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arXiv:0807.4727v2 [math.CO] 14 Aug 2008
THE GBG-RANK AND t-CORES I.
COUNTING AND 4-CORES
ALEXANDER BERKOVICH AND FRANK G. GARVAN
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) =
s−1
X
j=0
r
j
(π, s)e
I
2π
s
j
, I =
√
−1.
We will prove that for (s, t) = 1
ν(s, t) ≤
“
s + t
s
”
s + t
,
where ν(s, t) denotes a number of distinct values that 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 a smallest prime divisor of s.
We will show that the generating functions for 4-cores with prescribed GBG-
rank mod 3 value are all eta-products.
1. Introduction
A partition π is a nonincreasing s e quence
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| =
X
i=1
λ
i
.
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 ce ll 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-r esidue diag ram [7]. One can also label cells in the infinite column 0 and
the infinite row 0 in the same fashion. The resulting diagr am is called the extended
s-residue diagram o f π [4]. With each π we can associate the s-dimensional vector
r(π, s) = (r
0
, r
1
, . . . , r
s−1
),
Date: August 13, 2008.
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.
Research of both authors was supported in part by NSA grant MSPF-06G-150.
1
2 ALEXANDER BERKOVICH AND FRANK G. GARVAN
where r
i
, 0 ≤ i ≤ s −1 is the number of cells of π 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
− r
0
.
Note that
n · l
s
=
s−1
X
i=0
n
i
= 0,
where
l
s
= (1, 1, 1, . . . , 1) ∈ Z
s
.
If some cell of π s hares 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. We say that π is a t-core, denoted
π
t-core
, if its diagram has no rim hooks of length t [7]. The Durfee square of π,
denoted D(π), is the largest square that fits ins ide 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 [2] we defined a new partition statistic of π
(1.1) GBG-rank(π, s) :=
s−1
X
i=0
r
i
(π, s)ω
i
s
,
where
ω
s
= e
I
2π
s
and
I =
√
−1.
We refer to this statistic as the GBG-rank of π mod s. The special case s = 2 was
studied in great detail in [2] and [3]. In particular, we have shown in [2] that for
any odd t > 1
(1.2) GBG-rank(π
t-core
, 2) =
1 −
P
t−1
i=0
(−1)
i+n
i
(π
t-core
,t)
4
and that
(1.3) −
t − 1
4
≤ GBG-rank(π
t-core
, 2) ≤
t + 1
4
,
where ⌊x⌋ is the integer part of x. Our main object here is to prove the following
generalizatio ns of (1.2) and (1.3).
Theorem 1.1. Let t, s ∈ Z
>1
and (t, s) = 1. Then
(1.4) GBG-rank(π
t-core
, s) =
P
t−1
i=0
ω
i+1
s
(ω
tn
i
(π
t-core
,t)
s
− 1)
(1 − ω
s
)(1 − ω
t
s
)
THE GBG-RANK AND t-CORES I. COUNTING AND 4-CORES 3
Theorem 1.2. Let ν(s, t) denote the number of distinct values that GBG-rank of
π
t-core
mod s may assume. Then
(1.5) ν(s, t) ≤
t + s
t
t + s
,
provided (s, t) = 1.
Theorem 1.3. Let ν(s, t) be as in Theorem 1.2 and (s, t) = 1. Then
(1.6) ν(s, t) =
t + s
t
t + s
,
iff either s is prime or s is composite and t < 2p
s
, where p
s
is a smallest prime
divisor of s.
Our of 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
),
˜
j =
(
˜
j
0
,
˜
j
1
, . . . ,
˜
j
t−1
) be integer valued vectors such that
(1.7) 0 ≤ j
0
≤ j
1
≤ ··· ≤ j
t−1
≤ s − 1,
(1.8) 0 ≤
˜
j
0
≤
˜
j
1
≤ ··· ≤
˜
j
t−1
≤ s − 1,
and
(1.9)
t−1
X
i=0
ω
j
i
s
=
t−1
X
i=0
ω
˜
j
i
s
(1.10)
t−1
Y
i=0
ω
j
i
s
=
t−1
Y
i=0
ω
˜
j
i
s
.
Then
j =
˜
j,
iff either s is prime or s is composite such that t < 2p
s
, where p
s
is a smallest
prime divisor of s.
The rest of this paper is orga nis e d as follows. In Section 2, we collect some
necessary background on t-cores and prove Theorems 1.1 and Theorem 1.2. Section
1.3 is devoted to the proo f of Lemma 1.4 and Theore m 1.3 . Section 4 deals with
4-cores with prescribed values of GBG-rank mod 3. There we will provide new
combinatorial interpr etation and proof of the Hirshhorn-Sellers identities for 4-
cores [6]. We conclude with the remarks connecting this development and that of
[10] and [1].
4 ALEXANDER BERKOVICH AND FRANK G. GARVAN
2. Properties of the GBG-rank
We begin with some definitions from [4]. A region r in the extended t-residue
diagram of π is the se t of all cells (i, j) satisfying t(r − 1) ≤ j − i < tr. A cell of
π is ca lled 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 [4] 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 thr e e more useful facts from [4].
A.
(2.2)
π
t-core
=
t
2
|n|
2
+ b
t
· n,
where b
t
= (0, 1, 2, . . . , t − 1).
B.
(2.3)
X
i∈P
1
n
i
= −
X
i∈P
−1
n
i
= D(π
t-core
),
where P
α
= {i ∈ Z : 0 ≤ i ≤ t − 1, αn
i
> 0}, α = −1, 1.
C.
Under conjugation φ(π
t-core
) transforms as
(2.4) (n
0
, n
1
, n
2
, . . . , n
t−1
) → (−n
t−1
, −n
t−2
, . . . , −n
0
).
We beg in our proof of the Theorem 1.1 by observing tha t under conjugation
GBG-rank transforms as
(2.5) GBG-rank(π, s) =
s−1
X
i=0
r
i
ω
i
s
=⇒ GBG-rank(π
∗
, s) =
s−1
X
i=0
r
i
ω
−i
s
.
Next, we use that
(2.6) GBG-rank(π, s) = GBG-rank(π
1
, s) + GBG-rank(π
2
, s) − D.
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
THE GBG-RANK AND t-CORES I. COUNTING AND 4-CORES 5
cells strictly to the right of the main diagonal.
Recalling (2.1) and (2.3) we find that
(2.7) GBG-rank(π
1
, s) =
X
i∈P
1
n
i
X
k=1
i+t(k−1)
X
j=0
ω
j
s
=
D
1 − ω
s
−
X
i∈P
1
ω
i+1
s
(1 − ω
tn
i
s
)
(1 − ω
s
)(1 − ω
t
s
)
.
Analogously,
(2.8) GBG-rank(π
∗
2
, s) =
D
1 − ω
s
−
X
i∈P
−1
ω
t−i
s
(1 − ω
−tn
i
s
)
(1 − ω
s
)(1 − ω
t
s
)
,
where we made use of (2.4).
Clearly, (2.5) and (2.8) imply that
(2.9) GBG-rank(π
2
, s) = −
Dω
s
1 − ω
s
−
X
i∈P
−1
ω
1+i
s
(1 − ω
tn
i
s
)
(1 − ω
s
)(1 − ω
t
s
)
.
Next, we combine (2.6), (2.7) and (2.9) to find that
GBG-rank(π
t-core
, s) = −
X
i∈P
−1
S
P
1
ω
1+i
s
(1 − ω
tn
i
s
)
(1 − ω
s
)(1 − ω
t
s
)
=
t−1
X
i=0
ω
1+i
s
(ω
tn
i
s
− 1)
(1 − ω
s
)(1 − ω
t
s
)
,
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,
t−1
X
k=0
j
k
≡ r mod s.
Then
(2.10) ν(s, t) ≤ a
t(t+1)
2
(s, t),
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 distinct values of
t−1
X
i=0
ω
1+i+tn
i
s
,
where n ∈ Z
t
and n · 1
t
= 0. Given any such n-vector we reduce the exp onents
1 + i + tn
i
mod s and reorder to obtain a j-vector such that
t−1
X
k=0
j
k
≡
t−1
X
i=0
1 + i + tn
i
≡
t(t+1)
2
(mod s).
It follows tha t
ν(s, t) ≤ a
t(t+1)
2
(s, t).
6 ALEXANDER BERKOVICH AND FRANK G. GARVAN
Observation 2:
(2.11)
s−1
X
r=0
a
r
(s, t) =
t + s − 1
t
.
This result is well known and we omit the proof. Finally, we need
Observation 3:
If (s, t) = 1 then
(2.12) a
0
(s, t) = a
1
(s, t) = ··· = a
s−1
(s, t).
Proof.
There exists an integer T such that T ·t ≡ 1 mod s, bec ause s and t are coprime.
This implies that
t−1
X
i=0
(j
i
+ T ) ≡ 1 +
t−1
X
i=0
j
i
mod s.
Consequently, a
r
(s, t) = a
r+1
(s, t), as desired. Combining (2.1 0), (2.11) and (2.12)
we see that
ν(s, t) ≤ a
t(t+1)
2
=
s − 1 + t
t
s
=
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 pro of of Theore m 1.2 that
ν(s, t) =
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,
t−1
Y
i=0
ω
j
i
s
= ω
t(t+1)
2
s
is associated with a distinct co mplex number
P
t−1
i=0
ω
j
i
s
. Lemma 1.4 tells us when
this is exac tly 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 po ly nomial of ω
s
over Q. Let us now define
(3.1) p
1
(x) :=
t−1
X
i=0
(x
j
i
− x
˜
j
i
),
where j and
˜
j satisfy the constraints (1.7) - (1.10). It is clear tha t
p
1
(ω
s
) = 0,
THE GBG-RANK AND t-CORES I. COUNTING AND 4-CORES 7
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 =
˜
j, as desir e d.
Case 2.
Here s is c omposite, (s, t) = 1 and t < 2p
s
, where p
s
is a smallest prime divisor
of s. Once again (1.9) implies that
p
1
(ω
s
) = 0.
Moreover, the sth cyclotonic polynomial, defined as
(3.2) Φ
s
(x) :=
Y
0<j<s,
(j,s)=1
(x − ω
j
s
),
is a minimal po ly nomial of ω
s
over Q. This means that
(3.3) p
1
(ω
m
s
) = 0,
for any 1 ≤ m < s such that (s, m) = 1. In particular, we have that
(3.4) p
1
(ω
k
s
) = 0, 1 ≤ k ≤ p
s
− 1.
At this point, it is expedient to rewrite (3.4) as
(3.5) h
k
(ω
j
0
s
, . . . , ω
j
t−1
s
) = h
k
(ω
˜
j
0
s
, . . . , ω
˜
j
t−1
s
), 1 ≤ k ≤ p
s
− 1,
where
h
k
(x
1
, x
2
, . . . , x
t
) = x
k
1
+ x
k
2
+ ··· + x
k
t
.
Next, we use Newton’s theorem on symmetric polynomials to convert (3.5) into
p
s
− 1 identities
(3.6) σ
k
(ω
j
0
s
, . . . , ω
j
t−1
s
) = σ
k
(ω
˜
j
0
s
, . . . , ω
˜
j
t−1
s
), 1 ≤ k ≤ p
s
− 1,
where the kth elementary symmetric polynomials σ
k
’s in x
1
, x
2
, . . . , x
t
are defined
in a standard way as
(3.7) σ
k
(x
1
, x
2
, . . . , x
t
) =
X
1≤i
1
≤i
2
<...<i
k
≤t
x
i
1
x
i
2
···x
i
k
, 1 ≤ k ≤ t.
Note that we can rewrite (1.10) now as
(3.8) σ
t
(ω
j
0
s
, . . . , ω
j
t−1
s
) = σ
t
(ω
˜
j
0
s
, . . . , ω
˜
j
t−1
s
).
But
(3.9) σ
t
σ
∗
k
= σ
t−k
,
where
σ
∗
k
(x
1
, x
2
, . . . , x
t
) = σ
k
(x
−1
1
, x
−1
2
, . . . , x
−1
t
).
This fortunate fact enables us to convert (3.6) into p
s
− 1 identities
(3.10) σ
k
(ω
j
0
s
, . . . , ω
j
t−1
s
) = σ
k
(ω
˜
j
0
s
, . . . , ω
˜
j
t−1
s
), t − p
s
+ 1 ≤ k ≤ t − 1.
But t < 2p
s
, and so, t − p
s
+ 1 ≤ p
s
. This means that we have the following t
identities
(3.11) σ
k
(ω
j
0
s
, . . . , ω
j
t−1
s
) = σ
k
(ω
˜
j
0
s
, . . . , ω
˜
j
t−1
s
), 1 ≤ k ≤ t.
8 ALEXANDER BERKOVICH AND FRANK G. GARVAN
Consequently,
t−1
Y
i=0
(x − ω
j
i
s
) =
t−1
Y
i=0
(x − ω
˜
j
i
s
).
Recalling that j,
˜
j satisfy (1.7) and (1.8), we conclude that j =
˜
j.
Let us summarize. If s is a prime or if s is a composite number such that t < 2p
s
,
then j =
˜
j, provided that (s, t) = 1 and j,
˜
j satisfy (1.7)–(1.10).
It remains to show that j =
˜
j does not have to be true if s is a composite numb er
and t ≥ 2p
s
. To this end consider
j : = (0, 0, . . . , 0, 1 , 1, 3, 3) ∈ Z
t
,
˜
j : = (0, 0, . . . , 0, 0 , 0, 2, 2) ∈ Z
t
,
if s = 4, t ≥ 4,
j : = (0, 0, . . . , 0, 1 , 1, 4, 4) ∈ Z
t
,
˜
j : = (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
,
˜
j : = (0, 0, . . . , 0, 1 , 2, 4, 5, 7, 8) ∈ Z
t
,
if s = 9, t ≥ 6, respectively. It is not har d to verify that j,
˜
j satisfy (1.7)–(1.10) and
that j 6=
˜
j in these cases . It remains to consider the last case where s is a comp osite
number 6= 4, 6, 9, t ≥ 2p
s
. In this case s > 3p
s
. And, as a result,
3 +
s
p
s
(p
s
− 1) < s.
Let us now consider
j : = (0, 0, . . . , 0, 2 , 2, 2 +
s
p
s
, 2 +
s
p
s
, . . . , 2 +
s
p
s
(p
s
− 1), 2 +
s
p
s
(p
s
− 1)) ∈ Z
t
,
˜
j : = (0, 0, . . . , 0, 1 , 3, 1 +
s
p
s
, 3 +
s
p
s
, . . . , 1 +
s
p
s
(p
s
− 1), 3 +
s
p
s
(p
s
− 1)) ∈ Z
t
.
Again, it is straightforward to check that j,
˜
j satisfy (1.7)–(1.10) and tha t j 6=
˜
j.
This completes our proof of Lemma 1.4.
We have an immediate
Corollary 3.1. Theorem 1.3 holds true.
To illustrate the usefulness of Theorem 1.3, consider the following example:
s = 3, t = 4. In this c ase we should have exactly
4 + 3
3
4+3
= 5 distinct values
of GBG-rank(π
4-core
, 3). To determine these distinct values we substitute the fol-
lowing 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
2
3
, resp e ctively. To verify this we note that there
are exactly 27 vectors such that
n ∈ Z
4
3
and n · 1
4
≡ 0 mod 3.
In Table 1 we lis t all these vectors together with the assoc iated GBG-rank mod 3
values, determined by (1.4). These vectors will come in handy later.
THE GBG-RANK AND t-CORES I. COUNTING AND 4-CORES 9
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
2
3
n
25
= (−1, 1, 1, −1) −ω
2
3
n
26
= (−1, 2, 0, −1) −ω
2
3
n
27
= (0, 1, 0, −1) −ω
2
3
Table 1.
4. The GBG-rank of 4-cores mod 3
Let G
t
(q) denote the generating function for t-cores.
(4.1) G
t
(q) :=
X
π
t-core
q
|π
t-core
|
.
Let P be the set of all partitions a nd P
t-core
be the set of all t-cores. Ther e is a
well-known bijection
˜
φ : P → P
t-core
× P × P × P . . . × P
which goes back to D.E. Littlewood [9]
˜
φ(π) = (π
t-core
, ˆπ
0
, ˆπ
1
, . . . , ˆπ
t−1
)
such that
|π| = |π
t-core
| + t
t−1
X
i=0
|ˆπ
i
|.
10 ALEXANDER BERKOVICH AND FRANK G. GARVAN
The multipartition (ˆπ
0
, ˆπ
1
, . . . , ˆπ
t−1
) is ca lled the t-quotient of π. The immediate
corollary of the Littlewood bijection is
(4.2) G
t
(q) =
E
t
(q
t
)
E(q)
,
where
(4.3) E(q) :=
Y
j≥1
(1 − q
j
).
On the other hand, formula (2.2) s uggests [4] that
(4.4) G
t
(q) =
X
n∈Z
t
,
n·1
t
=0
q
t
2
|n|
2
+n·b
t
,
so that
(4.5)
X
n∈Z
t
,
n·1
t
=0
q
t
2
|n|
2
+n·b
t
=
E
t
(q
t
)
E(q)
.
The above identity was first obtained by Klyachko [8], who observed that it is a
sp e c ial case of A
t−1
MacDonald’s identity. An elementary proof of (4.5) ca n be
found in [2]. Next we define
(4.6) g
c
(q) =
X
π
4-core
,
GBG-rank(π
4-core
,3)=c
q
|π
4-core
|
In other words, g
c
(q) is the generating function for 4-cores with a given value c of
the GBG-ra nk mod 3. ¿From the discussion a t the end of the last section it is clear
that
(4.7)
E
4
(q
4
)
E(q)
= g
−1
(q) + g
0
(q) + g
1
(q) + g
−ω
3
(q) + g
−ω
2
3
(q).
It turns out that
g
−1
(q) = q
5
E
4
(q
36
)
E(q
9
)
,(4.8)
g
0
(q) =
E
6
(q
6
)E
2
(q
18
)
E
3
(q
3
)E(q
12
)E(q
36
)
,(4.9)
g
1
(q) = q
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
E
2
(q
9
)E(q
12
)E(q
36
)
E(q
3
)
,(4.11)
g
−ω
2
3
(q) = q
2
E
2
(q
9
)E(q
12
)E(q
36
)
E(q
3
)
.(4.12)
THE GBG-RANK AND t-CORES I. COUNTING AND 4-CORES 11
Hence
E
4
(q
4
)
E(q)
=
E
6
(q
6
)E
2
(q
18
)
E
3
(q
3
)E(q
12
)E(q
36
)
+ q
E
2
(q
9
)E
4
(q
12
)
E(q
3
)E(q
6
)E(q
18
)
+ 2q
2
E
2
(q
9
)E(q
12
)E(q
36
)
E(q
3
)
+ q
5
E
4
(q
36
)
E(q
9
)
.
(4.13)
We note that the identity
g
−ω
3
(q) = g
−ω
2
3
(q)
follows from (2 .5) a nd the fact that π is a t-core if and only if the conjugate π
∗
is. The identities equivalent to (4.13) were firs t proven by Hirschhorn and Sellers
[6]. However, combinatorial identities (4.7)–(4.12) given here are brand new. The
proof of (4.8) is rather simple. Indeed, data in Table 1, suggests that
g
−1
(q) =
X
n·1
4
=0,
n≡n
1
mod Z
4
3
q
2|n|
2
+b
4
·n
= q
5
X
˜n·1
4
=˜n
0
+˜n
1
+˜n
2
+˜n
3
=0
q
9(2|˜n|
2
−˜n
1
+2˜n
2
+˜n
3
)
= q
5
X
˜n·1
4
=0
q
9(2|˜n|
2
+˜n
0
+2˜n
3
+3˜n
2
)
= q
5
E
4
(q
36
)
E(q
9
)
(4.14)
where in the last s tep we relabelled the summation variables and use d (4.5) with
t = 4 and q → q
9
.
In what follows we shall require the J acobi triple product identity ([5],II.28)
(4.15)
∞
X
n=−∞
(−1)
n
q
n
2
z
n
= E(q
2
)[zq; q
2
]
∞
,
where
[z; q]
∞
:=
∞
Y
j=0
(1 − zq
j
)
1 −
q
1+j
z
and the formula ([5],ex.5.21)
(4.16)
ux,
u
x
, vy,
v
y
; q
∞
=
uy,
u
y
, vx,
v
x
; q
∞
+
v
x
xy,
x
y
, uv,
u
v
; q
∞
,
where
[z
1
, z
2
, . . . , z
n
; q]
∞
:=
n
Y
j=1
[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
(4.17) [q
2
, q
3
; q
12
]
∞
([q
5
; q
12
]
∞
− q[q; q
12
]
∞
) = [q, q
5
, q
6
; q
12
]
∞
.
12 ALEXANDER BERKOVICH AND FRANK G. GARVAN
Analogously, (4.1 6) with u = q
5
, v = q
2
, x = q, y = 1 and q → q
12
becomes
(4.18) [q
5
; q
12
]
∞
+ q[q; q
12
]
∞
=
[q
2
, q
2
, q
4
, q
6
; q
12
]
∞
[q, q
3
, q
5
; q
12
]
∞
.
Finally, setting u = q
6
, v = q
4
, x = q
3
, y = 1 and q → q
12
in (4.16) yields
(4.19) [q
3
, q
4
; q
12
]
2
∞
= [q, q
5
, q
6
, q
6
; q
12
]
∞
+ q[q
2
, q
3
; q
12
]
2
∞
.
Next, we use again Table 1 to rewrite (4.9) as
(4.20)
13
X
j=2
X
n·1
4
=0,
n≡n
j
mod Z
4
3
q
2|n|
2
+b
4
·n
= R
1
(q),
where
(4.21) R
1
(q) =
E
6
(q
6
)E
2
(q
18
)
E
3
(q
3
)E(q
12
)E(q
36
)
.
Remarkably, (4.20) is a constant term in z of the following identity
(4.22)
13
X
j=2
s
j
(z, q) = R
1
(q)
∞
X
n=−∞
q
9
n(n+1)
2
z
n
,
where
(4.23) s
j
(z, q) :=
X
n≡n
j
mod Z
4
3
q
2|n|
2
+b
4
·n
z
n·1
4
3
, j = 1, 2, . . . , 27.
Using s imple changes of variables, it is straightforward to check that
(4.24) zq
9
s
i
(zq
9
, q) = s
j
(z, q),
holds tr ue 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), (1 3, 10), and that
(4.25) zq
9
∞
X
n=−∞
q
9
n(n+1)
2
(zq
9
)
n
=
∞
X
n=−∞
q
9
n(n+1)
2
z
n
.
Consequently, both s ides of (4.22) satisfy the same first order functional equation
(4.26) zq
9
f(zq
9
, q) = f (z, q).
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
(4.27) s
4
(z
0
, q) = s
8
(z
0
, q) = s
11
(z
0
, q) = 0,
and that
(4.28) s
3
(z
0
, q) + s
9
(z
0
, q) = s
5
(z
0
, q) + s
12
(z
0
, q) = 0.
We see that (4.22) with z = z
0
becomes
(4.29)
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)
∞
X
n=−∞
q
9
n(n+1)
2
z
n
0
.
THE GBG-RANK AND t-CORES I. COUNTING AND 4-CORES 13
Upo n making r epeated 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
]
∞
=
E
2
(q
6
)E
6
(q
2
)
E
5
(q
12
)E(q
4
)E
2
(q)
.
(4.30)
We can simplify (4.30) with the aid of (4.17) as
(4.31) [q
4
, q
6
; q
12
]
∞
([q
5
; q
12
]
∞
+ q[q; q
12
]
∞
)
2
=
E
2
(q
6
)E
6
(q
2
)
E
5
(q
12
)E(q
4
)E
2
(q)
.
Next, we use (4.18) to reduce (4.31) to the following easily verifiable identity
(4.32)
[q
2
; q
12
]
4
∞
[q
4
, q
6
; q
12
]
3
∞
[q, q
3
, q
5
; q
12
]
2
∞
=
E
2
(q
6
)E
6
(q
2
)
E
5
(q
12
)E(q
4
)E
2
(q)
.
This completes our proof of (4.22), (4.20). We have (4 .9), as desire d.
The proof of (4.10) is analogous. Ag ain, we vie w this identity a s a constant term
in z of the following
(4.33)
19
X
j=14
s
j
(z, q) = R
2
(q)
∞
X
n=−∞
q
9
n(n+1)
2
z
n
,
where
(4.34) R
2
(q) = q
E
2
(q
9
)E
4
(q
12
)
E(q
3
)E(q
6
)E(q
18
)
.
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 o f (4.33) satisfy (4.26). Again it remains to show that (4.33)
holds at o ne nontrivial point, say z
1
= −q
−3
. Observing that
(4.35) s
14
(z
1
, q) = s
15
(z
1
, q) = s
19
(z
1
, q) = 0,
we find tha t (4.33) with z = z
1
becomes
(4.36) s
16
(z
1
, q) + s
17
(z
1
, q) + s
18
(z
1
, q) = R
2
(q)
∞
X
n=−∞
q
9
n(n+1)
2
z
n
1
,
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
]
∞
=
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
[q
2
;q
12
]
∞
[q
4
;q
12
]
∞
and take advantage of (4.17) we find
that
(4.38) [q, q
5
, q
6
, q
6
; q
12
]
∞
+ q[q
2
, q
3
; q
12
]
2
∞
=
E
4
(q
4
)E
2
(q
3
)
E
4
(q
12
)E(q
6
)E(q
2
)
[q
2
; q
12
]
∞
[q
4
; q
12
]
∞
,
14 ALEXANDER BERKOVICH AND FRANK G. GARVAN
which is easy to recognize as (4.19). This completes o ur proof of (4 .33) and (4.10).
To prove (4.11), (4.12) we will follow well trodden path and observe that these
identities are just constant terms in z of
(4.39)
23+4α
X
j=20+4α
s
j
(z, q) = q
2
E
2
(q
9
)E(q
12
)E(q
36
)
E(q
3
)
·
∞
X
n=−∞
q
9
n(n+1)
2
z
n
,
with α = 0 and 1, respectively.
To prove that both sides of (4.39) satisfy (4.2 6) we verify that (4.24) holds for the
following (i, j) pairs: (2 0 + 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
˜z
α
= −q
6(1−2α)
, α = 0, 1 .
Taking into account that
s
j+4α
(˜z
α
, q) = 0
for j = 20, 21, 22 and α = 0, 1, we find that
s
23+4α
(˜z
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) a nd (4.12).
5. Concluding Remarks
Making use of the Littlewood decomposition of π
t-core
into its s-core and s-
quotient,
˜
φ(π
t-core
) = (π
s-core
, ˆπ
0
, ˆπ
1
, . . . , ˆπ
s−1
),
together with
1 + ω
s
+ ω
2
s
+ ··· + ω
s−1
s
= 0,
it is not hard to see that
GBG-rank(π
t-core
, s) = GBG-rank(π
s-core
, s).
In a rece nt paper [10], 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.
In [1], Anderson established
Theorem 5.2. Let s, t be relatively prime positive integers, then the number of
partitions, which are simultaneously s-core and t-core is
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 a smallest prime divisor of s.
THE GBG-RANK AND t-CORES I. COUNTING AND 4-CORES 15
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 +
s
2
, t 6= s + 1. In this case partition [1
s
2
−1
, 2, 1 +
s
2
] and empty partition [ ]
are two distinct (s, t)-cores such that
GBG-rank
1
s
2
−1
, 2, 1 +
s
2
, s
= GBG-rank([ ], s) = 0.
Acknowledgement
We are grateful 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.
References
1. J. Anderson, Partitions which are simultaneously t
1
and t
2
-core, D iscrete Math. 248 (2002)
237–243.
2. A. Berkovich, F. G. Garvan The BG-rank of a partition and its applications, Adv. Appl.
Math. 40 (2008) 377–400.
3. A. Berkovich, H. Yesilyurt, New identities for 7-cores with prescribed BG-rank, preprint,
arXiv: math.NT/0603150, to appear in Discrete Math.
4. F. Garvan, D. Kim, D. Stanton, Cranks and t-cores, Invent. Math. 101 (1990), 1–17.
5. G. Gasper, M. Rahman, Basic hypergeometric series, Encyclopedia of Mathematics and its
applications v.35, Cambridge, 1990.
6. M. D. Hirshhorn, J. A. Sellers, Some amazing f acts about 4-cores, J. of Num. Theor. 60 (1996)
51–69.
7. G. James, A. Kerber, The Representation Theory of the Symmetric Group, Encyclopedia of
Mathematics and its A pplications v.16, Reading, MA, 1981.
8. A. A. Klyachko, Modular forms and representations of symmetric groups, J. Soviet Math. 26
(1984), 1879–1887.
9. D. E. Littlewoo d, Modular representations of symmetric groups, Proc. Roy. Soc. London Ser.
A. 209 (1951), 333–353.
10. J. B. Olsson, A theorem on the cores of partitions, preprint, arXiv:0801.4884.
Department of Mathematics, University of Florida, Gainesville, Florida 32611-8105
E-mail address: alexb@math.ufl.edu
Department of Mathematics, University of Florida, Gainesville, Florida 32611-8105
E-mail address: fgarvan@math.ufl.edu