r_{1}, …, r_{t−1})

the r-vector of λ. Bijection 2 is given by

(2.1)  ϕ_{2}(λ̃) = n⃗ = (r_{0} − r_{1}, r_{1} − r_{2}, …, r_{t−1} − r_{0}).

Let [x] denote the greatest integer ≤ x. We will need the following Lemma 2.1. Let λ : λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{m} be a partition and suppose

ϕ_{1}(λ) = (λ̃, λ̂_{0}, λ̂_{1}, …, λ̂_{t−1}).

Then

(2.2)  ∑_{i=0}^{t−1} |λ̂_{i}| = r_{0} − (∑_{i=0}^{t−1} r_{i}^{2} − r_{i}r_{i+1}) ,

and (2.3)

∑_{i=0}^{t−1} i|λ̂_{i}| ≡ ∑_{j=1}^{m}(λ_{j} − j) [\frac{λ_{j} − j}{t} ] − ∑_{i=1}^{t−1} id_{i} (\frac{1}{2}(d_{i} + 1) + [\frac{m − i − 1}{t}]) (mod t),

where d_{i} is the number of elements of the sequence

λ_{1} − 1, λ_{2} − 2, …, λ_{m} − m,

which are congruent to i (mod t).

Proof. For t-cores, we have

r_{0} = ∑_{i=0}^{t−1}(r_{i}^{2} − r_{i}r_{i+1}).

See [6, p.6]. Now suppose r⃗ is the r-vector of λ and r⃗′ is the r-vector of its t-core λ̃. The partition λ̃ is obtained from λ by the removal of rim hooks whose lengths are multiples of t. Each rim hook of length t contains cells with distinct t-residues. It follows that

r′_{i} + s = r_{i}

where

s = ∑_{j=0}^{t−1} |λ̂_{j}|.

Since r⃗′ is the r-vector of a t-core we have

r′_{0} = ∑_{i=0}^{t−1}(r′_{i}^{2} − r′_{i}r′_{i+1}),

r_{0} − s = ∑_{i=0}^{t−1}((r_{i} − s)^{2} − (r_{i} − s)(r_{i+1} − s)),

= ∑_{i=0}^{t−1}(r_{i}^{2} − r_{i}r_{i+1}),

and (2.2) follows.

We add t dummy zeros to the parts of λ:

λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{m} ≥ 0 ≥ ⋯ ≥ 0,

and form the sequence

λ̄ : λ_{1} − 1 > λ_{2} − 2 > ⋯ > λ_{m} − m > −m − 1 > ⋯ > −m − t.

Let

μ̄_{i} : μ_{i,1} > μ_{i,2} > ⋯ > μ_{i,k_{i}}

be the terms of the sequence λ̄ that are congruent to i (mod t). Here k_{i} is the terms so that d_{i} = k_{i} − 1. Each μ_{i,k} corresponds to an exposed cell labeled i in region [μ_{i,k}/t] + 1. In Bijection 1, the numbers

n_{i,k} = [\frac{μ_{i,k}}{t} ] − [\frac{μ_{i,k+1}}{t} ] − 1

correspond to a string of n_{i,k} consecutive N’s in the word W_{i}. Since these N’s are shifted as far as possible to the right we find that the sum of parts of the (i + 1)th component

|λ̂_{i}| = n_{i,1} + 2n_{i,2} + ⋯ (k_{i} − 1)n_{i,k−1}

= [\frac{μ_{i,1}}{t} ] + ⋯ + [\frac{μ_{i,k_{i}−1}}{t} ]

− \frac{1}{2}k_{i}(k_{i} − 1) − (k_{i} − 1) [\frac{μ_{i,k_{i}}}{t} ] .

Since μ_{i,k} ≡ i (mod t) we find that

∑_{i=0}^{t−1} i ([\frac{μ_{i,1}}{t} ] + ⋯ + [\frac{μ_{i,k_{i}−1}}{t} ])

≡ ∑_{j=1}^{m}(λ_{j} − j) [\frac{λ_{j} − j}{t} ] (mod t).

The desired result (2.3) follows from the fact that d_{i} = k_{i} − 1 and that

[\frac{μ_{i,k_{i}}}{t} ] = [\frac{m − i − 1}{t}] .

□

3. Cranks for t-cores and partitions

We need the crank results in [6]. The following theorem follows from [6, Theorem 1]. Theorem 3.1. ([6]) If (t, δ) = (5, 4), (7, 5) or (11, 6), then

∑_{n≥0} a_{t}(tn + δ)q^{n+1} = ∑_{α⃗∈ℤ^{t},α⃗·1=1} q^{Q(α⃗)},

where

Q(α⃗) = ||α⃗||^{2} − ∑_{i=0}^{t−1} α_{i}α_{i+1}.

The form Q(α⃗) remains invariant under a cyclic permutation of the α_{i}. This induces a t-cycle on t-cores of tn + δ, which in turn induces a t-cycle on partitions of tn + δ via Bijection 1. For the form Q(α⃗) the associated crank statistic is ∑_{i=0}^{t−1} iα_{i}. This leads to crank statistics for t-cores of tn + δ, and for partitions of tn + δ.

3.1. Cranks for partitions of 5n + 4 and 25n + 24. Theorem 3.2. ([6, p.7]) Let r⃗ = (r_{0}, r_{1}, …, r_{4}) be the r-vector of λ, a 5-core of 5n + 4. Then

(3.1)  c_{1}(λ) := 2r_{1} − r_{2} + r_{3} − 2r_{4} (mod 5) ∈ ℤ_{5}

is a crank for 5-cores of 5n + 4.

We make explicit the 5-cycle σ that acts of 5-cores of 5n + 4. We let P_{t-core}(m) denote the set of t-cores of m. For 0 ≤ j ≤ 4 we let P_{t-core}^{j}(m) denote the set of t-cores λ̃ of m, with crank c_{1}(λ̃) ≡ j (mod 5). For a t-core λ̃ we call n⃗ = ϕ_{2}(λ̃) its n-vector. We define the 5-cycle σ in terms of n-vectors. The map

σ : P_{5-core}(5n + 4) ⟶ P_{5-core}(5n + 4)

is defined by

n⃗ ↦ (−\frac{2 n_{0}}{5} + \frac{n_{1}}{5} + \frac{4 n_{2}}{5} + \frac{2 n_{3}}{5} + \frac{3}{5}, −n_{3}, −\frac{3 n_{0}}{5} − \frac{6 n_{1}}{5} − \frac{4 n_{2}}{5} − \frac{2 n_{3}}{5} + \frac{2}{5},

−\frac{n_{0}}{5} + \frac{3 n_{1}}{5} − \frac{3 n_{2}}{5} + \frac{n_{3}}{5} − \frac{1}{5}, \frac{6 n_{0}}{5} + \frac{2 n_{1}}{5} + \frac{3 n_{2}}{5} + \frac{4 n_{3}}{5} − \frac{4}{5})

For each 0 ≤ j ≤ 4, the map

σ : P_{5-core}^{j}(5n + 4) ⟶ P_{5-core}^{j+1}(5n + 4)

is a bijection.

The key to finding a crank for partitions of 25n + 24 in [6] was a bijective proof of the identity

(3.2)  a_{5}(5n + 4) = 5a_{5}(n).

The map

θ : P_{5-core}(n) ⟶ P_{5-core}^{0}(5n + 4)

defined by

n⃗ ↦ (n_{1} + 2n_{2} + 2n_{4} + 1, −n_{1} − n_{2} + n_{3} + n_{4} + 1, 2n_{1} + n_{2} + 2n_{3},
− 2n_{2} − 2n_{3} − n_{4} − 1, −2n_{1} − n_{3} − 2n_{4} − 1)

is a bijection. See [6, p.8]. This together with Theorem 3.2 yields a combinatorial proof of (3.2).

We now describe the crank for 5-cores of 25n + 24 found in [6]. For λ ∈ P_{5-core}(25n + 24) choose the unique λ′ ∈ P_{5-core}^{0}(25n + 24) which is in the same orbit as λ under the 5-cycle σ. Define

(3.3)  c_{2}(λ) := c_{1}(θ^{−1}(λ′)).

Let n⃗ = θ^{−1}(λ′). By (2.1)

c_{2}(λ′) = c_{1}(n⃗) = 2n_{1} + n_{2} + 2n_{3}.

Observe that this is the third component in the n-vector of θ(n⃗) = λ′. It follows that

(3.4)  c_{2}(λ′) = r_{2} − r_{3},

where r⃗ is the r-vector of λ′. Unfortunately, it is not true in general that c_{2}(λ′) ≡ c_{2}(λ) (mod 5). Nonetheless we can find a crank for 5-cores of 25n+24 independent of the two maps σ and θ. We have the following Theorem 3.3. Let r⃗ = (r_{0}, r_{1}, …, r_{4}) be the r-vector of λ, a 5-core of 25n + 24. Then

c(λ) := (c_{1}(λ), c_{2}(λ))

= (2r_{1} − r_{2} + r_{3} − 2r_{4}, r_{2} − r_{3}) (mod 5) ∈ ℤ_{5} × ℤ_{5} (3.5)

is a crank for 5-cores of 25n + 24.

Proof. For each (i, j) in ℤ_{5} × ℤ_{5}, we let P_{5-core}^{i,j}(25n + 24) be the set of 5-cores λ of 25n + 24 such that c(λ) ≡ (i, j) (mod 5). The map

Ψ = θσθ^{−1} : P_{5-core}^{0,j}(25n + 24) ⟶ P_{5-core}^{0,j+1}(25n + 24)

is a bijection. We have calculated the effect σ has on our crank statistics c_{1}, c_{2}. A calculation shows that the map

σ : P_{5-core}^{i,j}(25n + 24) ⟶ P_{5-core}^{i+1,i^{2}+i+j+2}(25n + 24)

is a bijection. We omit the details. We note that the indices are reduced mod 5. Using the maps Ψ and σ we find that

|P_{5-core}^{i,j}(25n + 24)| = |P_{5-core}^{0,0}(25n + 24)| = \frac{1}{25} a_{5}(25n + 24),

for 0 ≤ i, j ≤ 4. Hence c = (c_{1}, c_{2}) (mod 5) is a crank for 5-cores of 25n + 24 (mod 25). □

A crank for partitions of 25n + 24 is given in [6, Theorem 6]. This crank is algorithmic in nature. It depends on Bijection 1, and the map θ. In view of Lemma 2.1 and Theorem 3.3, we may define a crank independent of these maps. For a partition λ : λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{m}, with r-vector r⃗ = (r_{0}, …, r_{t−1}), the definition of c_{1}(λ) and c_{2}(λ) is analogous to that given for t-cores in (3.1), (3.4) respectively. We need two more statistics. We define

(3.6)  s(λ) := r_{0} − (∑_{i=0}^{t−1} r_{i}^{2} − r_{i}r_{i+1}) ,

and

(3.7)  c_{3}(λ) := ∑_{j=1}^{m}(λ_{j} − j) [\frac{λ_{j} − j}{t} ] − ∑_{i=1}^{t−1} id_{i} (\frac{1}{2}(d_{i} + 1) + [\frac{m − i − 1}{t}]) ,

where d_{i}(λ) is the number of elements of the sequence

λ_{1} − 1, λ_{2} − 2, …, λ_{m} − m,

which are congruent to i (mod t). Now let λ be any partition of 25n + 24, and suppose

ϕ_{1}(λ) = (λ̃, λ̂_{0}, λ̂_{1}, …, λ̂_{4}).

Then by Lemma 2.1,

∑_{i=0}^{4} |λ̂_{i}| = s(λ),

and

∑_{i=0}^{4} i|λ̂_{i}| ≡ c_{3}(λ) (mod 5).

If s(λ) ≡ 0 (mod 5), then λ̃ is a 5-core with |λ̃| ≡ 24 (mod 25) and

c(λ) := (c_{1}(λ), c_{2}(λ)) ≡ (c_{1}(λ̃), c_{2}(λ̃)) (mod 5),

since the sum of the coefficients in the definitions of c_{1}, c_{2} is zero. By rewriting [6, Theorem 6] in terms of our new statistics we obtain a bijection independent crank. Theorem 3.4. Let r⃗ = (r_{0}, r_{1}, …, r_{4}) be the r-vector of a partition λ of 25n + 24. We define a crank c(λ) ∈ ℤ_{5} × ℤ_{5} as follows.

If s(λ) ≡ 0 (mod 5) we define

c(λ) := (c_{1}(λ), c_{2}(λ))

= (2r_{1} − r_{2} + r_{3} − 2r_{4}, r_{2} − r_{3}). (3.8)

If s(λ) ≢ 0 (mod 5) we define

(3.9)  c(λ) := (c_{1}(λ), c_{3}(λ)).

Then c(λ) is a crank for the partitions of 25n + 24 mod 25.

The proof utilises Theorem 3.3 and follows from [6, Theorem 6].

3.2. Cranks for partitions of 7n + 5 and 49n + 47. For 7-cores of 7n + 5 there is no analog of (3.2) and so there is no analog of the map θ. Nonetheless we are able to find a crank c(λ) ∈ ℤ_{7} × ℤ_{7} for the partitions of 49n + 47. Theorem 3.5. ([6, p.7]) Let r⃗ = (r_{0}, r_{1}, …, r_{6}) be the r-vector of λ, a 7-core of 7n + 5. Then

(3.10)  c_{1}(λ) := 5r_{1} − r_{2} − r_{3} + r_{4} + r_{5} − 5r_{6} (mod 7) ∈ ℤ_{7}

is a crank for 7-cores of 7n + 5.

We make explicit the 7-cycle σ that acts of 7-cores of 7n + 5. We define the 7-cycle σ in terms of n-vectors. Since ∑_{i=0}^{6} n_{i} = 0, we omit the last component n_{6}, and let n⃗ = (n_{0}, n_{1}, …, n_{5})^{T}.The map

σ : P_{7-core}(7n + 5) ⟶ P_{7-core}(7n + 5)

is defined by

σ(n⃗) = M n⃗ + τ⃗,

where

M = \frac{1}{7} (\begin{matrix}−8 & −2 & −3 & −4 & −5 & −6 \\ 1 & 2 & 3 & 4 & −2 & 6 \\ 3 & −1 & 2 & 5 & 1 & −3 \\ −2 & −4 & 1 & −1 & 4 & 2 \\ 0 & 0 & −7 & 0 & 0 & 0 \\ 2 & −3 & −1 & −6 & −4 & −2 \end{matrix}) , τ⃗ = \frac{1}{7} (\begin{matrix}5 \\ 2 \\ −1 \\ 0 \\ −3 \end{matrix}) .

We have the following Theorem 3.6. Let r⃗ = (r_{0}, r_{1}, …, r_{6}) be the r-vector of λ, a 7-core of 49n + 47. Then

c(λ) := (c_{1}(λ), c_{2}(λ))
(3.11) = (5r_{1} − r_{2} − r_{3} + r_{4} + r_{5} − 5r_{6}, r_{3} + 4r_{4} − 4r_{5} − r_{6}) (mod 7) ∈ ℤ_{7} × ℤ_{7}

is a crank for 7-cores of 49n + 47.

Proof. For each (i, j) in ℤ_{7} × ℤ_{7}, we let P_{7-core}^{i,j}(49n + 47) be the set of 7-cores λ of 49n + 47 such that c(λ) ≡ (i, j) (mod 7). We construct 7 bijections

Ψ_{j} : P_{7-core}^{0,j}(49n + 47) ⟶ P_{7-core}^{0,j+1}(49n + 47), 0 ≤ j ≤ 6.

Each map Ψ_{j} has the form

Ψ_{j}(n⃗) = M_{j} n⃗ + τ⃗_{j},

where M_{j} is a 6 × 6 matrix, and τ⃗_{j} is a constant vector, and which are given below.

M_{0} = \frac{1}{49} (\begin{matrix}−24 & −36 & −2 & −5 & 13 & −4 \\ 40 & 17 & −20 & −1 & 4 & 2 \\ −15 & −37 & −17 & −18 & 23 & −13 \\ 0 & 0 & 0 & 49 & 0 & 0 \\ 15 & 2 & 38 & −3 & −2 & −22 \\ −40 & −38 & −36 & −41 & −60 & −23 \end{matrix}) , τ⃗_{0} = \frac{1}{49} (\begin{matrix}22 \\ −4 \\ 26 \\ 0 \\ −26 \\ 4 \end{matrix})

M_{1} = \frac{1}{49} (\begin{matrix}32 & 1 & −30 & −5 & 6 & 10 \\ −9 & −46 & 8 & −15 & 4 & 2 \\ −36 & 12 & −17 & −11 & 16 & 8 \\ 0 & 0 & 0 & 49 & 0 & 0 \\ 36 & 44 & 52 & 25 & 19 & 48 \\ 9 & 11 & 13 & −6 & 17 & −37 \end{matrix}) , τ⃗_{1} = \frac{1}{49} (\begin{matrix}15 \\ 31 \\ 26 \\ 0 \\ −26 \\ −31 \end{matrix})

M_{2} = \frac{1}{49} (\begin{matrix}36 & 58 & 24 & 18 & 12 & 27 \\ 24 & 6 & 16 & 12 & 57 & 18 \\ −23 & −18 & −48 & −36 & −24 & −54 \\ 0 & 0 & 0 & 49 & 0 & 0 \\ 23 & −31 & −1 & −13 & −25 & 5 \\ −24 & −6 & 33 & −12 & −8 & −18 \end{matrix}) , τ⃗_{2} = \frac{1}{49} (\begin{matrix}−5 \\ 13 \\ 10 \\ 0 \\ −10 \\ −13 \end{matrix})

M_{3} = \frac{1}{49} (\begin{matrix}4 & 8 & −44 & −19 & −8 & −25 \\ −9 & −4 & 22 & −15 & −10 & −40 \\ 48 & 54 & 46 & 31 & 37 & 50 \\ 0 & 0 & 0 & 49 & 0 & 0 \\ −48 & 2 & −11 & −17 & −2 & 6 \\ 9 & −31 & −1 & −6 & 31 & 5 \end{matrix}) , τ⃗_{3} = \frac{1}{49} (\begin{matrix}15 \\ 3 \\ −16 \\ 0 \\ 16 \\ −3 \end{matrix})

M_{4} = \frac{1}{49} (\begin{matrix}−12 & 39 & 6 & −6 & −18 & −2 \\ −36 & −30 & 18 & −18 & −5 & −6 \\ 31 & −15 & 9 & −9 & −27 & −3 \\ 0 & 0 & 0 & 49 & 0 & 0 \\ −31 & −34 & −58 & −40 & −22 & −46 \\ 36 & 30 & 31 & 18 & 54 & 6 \end{matrix}) , τ⃗_{4} = \frac{1}{49} (\begin{matrix}11 \\ 33 \\ −8 \\ 0 \\ 8 \\ −33 \end{matrix})

M_{5} = \frac{1}{49} (\begin{matrix}−31 & −34 & −58 & −40 & −22 & −46 \\ 12 & 10 & −6 & 6 & 18 & 51 \\ −36 & −30 & 18 & −18 & −5 & −6 \\ 0 & 0 & 0 & 49 & 0 & 0 \\ 36 & 30 & 31 & 18 & 54 & 6 \\ −12 & 39 & 6 & −6 & −18 & −2 \end{matrix}) , τ⃗_{5} = \frac{1}{49} (\begin{matrix}36 \\ 24 \\ 26 \\ 0 \\ −26 \\ −24 \end{matrix})

M_{6} = \frac{1}{49} (\begin{matrix}36 & 44 & 52 & 25 & 19 & 48 \\ −32 & −22 & −26 & −37 & −62 & −31 \\ −9 & −46 & 8 & −15 & 4 & 2 \\ 0 & 0 & 0 & 49 & 0 & 0 \\ 9 & 11 & 13 & −6 & 17 & −37 \\ 32 & 1 & −30 & −5 & 6 & 10 \end{matrix}) , τ⃗_{6} = \frac{1}{49} (\begin{matrix}2 \\ 20 \\ 24 \\ 0 \\ −24 \\ −20 \end{matrix})

Let

(3.12)  w(n⃗) := w(n_{0}, n_{1}, …, n_{5})
= \frac{7}{2}(n_{0}^{2} + ⋯ + n_{5}^{2} + (n_{0} + ⋯ + n_{5})^{2})
+ n_{1} + 2n_{2} + ⋯ + 5n_{5} − 6(n_{0} + ⋯ + n_{5}).

In terms of the n-vector c_{1}, c_{2} are given by

c_{1}(n⃗) = 5n_{1} + 4n_{2} + 3n_{3} + 4n_{4} + 5n_{5},
c_{2}(n⃗) = n_{3} + 5n_{4} + n_{5}.

In order to show the Ψ_{j} are bijections, we have used computer algebra to show for each j, (i) Ψ_{j} preserves the form w, (ii) det(M_{j}) = ±1, and (iii) n⃗ ∈ ℤ^{6}, (c_{1}(n⃗), c_{2}(n⃗)) = (0, j), w(n⃗) ≡ 47 (mod 49) implies Ψ(n⃗) ∈ ℤ^{6} and (c_{1}(n⃗), c_{2}(n⃗)) = (0, j + 1). We have calculated the effect of the 7-cycle σ has on our crank statistics c_{1}, c_{2}. A calculation shows that the map

σ : P_{7-core}^{i,j}(49n + 47) ⟶ P_{7-core}^{i+1,4i+j}(49n + 47)

is a bijection. We omit the details. We note that the indices are reduced mod 7. Using the seven maps Ψ_{j} and the 7-cycle σ we find that

|P_{7-core}^{i,j}(49n + 47)| = |P_{7-core}^{0,0}(49n + 47)| = \frac{1}{49} a_{7}(49n + 47),

for 0 ≤ i, j ≤ 6. Hence c = (c_{1}, c_{2}) (mod 7) is a crank for 7-cores of 49n + 47 (mod 49).

□

Corollary 3.1. Let r⃗ = (r_{0}, r_{1}, …, r_{6}) be the r-vector of a partition λ of 49n + 47. We define a crank c(λ) ∈ ℤ_{7} × ℤ_{7} as follows.

If s(λ) ≡ 0 (mod 7) we define

c(λ) := (c_{1}(λ), c_{2}(λ))

= (5r_{1} − r_{2} − r_{3} + r_{4} + r_{5} − 5r_{6}, r_{3} + 4r_{4} − 4r_{5} − r_{6}). (3.13)

If s(λ) ≢ 0 (mod 7) we define

(3.14)  c(λ) := (c_{1}(λ), c_{3}(λ)),

where c_{3} is defined in (3.7).

Then c(λ) is a crank for the partitions of 49n + 47 mod 49.

The proof is analogous to that of Theorem 3.4.

4. Remarks

Our cranks for the partitions of 25n+24 and 49n+47 depend crucially on finding the two crank functions c_{1} and c_{2}. The first crank function c_{1} arises naturally from the t-cycle one gets from Theorem 3.1. For 5-cores the second crank function c_{2} arises from the map θ. We describe another way the second crank function arises. Let w(n⃗) be defined as in (3.12). Then since w(n⃗) ≡ 5 (mod 7) and assuming c_{1}(n⃗) ≡ 0 (mod 7), there are integers k, ℓ such that

n_{0} = 7k + 5 − 2n_{1} − 3n_{2} − 4n_{3} − 5n_{4} − 6n_{5},
n_{1} = 7ℓ − n_{5} − 5n_{2} − 2n_{3} − 5n_{4}.

Now assume the second crank function takes the form

c_{2}(n⃗) = ab_{2} + n_{3} + bn_{4} + cn_{5},

for some integers a, b, c. If we assume c_{2}(n⃗) ≡ 0 (mod 7), then there is an integer m such that

n_{3} = 7m − an_{2} − bn_{4} − cn_{5}.

We want w(n⃗) to be a linear form mod 49 in the remaining variables n_{2}, n_{4}, n_{5}. A calculation shows that this can only happen if

(a, b, c) ≡ (0, 5, 1) (mod 7),

which nails down the second crank function c_{2}. We have considered the analogous problem for 11-cores of 121n + 116, and found there is no second crank function of a similar form which makes the corresponding w(n⃗) linear mod 121. So if there is a crank for 11-cores of 121n + 116 it must be more complicated.

It would be interesting to find other occurrences of pairs of crank functions (c_{1}, c_{2}) which give combinatorial congruences. Zoltan Reti [8] found a pair of crank functions which explains that the congruence

s(9n + 8) ≡ 0 (mod 9),

where s(n) is the number of partitions of n in which an even part may have two colors. It was Reti’s result which led us to search for a function c_{2} for 7-cores of 49n + 47.

References

[1] G.E. Andrews, The theory of partitions, Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976.

[2] G.E. Andrews and F.G. Garvan, Dyson’s crank of a partition, Bull. Amer. Math. Soc., 18 (1988), 167–171.

[3] A.O.L. Atkin, Proof of a conjecture of Ramanujan, Glasgow Math. J., 8 (1967), 14–32.

[4] A.O.L. Atkin and H.P.F. Swinnerton-Dyer, Some properties of partitions, Proc. London Math. Soc. (3) 4 (1954), 84–106.

[5] F. Dyson, Some guesses in the theory of partitions, Eureka (Cambridge), 8 (1944), 10–15.

[6] F.G. Garvan, D. Kim and D. Stanton, Cranks and t-cores, Inventiones math. 101 (1990), 1–17.

[7] J.G. Kerber, The Representation Theory of the Symmetric Group, Addison-Wesley, Reading, MA, 1981.

[8] Z. Reti, Five problems in combinatorial number theory, Ph.D. thesis, University of Florida, [9] G.N. Watson, Ramanujans Vermutung über Zerfällungsanzahlen, J. Reine. Angew. Math.,

179 (1938), 97–128.

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

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