Journal of Combinatorics and Number Theory JCNT 2009, Volume 1, Issue # 3, pp. 49-64

ISSN 1942-5600

\begin{matrix}c \\ ◯ \end{matrix} 2009 Nova Science Publishers, Inc.

The GBG-rank and t-cores I. Counting and 4-cores

Alexander Berkovich^{∗}and Frank G. Garvan^{†} Department of Mathematics, University of Florida, Gainesville, Florida 32611-8105

October 11, 2009

Abstract

Let r_{j}(π,s) denote the number of cells, colored j, in the s-residue diagram of partition π. The GBG-rank of π mod s is defined as

GBG-rank(π,s) = ∑_{j=0}^{s−1} r_{j}(π,s)e^{\frac{2πi}{s} j}.

We prove that for (s,t) = 1

ν(s,t) ≤ \frac{\binom{s + t}{s}}{s + t} ,

where ν(s,t) denotes the number of distinct values that the GBG-rank of a t-core mod s may assume. The above inequality becomes an equality when s is prime or when s is composite and t ≤ 2p_{s}, where p_{s} is the smallest prime divisor of s. We show that the generating functions for 4-cores with prescribed GBG-rank mod 3 value are all eta-quotients.

2000 Mathematics Subject Classification: Primary 11P81, 11P83; Secondary 05A17, 05A19.

Key words and phrases: Partitions, t-cores, (s,t)-cores, GBG-rank, eta-quotients, multiple theta series.

1 Introduction

A partition π is a nonincreasing sequence

π = (λ_{1},λ_{2},...,λ_{ν})

of positive integers (parts) λ_{1} ≥ λ_{2} ≥ λ_{3} ≥ ···≥ λ_{ν} > 0. The norm of π, denoted |π|, is defined as

|π| = ∑_{i=1} λ_{i}.

Received August 11, 2008; Revised November 5, 2008 ∗E-mail address: alexb@ufl.edu †E-mail address: fgarvan@ufl.edu

If |π| = n, we say that π is a partition of n. The (Young) diagram of π is a convenient way to represent π graphically: the parts of π are shown as rows of unit squares (cells). Given the diagram of π we label a cell in the i-th row and j-th column by the least nonnegative integer ≡ j − i mod s. The resulting diagram is called an s-residue diagram [8]. One can also label cells in the infinite column 0 and the infinite row 0 in the same fashion. The resulting diagram is called the extended s-residue diagram of π [5]. With each π we can associate the s-dimensional vector

r(π,s) = (r_{0},r_{1},...,r_{s−1} ),

where r_{i} = r_{i}(π,s) (0 ≤ i ≤ s − 1) is the number of cells labelled i in the s-residue diagram of π. We shall also require

n(π,s) = (n_{0},n_{1},...,n_{s−1} ),

where for 0 ≤ i ≤ s − 2

n_{i} = r_{i}− r_{i+1} ,

and

n_{s−1} = r_{s−1} − r_{0}.

Note that

n · l_{s} = ∑_{i=0}^{s−1} n_{i} = 0,

where

l_{s} = (1,1,1,...,1) ∈ ℤ^{s}.

We recall the notions of rim hook and t-core [8]. If some cell of π shares a vertex or edge with the rim of the diagram of π, we call this cell a rim cell of π. A connected collection of rim cells of π is called a rim hook of π if π\(rim hook) is a legitimate partition. A partition is a t-core if it has no rim hooks of length t. Throughout this paper we usually denote a generic t-core by π_{t-core}. We will also use this notation in a different way, but the context should be clear. Any partition π has a uniquely determined t-core which we will also denote by π_{t-core}. This partition π_{t-core}is called the t-core of π. One can obtain π_{t-core}from π by the successive removal of rim hooks of length t. The t-core π_{t-core}is independent of the manner in which hooks removed. The Durfee square of π, denoted D(π), is the largest square that fits inside the diagram of π. Reflecting the diagram of π about its main diagonal, one gets the diagram of π^{∗} (the conjugate of π). More formally,

π^{∗} = (λ_{1}^{∗},λ_{2}^{∗},...)

with λ_{i}^{∗} being the number of parts of π ≥ i. Clearly,

D(π) = D(π^{∗}).

In [3] we defined a new partition statistic of π

GBG-rank(π,s) := ∑_{j=0}^{s−1} r_{j}(π,s)ω_{s}^{j},  (1.1)

The GBG-rank and t-cores I. Counting and 4-cores  51

where

ω_{s} = e^{\frac{2πi}{s}}.

We refer to this statistic as the GBG-rank of π mod s. The special case s = 2 was studied in great detail in [3] and [4]. In particular, we have shown in [3] that for any odd t > 1

GBG-rank(π_{t-core},2) = \frac{1 − ∑ _{i=0}^{t−1}(−1)^{i+n_{i}(πt-core,t)}}{4}  (1.2)

and that

−⌊ \frac{t − 1}{4} ⌋ ≤ GBG-rank(π_{t-core},2) ≤ ⌊ \frac{t + 1}{4} ⌋,  (1.3)

where ⌊x⌋ is the integer part of x. Our main object here is to prove the following generalisations of (1.2) and (1.3).

Theorem 1.1. Let t,s ∈ ℤ_{>1} and (t,s) = 1. Then

GBG-rank(π_{t-core},s) = \frac{∑ _{j=0}^{t−1} ω_{s}^{j+1} (ω_{s}^{tn_{j}(πt-core,t)}− 1)}{(1 − ω_{s})(1 − ω_{s}^{t})}  (1.4)

Theorem 1.2. Let ν(s,t) denote the number of distinct values that GBG-rankof π_{t-core} mods may assume. Then

ν(s,t) ≤ \frac{\binom{t + s}{t}}{t + s} ,  (1.5)

provided (s,t) = 1.

Theorem 1.3. Let ν(s,t) be as in Theorem 1.2 and (s,t) = 1. Then

ν(s,t) = \frac{\binom{t + s}{t}}{t + s} ,  (1.6)

iff either s is prime or s is composite and t < 2p_{s}, where p_{s} is the smallest prime divisor of s.

Our proof of this theorem depends crucially on the following

Lemma 1.4. Let s,t ∈ ℤ_{>1} and (s,t) = 1. Let j = (j_{0},j_{1},...,j_{t−1}), j̃= (j̃_{0},j̃_{1},...,j̃_{t−1} ) be integer valued vectors such that

0 ≤ j_{0} ≤ j_{1} ≤ ···≤ j_{t−1} ≤ s − 1,  (1.7)
0 ≤ j̃_{0} ≤ j̃_{1} ≤ ···≤ j̃_{t−1} ≤ s − 1,  (1.8)

and

∑_{i=0}^{t−1} ω_{s}^{j_{i}} = ∑_{i=0}^{t−1} ωj̃_{i} ; s  (1.9)

∏_{i=0}^{t−1} ω_{s}^{j_{i}} = ∏_{i=0}^{t−1} ωj̃_{i} ; s.  (1.10)

Then

j = j̃,

iff either s is prime or s is composite such that t < 2p_{s}, where p_{s} is the smallest prime divisor of s.

The rest of this paper is organised as follows. In Section 2, we collect some necessary background on t-cores and prove Theorems 1.1 and 1.2. Section 1.3 is devoted to the proof of Lemma 1.4 and Theorem 1.3. Section 4 deals with 4-cores with prescribed values of GBG-rank mod 3. There we will provide new combinatorial interpretation and proof of the Hirschhorn-Sellers identities for 4-cores [7]. We conclude with remarks connecting this development and that of [11] and [2].

2 Properties of the GBG-rank

We begin with some definitions from [5]. A region r in the extended t-residue diagram of π is the set of all cells (i,j) satisfying t(r − 1) ≤ j − i < tr. A cell of π is called exposed if it is at the end of a row of π. One can construct t bi-infinite words W_{0},W_{1},...,W_{t−1} of two letters N,E as follows: The rth letter of W_{i} is E if there is an exposed cell labelled i in the region r of π, otherwise the rth letter of W_{i} is N. It is easy to see that the word set {W_{0},W_{1},...,W_{t−1}} fixes π uniquely. It was shown in [5] that π is a t-core iff each word of π is of the form:

Region : ··· ···n_{i−1} n_{i} n_{i+1} n_{i+2} ··· ···
W_{0}: ··· ··· E E  N  N ··· ···.  (2.1)

For example, the word image of π_{3-core} = (4,2) is

Region : ··· ···− 1 0 1 2 3 ··· ···
W_{0}: ··· ··· E E E E N ··· ···
W_{1}: ··· ··· E N N N N ··· ···
W_{2}: ··· ··· E N N N N ··· ···,

while the associated r and n vectors are r = (r_{0},r_{1},r_{2}) = (3,1,2), n = (n_{0},n_{1},n_{2}) = (2,−1,−1), respectively. In general, the map

ϕ(π_{t-core}) = n(π_{t-core},t) = (n_{0},n_{1},...,n_{t−1})

is a bijection from the set of t-cores to the set

{n ∈ ℤ^{t} : n · 1 = 0}.

Next, we mention three more useful facts from [5].

(A) We have

|π_{t-core}| = \frac{t}{2}|n|^{2}+ b_{t}· n,  (2.2)

where b_{t} = (0,1,2,...,t − 1).

(B) We have

∑_{i∈P_{1}}n_{i} = − ∑_{i∈P_{−1}} n_{i} = |D(π_{t-core})|,  (2.3)

where P_{α} = {i ∈ ℤ: 0 ≤ i ≤ t − 1,αn_{i} > 0}, α = −1,1, and |D(π_{t-core})| denotes the number of cells on the main diagonal of the Durfee square.

(C) Under conjugation, ϕ(π_{t-core}) transforms as

(n_{0},n_{1},n_{2},...,n_{t−1}) → (−n_{t−1},−n_{t−2},...,−n_{0}).  (2.4)

We begin our proof of the Theorem 1.1 by observing that under conjugation GBG-rank transforms as

GBG-rank(π,s) = ∑_{i=0}^{s−1} r_{i}ω_{s}^{i} ⟹ GBG-rank(π^{∗},s) = ∑_{i=0}^{s−1} r_{i}ω_{s}^{−i} .  (2.5)

Next, we use that

GBG-rank(π,s) = GBG-rank(π_{1},s) + GBG-rank(π_{2},s) − D.  (2.6)

Here, π_{1} is obtained from the diagram of π_{t-core}by removing all cells strictly below the main diagonal of π_{t-core}. Similarly, π_{2} is obtained from π_{t-core}by removing the cells strictly to the right of the main diagonal.

Recalling (2.1) and (2.3) we find that

GBG-rank(π_{1},s) = ∑_{i∈P_{1}} ∑_{k=1}^{n_{i}} ∑_{j=0}^{i+t(k−1)} ω_{s}^{j} = \frac{D}{1 − ω_{s}} − ∑_{i∈P_{1}} \frac{ω_{s}^{i+1} (1 − ω_{s}^{tn_{i}})}{(1 − ω_{s})(1 − ω_{s}^{t})} .  (2.7)

Analogously,

GBG-rank(π_{2}^{∗},s) = \frac{D}{1 − ω_{s}} − ∑_{i∈P_{−1}} \frac{ω_{s}^{t−i}(1 − ω_{s}^{−tn_{i}})}{(1 − ω_{s})(1 − ω_{s}^{t})} ,  (2.8)

where we made use of (2.4).

Clearly, (2.5) and (2.8) imply that

GBG-rank(π_{2},s) = −\frac{Dω_{s}}{1 − ω_{s}} − ∑_{i∈P_{−1}} \frac{ω_{s}^{1+i} (1 − ω_{s}^{tn_{i}})}{(1 − ω_{s})(1 − ω_{s}^{t})} .  (2.9)

Next, we combine (2.6), (2.7) and (2.9) to find that

GBG-rank(π_{t-core},s) = − ∑ ; i∈P_{−1} ⋃ P_{1} \frac{ω_{s}^{1+i} (1 − ω_{s}^{tn_{i}})}{(1 − ω_{s})(1 − ω_{s}^{t})} = ∑_{i=0}^{t−1} \frac{ω_{s}^{1+i} (ω_{s}^{tn_{i}}− 1)}{(1 − ω_{s})(1 − ω_{s}^{t})} ,

as desired.

Our proof of Theorem 1.2 involves three observations, which we now proceed to discuss.

Observation 1. Let a_{r}(s,t) denote the number of vectors j = (j_{0},j_{1},...,j_{t−1}) such that

0 ≤ j_{0} ≤ j_{1} ≤ j_{2} ≤ ···≤ j_{t−1} < s,

∑_{k=0}^{t−1} j_{k} ≡ r mods.

Then

ν(s,t) ≤ a_{\frac{t(t+1)}{2}}(s,t),  (2.10)

provided (s,t) = 1.

Proof. Suppose (s,t) = 1. It is clear that the number of values of the GBG-rank of t-cores mod s is the number of distinct values of

∑_{i=0}^{t−1} ω_{s}^{1+i+tn_{i}} ,

where n ∈ ℤ^{t} and n · 1_{t} = 0. Given any such n-vector we reduce the exponents 1 + i + tn_{i} mod s and reorder to obtain a j-vector such that

∑_{k=0}^{t−1} j_{k} ≡ ∑_{i=0}^{t−1} 1 + i + tn_{i} ≡ \frac{t(t+1)}{2} (mod s).

It follows that

ν(s,t) ≤ a_{\frac{t(t+1)}{2}}(s,t).

Observation 2. We have

∑_{r=0}^{s−1} a_{r}(s,t) = \binom{t + s − 1}{t} .  (2.11)

This result is well known and we omit the proof. Finally, we need

Observation 3. If (s,t) = 1 then

a_{0}(s,t) = a_{1}(s,t) = ···= a_{s−1} (s,t).  (2.12)

Proof. There exists an integer T such that T ·t ≡ 1 mod s, because s and t are coprime. This implies that

∑_{i=0}^{t−1}(j_{i}+ T) ≡ 1 + ∑_{i=0}^{t−1} j_{i} mod s.

Consequently, a_{r}(s,t) = a_{r+1} (s,t), as desired. Combining (2.10), (2.11) and (2.12) we see that

ν(s,t) ≤ a_{\frac{t(t+1)}{2}} = \frac{\binom{s − 1 + t}{t}}{s} = \frac{\binom{s + t}{t}}{s + t} ,

and we have Theorem 1.2.

3 Roots of Unity and the Number of Values of the GBG-rank

It is clear from our proof of Theorem 1.2 that

ν(s,t) = \frac{\binom{s + t}{t}}{s + t} .

iff each j = (j_{0},j_{1},...,j_{t−1}) such that

0 ≤ j_{0} ≤ j_{1} ≤ ···≤ j_{t−1} < s,

∏_{i=0}^{t−1} ω_{s}^{j_{i}} = ωs^{\frac{t(t+1)}{2}}

is associated with a distinct complex number ∑ _{i=0}^{t−1} ω_{s}^{j_{i}}. Lemma 1.4 tells us when this is exactly the case. This means that Theorem 1.3 is an immediate corollary of this Lemma. To prove it we need to consider six cases.

Case 1. s is prime, (s,t) = 1.

Note that

Φ_{s}(x) := 1 + x + x^{2}+ ··· + x^{s−1}

is a minimal polynomial of ω_{s} over ℚ. Let us now define

p_{1}(x) := ∑_{i=0}^{t−1}(x^{j_{i}} − xj̃_{i}),  (3.1)

where j and j̃satisfy the constraints (1.7) - (1.10). It is clear that

p_{1}(ω_{s}) = 0,
p_{1}(1) = 0,

and that deg(p_{1}(x)) < s. But (x − 1)Φ_{s}(x) divides p_{1}(x). This implies that p_{1}(x) is identically zero and j = j̃, as desired.

Case 2. s is composite, (s,t) = 1 and t < 2p_{s}, where p_{s} is the smallest prime divisor of s.

Once again (1.9) implies that

p_{1}(ω_{s}) = 0.

Moreover, the sth cyclotomic polynomial, defined as

Φ_{s}(x) := ∏_{\begin{matrix}0<j<s, \\ (j,s)=1 \end{matrix}} (x − ω_{s}^{j}),  (3.2)

is a minimal polynomial of ω_{s} over ℚ. This means that

p_{1}(ω_{s}^{m}) = 0,  (3.3)

for any 1 ≤ m < s such that (s,m) = 1. In particular, we have that

p_{1}(ω_{s}^{k}) = 0,  1 ≤ k ≤ p_{s}− 1.  (3.4)

At this point, it is expedient to rewrite (3.4) as

h_{k}(ω_{s}^{j_{0}},...,ω_{s}^{j_{t−1}} ) = h_{k}(ωj̃_{0} ; s,...,ωj̃_{t−1} ; s ),  1 ≤ k ≤ p_{s}− 1,  (3.5)

where

h_{k}(x_{1},x_{2},...,x_{t}) = x_{1}^{k}+ x_{2}^{k}+ ··· + x_{t}^{k}.

Next, we use Newton’s theorem on symmetric polynomials to convert (3.5) into p_{s} − 1 identities

σ_{k}(ω_{s}^{j_{0}},...,ω_{s}^{j_{t−1}} ) = σ_{k}(ωj̃_{0} ; s,...,ωj̃_{t−1} ; s ),  1 ≤ k ≤ p_{s}− 1,  (3.6)

where the kth elementary symmetric polynomials σ_{k}’s in x_{1},x_{2},...,x_{t} are defined in a standard way as

σ_{k}(x_{1},x_{2},...,x_{t}) = ∑_{1≤i_{1}≤i_{2}<...<i_{k}≤t} x_{i_{1}}x_{i_{2}} ··· x_{i_{k}},  1 ≤ k ≤ t.  (3.7)

Note that we can rewrite (1.10) now as

σ_{t}(ω_{s}^{j_{0}},...,ω_{s}^{j_{t−1}} ) = σ_{t}(ωj̃_{0} ; s,...,ωj̃_{t−1} ; s ).  (3.8)

But

σ_{t}σ_{k}^{∗} = σ_{t−k},  (3.9)

where

σ_{k}^{∗}(x_{1},x_{2},...,x_{t}) = σ_{k}(x_{1}^{−1} ,x_{2}^{−1} ,...,x_{t}^{−1} ).

This fortunate fact enables us to convert (3.6) into p_{s}− 1 identities

σ_{k}(ω_{s}^{j_{0}},...,ω_{s}^{j_{t−1}} ) = σ_{k}(ωj̃_{0} ; s,...,ωj̃_{t−1} ; s ),  t − p_{s}+ 1 ≤ k ≤ t − 1.  (3.10)

But t < 2p_{s}, and so, t − p_{s}+ 1 ≤ p_{s}. This means that we have the following t identities

σ_{k}(ω_{s}^{j_{0}},...,ω_{s}^{j_{t−1}} ) = σ_{k}(ωj̃_{0} ; s,...,ωj̃_{t−1} ; s ),  1 ≤ k ≤ t.  (3.11)

Consequently,

∏_{i=0}^{t−1}(x − ω_{s}^{j_{i}}) = ∏_{i=0}^{t−1}(x − ωj̃_{i} ; s).

Recalling that j,j̃satisfy (1.7) and (1.8), we conclude that j = j̃.

Let us summarize. If s is a prime or if s is a composite number such that t < 2p_{s}, then j = j̃, provided that (s,t) = 1 and j,j̃satisfy (1.7)–(1.10).

It remains to show that j = j̃does not have to be true if s is a composite number and t ≥ 2p_{s}. To this end consider

j : = (0,0,...,0,1,1,3,3) ∈ ℤ^{t},
j̃: = (0,0,...,0,0,0,2,2) ∈ ℤ^{t},

if s = 4, t ≥ 4,

j : = (0,0,...,0,1,1,4,4) ∈ ℤ^{t},
j̃: = (0,0,...,0,0,2,3,5) ∈ ℤ^{t},

if s = 6, t ≥ 4 and

j : = (0,0,...,0,3,3,6,6) ∈ ℤ^{t},
j̃: = (0,0,...,0,1,2,4,5,7,8) ∈ ℤ^{t},

if s = 9, t ≥ 6, respectively. It is not hard to verify that j,j̃satisfy (1.7)–(1.10) and that j ≠ j̃ in these cases. It remains to consider the last case where s is a composite number ≠ 4,6,9, t ≥ 2p_{s}. In this case s > 3p_{s}. And, as a result,

3 + \frac{s}{p_{s}}(p_{s}− 1) < s.

Let us now consider

j : = ( 0,0,...,0,2,2,2 + \frac{s}{p_{s}},2 + \frac{s}{p_{s}},...,2 + \frac{s}{p_{s}}(p_{s}− 1),2 + \frac{s}{p_{s}}(p_{s}− 1)) ∈ ℤ^{t},
j̃: = ( 0,0,...,0,1,3,1 + \frac{s}{p_{s}},3 + \frac{s}{p_{s}},...,1 + \frac{s}{p_{s}}(p_{s}− 1),3 + \frac{s}{p_{s}}(p_{s}− 1)) ∈ ℤ^{t}.

Again, it is straightforward to check that j,j̃satisfy (1.7)–(1.10) and that j ≠ j̃. This completes our proof of Lemma 1.4.

It is clear that Theorem 1.3 follows from the above.

To illustrate the usefulness of Theorem 1.3, consider the following example: s = 3, t = 4. In this case we should have exactly \binom{4+3}{3}/(4 + 3) = 5 distinct values of GBG-rank (π_{4-core},3). To determine these distinct values we substitute the following n-vectors (0,−1,1,0), (0,0,0,0), (−1,0,0,1), (0,0,−1,1), (−1,1,0,0) into (1.4) to obtain −1,0,1, −ω_{3},−ω_{3}^{2}, respectively. To verify this we note that there are exactly 27 vectors such that

n ∈ ℤ_{3}^{4} and n · 1_{4} ≡ 0 mod 3.

In Table 1 we list all these vectors together with the associated GBG-rank mod 3 values, determined by (1.4). These vectors will come in handy later.

4 The GBG-rank of 4-cores mod 3

Let G_{t}(q) denote the generating function for t-cores.

G_{t}(q) := ∑_{πt-core}q^{|πt-core|}.  (4.1)

Let P be the set of all partitions and P_{t-core}be the set of all t-cores. There is a well-known bijection

ϕ̃: P → P_{t-core}× P × P × P ...× P

58  Alexander Berkovich and Frank G. Garvan
n vectors  GBG-rank values
n_{1} = (0,−1,1,0)  -1
n_{2} = (0,0,0,0)  0
n_{3} = (1,1,−2,0)  0
n_{4} = (−1,−1,1,1)  0
n_{5} = (0,−1,−1,2)  0
n_{6} = (1,−1,0,0)  0
n_{7} = (0,1,−2,1)  0
n_{8} = (2,−1,−1,0)  0
n_{9} = (0,0,1,−1)  0
n_{10} = (0,1,−1,0)  0
n_{11} = (−1,0,1,0)  0
n_{12} = (1,−1,1,−1)  0
n_{13} = (0,−1,0,1)  0
n_{14} = (1,1,0,−2)  1
n_{15} = (−1,1,−1,1)  1
n_{16} = (2,0,−1,−1)  1
n_{17} = (1,0,0,−1)  1
n_{18} = (1,1,−1,−1)  1
n_{19} = (−1,0,0,1)  1
n_{20} = (1,0,−1,0)  −ω_{3}
n_{21} = (1,0,−2,1)  −ω_{3}
n_{22} = (1,−1,−1,1)  −ω_{3}
n_{23} = (0,0,−1,1)  −ω_{3}
n_{24} = (−1,1,0,0)  −ω_{3}^{2}
n_{25} = (−1,1,1,−1)  −ω_{3}^{2}
n_{26} = (−1,2,0,−1)  −ω_{3}^{2}
n_{27} = (0,1,0,−1)  −ω_{3}^{2}

Table 1:

which goes back to D.E. Littlewood [10]

ϕ̃(π) = (π_{t-core},π̂_{0},π̂_{1},...,π̂_{t−1})

such that

|π| = |π_{t-core}| + t ∑_{i=0}^{t−1} |π̂_{i}|.

The multipartition (π̂_{0},π̂_{1},...,π̂_{t−1}) is called the t-quotient of π. An immediate corollary of the Littlewood bijection is

G_{t}(q) = \frac{E^{t}(q^{t})}{E(q)},  (4.2)

where

E(q) := ∏_{j≥1} (1 − q^{j}).  (4.3)

The function E(q) is related to Dedekind’s eta-function η(τ) by

η(τ) = q^{\frac{1}{24}}E(q),

where q = exp(2πiτ) and ℑ(τ) > 0. Quotients of the functions E(q^{a}) are called etaquotients. On the other hand, formula (2.2) implies [5] that

G_{t}(q) = ∑_{\begin{matrix}n∈ℤ^{t}, \\ n·1_{t}=0 \end{matrix}}q^{\frac{t}{2}|n|^{2}+n·b_{t}} ,  (4.4)

so that

∑_{\begin{matrix}n∈ℤ^{t}, \\ n·1_{t}=0 \end{matrix}}q^{\frac{t}{2}|n|^{2}+n·b_{t}} = \frac{E^{t}(q^{t})}{E(q)}.  (4.5)

The above identity was first obtained by Klyachko [9], who observed that it is a special case of Macdonald’s identity for the root system A_{t−1}. An elementary proof of (4.5) can be found in [3]. Next we define

g_{c}(q) = ∑_{π4-core, GBG-rank(π4-core,3)=c} q^{|π4-core|}  (4.6)

In other words, g_{c}(q) is the generating function for 4-cores with a given value c of the GBG-rank mod 3. From the discussion at the end of the last section it is clear that

\frac{E^{4}(q^{4})}{E(q)} = g_{−1} (q) + g_{0}(q) + g_{1}(q) + g_{−ω_{3}} (q) + g_{−ω_{3}^{2}} (q).  (4.7)

It turns out that each g_{c}(q) is an eta-quotient.

g_{−1} (q)= q^{5} \frac{E^{4}(q^{36})}{E(q^{9})},  (4.8)
g_{0}(q)= \frac{E^{6}(q^{6})E^{2}(q^{18})}{E^{3}(q^{3})E(q^{12})E(q^{36})},  (4.9)
g_{1}(q)= q\frac{E^{2}(q^{9})E^{4}(q^{12})}{E(q^{3})E(q^{6})E(q^{18})},  (4.10)
g_{−ω_{3}} (q)= q^{2} \frac{E^{2}(q^{9})E(q^{12})E(q^{36})}{E(q^{3})},  (4.11)
g_{−ω_{3}^{2}} (q)= q^{2} \frac{E^{2}(q^{9})E(q^{12})E(q^{36})}{E(q^{3})}.  (4.12)

Hence

\frac{E^{4}(q^{4})}{E(q)} = \frac{E^{6}(q^{6})E^{2}(q^{18})}{E^{3}(q^{3})E(q^{12})E(q^{36})}+ q\frac{E^{2}(q^{9})E^{4}(q^{12})}{E(q^{3})E(q^{6})E(q^{18})}
+ 2q^{2} \frac{E^{2}(q^{9})E(q^{12})E(q^{36})}{E(q^{3})}+ q^{5} \frac{E^{4}(q^{36})}{E(q^{9})}.  (4.13)

We note that the identity

g_{−ω_{3}} (q) = g_{−ω_{3}^{2}} (q)

follows from (2.5) and the fact that π is a t-core if and only if the conjugate π^{∗} is a tcore. The identities equivalent to (4.13) were first proven by Hirschhorn and Sellers [7]. However, the combinatorial identities (4.7)–(4.12) given here are brand new. The proof of (4.8) is rather simple. Indeed, data in Table 1, implies that

g_{−1} (q)= ∑_{\begin{matrix}n·1_{4}=0, \\ n≡n_{1}mod 3 \end{matrix}} q^{2|n|^{2}+b_{4}·n}

= q^{5} ∑_{ñ·1_{4}=ñ_{0}+ñ_{1}+ñ_{2}+ñ_{3}=0} q^{9(2|ñ|^{2}−ñ_{1}+2ñ_{2}+ñ_{3})}

= q^{5} ∑_{ñ·1_{4}=0}q^{9(2|ñ|^{2}+ñ_{0}+2ñ_{3}+3ñ_{2})}

= q^{5} \frac{E^{4}(q^{36})}{E(q^{9})}  (4.14)

where in the last step we relabelled the summation variables and used (4.5) with t = 4 and q → q^{9}.

In what follows we shall require the Jacobi triple product identity ([6],II.28)

∑_{n=−∞}^{∞} (−1)^{n}q^{n^{2}}z^{n} = E(q^{2})[zq;q^{2}]_{∞} ,  (4.15)

where

[z;q]_{∞} := ∏_{j=0}^{∞} (1 − zq^{j})( 1 − \frac{q^{1+j}}{z} )

and the formula ([6],ex.5.21)

[ux, \frac{u}{x},vy, \frac{v}{y};q]_{∞} = [uy, \frac{u}{y},vx, \frac{v}{x};q]_{∞} + \frac{v}{x}[xy, \frac{x}{y},uv, \frac{u}{v};q]_{∞} ,  (4.16)

where

[z_{1},z_{2},...,z_{n};q]_{∞} := ∏_{j=1}^{n} [z_{i};q]_{∞} .

Setting u = q^{5}, v = q^{3}, x = q^{2}, y = q and replacing q by q^{12} in (4.16) we find that

[q^{2},q^{3};q^{12}]_{∞} ([q^{5};q^{12}]_{∞} − q[q;q^{12}]_{∞}) = [q,q^{5},q^{6};q^{12}]_{∞} .  (4.17)

Analogously, (4.16) with u = q^{5}, v = q^{2}, x = q, y = 1 and q → q^{12} becomes

[q^{5};q^{12}]_{∞} + q[q;q^{12}]_{∞} = \frac{[q^{2},q^{2},q^{4},q^{6};q^{12}]_{∞}}{[q,q^{3},q^{5};q^{12}]_{∞}}.  (4.18)

Finally, setting u = q^{6}, v = q^{4}, x = q^{3}, y = 1 and q → q^{12} in (4.16) yields

[q^{3},q^{4};q^{12}]_{∞}^{2} = [q,q^{5},q^{6},q^{6};q^{12}]_{∞} + q[q^{2},q^{3};q^{12}]_{∞}^{2} .  (4.19)

Next, we use again Table 1 to rewrite (4.9) as

∑_{j=2}^{13} ∑_{\begin{matrix}n·1_{4}=0, \\ n≡n_{j}mod 3 \end{matrix}} q^{2|n|^{2}+b_{4}·n} = R_{1}(q),  (4.20)

where

R_{1}(q) = \frac{E^{6}(q^{6})E^{2}(q^{18})}{E^{3}(q^{3})E(q^{12})E(q^{36})}.  (4.21)

Remarkably, (4.20) is the constant term in z of the following identity

∑_{j=2}^{13} s_{j}(z,q) = R_{1}(q) ∑_{n=−∞}^{∞} q^{9 \frac{n(n+1)}{2}} z^{n},  (4.22)

where

s_{j}(z,q) := ∑_{n≡n_{j}mod 3} q^{2|n|^{2}+b_{4}·n} z^{\frac{n·1_{4}}{3}},  j = 1,2,...,27.  (4.23)

Using simple changes of variables, it is straightforward to check that

zq^{9}s_{i}(zq^{9},q) = s_{j}(z,q),  (4.24)

holds true for the following (i,j) pairs: (2,3), (3,4), (4,5), (5,2), (6,7), (7,8), (8,9), (9,6), (10,11), (11,12), (12,13), (13,10), and that

zq^{9} ∑_{n=−∞}^{∞} q^{9 \frac{n(n+1)}{2}} (zq^{9})^{n} = ∑_{n=−∞}^{∞} q^{9 \frac{n(n+1)}{2}} z^{n}.  (4.25)

Consequently, both sides of (4.22) satisfy the same first order functional equation

zq^{9}f(zq^{9},q) = f(z,q).  (4.26)

Thus to prove (4.22) it is sufficient to verify it at one nontrivial point, say z = z_{0} := −q^{−6} . It is not hard to check that

s_{4}(z_{0},q) = s_{8}(z_{0},q) = s_{11}(z_{0},q) = 0,  (4.27)

and that

s_{3}(z_{0},q) + s_{9}(z_{0},q) = s_{5}(z_{0},q) + s_{12}(z_{0},q) = 0.  (4.28)

We see that (4.22) with z = z_{0} becomes

s_{2}(z_{0},q) + s_{6}(z_{0},q) + s_{7}(z_{0},q) + s_{10}(z_{0},q) + s_{13}(z_{0},q) = R_{1}(q) ∑_{n=−∞}^{∞} q^{9 \frac{n(n+1)}{2}} z_{0}^{n}.  (4.29)

Upon making repeated use of (4.15) and replacing q^{3} by q we find that (4.9) is equivalent to

[q^{4},q^{5},q^{5},q^{6};q^{12}]_{∞}+ q[q^{2},q^{3},q^{4};q^{12}]_{∞} ([q^{5};q^{12}]_{∞} − q[q;q^{12}]_{∞})
+ q[q,q^{4},q^{5},q^{6};q^{12}]_{∞} + q^{2}[q,q,q^{4},q^{6};q^{12}]_{∞}
= \frac{E^{2}(q^{6})E^{6}(q^{2})}{E^{5}(q^{12})E(q^{4})E^{2}(q)}.  (4.30)

eq We can simplify (4.30) with the aid of (4.17) as

[q^{4},q^{6};q^{12}]_{∞} ([q^{5};q^{12}]_{∞} + q[q;q^{12}]_{∞})^{2} = \frac{E^{2}(q^{6})E^{6}(q^{2})}{E^{5}(q^{12})E(q^{4})E^{2}(q)}.  (4.31)

Next, we use (4.18) to reduce (4.31) to the following easily verifiable identity

\frac{[q^{2};q^{12}]_{∞}^{4} [q^{4},q^{6};q^{12}]_{∞}^{3}}{[q,q^{3},q^{5};q^{12}]_{∞}^{2}}= \frac{E^{2}(q^{6})E^{6}(q^{2})}{E^{5}(q^{12})E(q^{4})E^{2}(q)}.  (4.32)

This completes our proof of (4.22), (4.20). We have (4.9), as desired.

The proof of (4.10) is analogous. Again, we view this identity as the constant term in z of the following

∑_{j=14}^{19} s_{j}(z,q) = R_{2}(q) ∑_{n=−∞}^{∞} q^{9 \frac{n(n+1)}{2}} z^{n},  (4.33)

where

R_{2}(q) = q\frac{E^{2}(q^{9})E^{4}(q^{12})}{E(q^{3})E(q^{6})E(q^{18})}.  (4.34)

Again, (4.24) holds true for the following (i,j) pairs: (14,15), (15,16), (16,17), (17,14), (18,19), (19,18). And so, both sides of (4.33) satisfy (4.26). Again it remains to show that (4.33) holds at one nontrivial point, say z_{1} = −q^{−3} . Observing that

s_{14}(z_{1},q) = s_{15}(z_{1},q) = s_{19}(z_{1},q) = 0,  (4.35)

we find that (4.33) with z = z_{1} becomes

s_{16}(z_{1},q) + s_{17}(z_{1},q) + s_{18}(z_{1},q) = R_{2}(q) ∑_{n=−∞}^{∞} q^{9 \frac{n(n+1)}{2}} z_{1}^{n},  (4.36)

Again, making repeated use of (4.15) and replacing q^{3} by q, we can rewrite (4.36) as

[q^{3},q^{4},q^{6};q^{12}]_{∞} ([q^{5};q^{12}]_{∞}− q[q;q^{12}]_{∞}) + q[q^{2},q^{3},q^{3},q^{4};q^{12}]_{∞}
= \frac{E^{4}(q^{4})E^{2}(q^{3})}{E^{4}(q^{12})E(q^{6})E(q^{2})}.  (4.37)

If we multiply both sides of (4.37) by \frac{[q^{2};q^{12}]_{∞}}{[q^{4};q^{12}]_{∞}} and take advantage of (4.17) we find that

[q,q^{5},q^{6},q^{6};q^{12}]_{∞} + q[q^{2},q^{3};q^{12}]_{∞}^{2} = \frac{E^{4}(q^{4})E^{2}(q^{3})}{E^{4}(q^{12})E(q^{6})E(q^{2})}\frac{[q^{2};q^{12}]_{∞}}{[q^{4};q^{12}]_{∞}},  (4.38)

which is easy to recognize as (4.19). This completes our proof of (4.33) and (4.10).

To prove (4.11), (4.12) we will follow a well trodden path and observe that these identities are just constant terms in z of

∑_{j=20+4α}^{23+4α} s_{j}(z,q) = q^{2} \frac{E^{2}(q^{9})E(q^{12})E(q^{36})}{E(q^{3})}· ∑_{n=−∞}^{∞} q^{9 \frac{n(n+1)}{2}} z^{n},  (4.39)

with α = 0 and 1, respectively.

To prove that both sides of (4.39) satisfy (4.26) we verify that (4.24) holds for the following (i,j) pairs: (20 + 4α,21 + 4α), (21 + 4α,22 + 4α), (22 + 4α,23 + 4α), (23 + 4α,20 + 4α) with α = 0,1. It remains to verify (4.39) at

z̃_{α} = −q^{6(1−2α)} , α = 0,1.

Taking into account that

s_{j+4α} (z̃_{α},q) = 0

for j = 20,21,22 and α = 0,1, we find that

s_{23+4α} (z̃_{α},q) = (−1)^{α+1} q^{4+6α} E^{2}(q^{9})E(q^{12})E(q^{36}),

which is easy to prove with the aid of (4.15).

This completes our proof of (4.11) and (4.12).

5 Concluding Remarks

Making use of the Littlewood decomposition of π_{t-core}into its s-core and s-quotient,

ϕ̃(π_{t-core}) = (π_{s-core},π̂_{0},π̂_{1},...,π̂_{s−1} ),

together with

1 + ω_{s}+ ω_{s}^{2}+ ··· + ω_{s}^{s−1} = 0,

it is not hard to see that

GBG-rank(π_{t-core},s) = GBG-rank(π_{s-core},s).

In a recent paper [11], Olsson proved a somewhat unexpected result:

Theorem 5.1. Let s,t be relatively prime positive integers. Then the s-core of a t-core is, again, a t-core.

A partition is called an (s,t)-core if it is simultaneously an s-core and a t-core.

In [2], Anderson established

Theorem 5.2. Let s,t be relatively prime positive integers. Then the number of (s,t)-cores is \binom{s+t}{s}/(s + t).

Remarkably, the three observations above imply our Theorem 1.2. Moreover, our Theorem 1.3 implies

Corollary 5.3. Let s,t be relatively prime positive integers. Then no two distinct (s,t)- cores share the same value of GBG-rank mod s, when s is prime, or when s is composite and t < 2p_{s}, where p_{s} is the smallest prime divisor of s.

On the other hand, when the conditions on s and t in the corollary above are not met, two distinct (s,t)-cores may, in fact, share the same value of GBG-rank mod s. For example, consider two relatively prime integers s and t such that 2 | s, s > 2, t > 1 + \frac{s}{2}, t ≠ s + 1. In this case partition [1^{\frac{s}{2}−1} ,2,1 + \frac{s}{2}] and empty partition [ ] are two distinct (s,t)-cores such that

GBG-rank( [1^{\frac{s}{2}−1} ,2,1 + \frac{s}{2}],s) = GBG-rank([ ],s) = 0.

Here we are using the partition notation of Andrews [1, p.1].

Acknowledgments

We are indebted to Hendrik Lenstra for his contribution to the proof of Lemma 1.4. We would like to thank Christian Krattenthaler, Jorn Olsson and Peter Paule for their kind interest and helpful discussions.

Research of both authors was supported in part by NSA grant H98230-07-1-0011.

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