Background
Borwein-Like Theorem
Proof
Final Thoughts
A Borwein-Like Theorem for Overpartitions
Conrad Parker and Melanie Stevenson
Abelian Group
Department of Mathematics, Statistics, and Computer Science
St. Olaf College
Northfield, MN 55057
Oct. 13, 2016
Abelian Group
A Borwein-Like Theorem for Overpartitions
Background
Borwein-Like Theorem
Proof
Final Thoughts
Integer Partitions
Definition
A partition λ of a non-negative integer n is a non-increasing
P
sequence of positive integers λ1 , λ2 , . . . , λk such that ki=1 λi = n.
The λi are called the parts of the partition. We write p(n) for the
number of partitions of a non-negative integer n.
Abelian Group
A Borwein-Like Theorem for Overpartitions
Background
Borwein-Like Theorem
Proof
Final Thoughts
Integer Partitions
Definition
A partition λ of a non-negative integer n is a non-increasing
P
sequence of positive integers λ1 , λ2 , . . . , λk such that ki=1 λi = n.
The λi are called the parts of the partition. We write p(n) for the
number of partitions of a non-negative integer n.
Example
∅
1
2, 11
3, 21, 111
Abelian Group
p(0) = 1
p(1) = 1
p(2) = 2
p(3) = 3
A Borwein-Like Theorem for Overpartitions
Background
Borwein-Like Theorem
Proof
Final Thoughts
Integer Partitions
Definition
A partition λ of a non-negative integer n is a non-increasing
P
sequence of positive integers λ1 , λ2 , . . . , λk such that ki=1 λi = n.
The λi are called the parts of the partition. We write p(n) for the
number of partitions of a non-negative integer n.
Example
∅
1
2, 11
3, 21, 111
4, 31, 22, 211, 1111
Abelian Group
p(0) = 1
p(1) = 1
p(2) = 2
p(3) = 3
p(4) = 5
A Borwein-Like Theorem for Overpartitions
Background
Borwein-Like Theorem
Proof
Final Thoughts
Counting Partitions
Definition
We define the q-Pochhammer symbol (a; q)∞ by
(a; q)∞ =
∞
Y
(1 − aq k ).
k=0
Abelian Group
A Borwein-Like Theorem for Overpartitions
Background
Borwein-Like Theorem
Proof
Final Thoughts
Counting Partitions
Definition
We define the q-Pochhammer symbol (a; q)∞ by
(a; q)∞ =
∞
Y
(1 − aq k ).
k=0
Theorem (Euler)
The generating function for the numbers p(n) is given by:
∞
X
n=0
p(n)q n =
∞
Y
1
1
=
(q; q)∞
1 − qn
n=1
= 1 + q + 2q 2 + 3q 3 + 5q 4 + 7q 5 + · · ·
Abelian Group
A Borwein-Like Theorem for Overpartitions
Background
Borwein-Like Theorem
Proof
Final Thoughts
Overpartitions
Definition
An overpartition λ of a non-negative integer n is a non-increasing
P
sequence of positive integers λ1 , λ2 , . . . , λk such that ki=1 λi = n.
The λi are called the parts of the partition and the first occurrence
of a part of size j may be overlined. We write p(n) for the number
of overpartitions of n.
Abelian Group
A Borwein-Like Theorem for Overpartitions
Background
Borwein-Like Theorem
Proof
Final Thoughts
Overpartitions
Definition
An overpartition λ of a non-negative integer n is a non-increasing
P
sequence of positive integers λ1 , λ2 , . . . , λk such that ki=1 λi = n.
The λi are called the parts of the partition and the first occurrence
of a part of size j may be overlined. We write p(n) for the number
of overpartitions of n.
Example
∅
1, 1
2, 2, 11, 11
3, 3, 21, 21, 21, 21, 111, 111
Abelian Group
p(0) = 1
p(1) = 2
p(2) = 4
p(3) = 8
A Borwein-Like Theorem for Overpartitions
Background
Borwein-Like Theorem
Proof
Final Thoughts
Overpartitions
Definition
An overpartition λ of a non-negative integer n is a non-increasing
P
sequence of positive integers λ1 , λ2 , . . . , λk such that ki=1 λi = n.
The λi are called the parts of the partition and the first occurrence
of a part of size j may be overlined. We write p(n) for the number
of overpartitions of n.
Example
∅
p(0) = 1
1, 1
p(1) = 2
2, 2, 11, 11
p(2) = 4
3, 3, 21, 21, 21, 21, 111, 111
p(3) = 8
4, 4, 31, 31, 31, 31, 22, 22, 211, 211, 211, 211, 1111, 1111 p(4) = 14
Abelian Group
A Borwein-Like Theorem for Overpartitions
Background
Borwein-Like Theorem
Proof
Final Thoughts
Overpartition Generating Function
We have the generating function
∞
X
n=0
p(n)q n =
(−q; q)∞
(q; q)∞
∞
Y
1 + qn
=
1 − qn
n=1
= 1 + 2q + 4q 2 + 8q 3 + 14q 4 · · ·
Abelian Group
A Borwein-Like Theorem for Overpartitions
Background
Borwein-Like Theorem
Proof
Final Thoughts
Pentagonal Numbers
Defined by Pn =
3n2 −n
2
for any integer n.
Pentagonal numbers have the useful property
Pn ≡ n
(mod 3)
because
2Pn ≡ 3n2 − n
(mod 3)
≡ −n
(mod 3)
≡ 2n
(mod 3),
so Pn ≡ n (mod 3).
Abelian Group
A Borwein-Like Theorem for Overpartitions
Background
Borwein-Like Theorem
Proof
Final Thoughts
Borwein’s Conjecture (the + - - conjecture)
Given
Hn (q) =
n
∞
Y
X
(1 − q 3i−2 )(1 − q 3i−1 ) =
a(j)q j ,
i=1
(1)
j=0
Conjecture
If a(j) is defined by (1), then for all j ≥ 0,
a(3j) ≥ 0,
a(3j + 1) ≤ 0,
Abelian Group
a(3j + 2) ≤ 0.
A Borwein-Like Theorem for Overpartitions
(2)
Background
Borwein-Like Theorem
Proof
Final Thoughts
Partition Interpretation
We may consider Hn (q) as a generating function for partitions into
distinct parts, none of which are congruent to 0 mod 3, which
subtracts the number of such partitions with an odd number of
parts from those with an even number of parts.
Example
6
7
8
9
10
11
5 1
4 2
7
5 2
4 2 1
8
7 1
5 2 1
8 1
7 2
5 4
10
8 2
7 2 1
5 4 1
11
10 1
8 2 1
7 4
5 4 2
+2
−1
+3
−2
−1
−1
Abelian Group
A Borwein-Like Theorem for Overpartitions
Background
Borwein-Like Theorem
Proof
Final Thoughts
Borwein-Like Theorem
Theorem
Defining the sequence b(n) by
F (q) =
=
∞
X
n=0
∞
Y
k=1
b(n)q n
(1 − q 3k−1 )(1 − q 3k−2 )
(1 + q 3k−1 )(1 + q 3k−2 )
we will have b(3j + 2) = 0 for all integers j ≥ 0.
Abelian Group
A Borwein-Like Theorem for Overpartitions
Background
Borwein-Like Theorem
Proof
Final Thoughts
Borwein-Like Conjecture
Conjecture (+ − 0)
In the series F (q) =
b(3n) ≥ 0,
P∞
n=0 b(n)q
n,
b(3n + 1) ≤ 0,
we will have
and
b(3n + 2) = 0
for all n ≥ 0.
Abelian Group
A Borwein-Like Theorem for Overpartitions
Background
Borwein-Like Theorem
Proof
Final Thoughts
Borwein-Like Theorem
Theorem (equivalent form)
Let n ≡ 2 (mod 3). Consider overpartitions of n into parts not
divisible by 3. Then the number of such overpartitions into an even
number of parts equals the number of such overpartitions into an
odd number of parts.
Abelian Group
A Borwein-Like Theorem for Overpartitions
Background
Borwein-Like Theorem
Proof
Final Thoughts
Borwein-Like Theorem
Theorem (equivalent form)
Let n ≡ 2 (mod 3). Consider overpartitions of n into parts not
divisible by 3. Then the number of such overpartitions into an even
number of parts equals the number of such overpartitions into an
odd number of parts.
Example (n = 5)
5
221̄
5̄ 2̄21̄
221
41
2111
4̄1 2̄111
11111
41̄
21̄11
2̄21 1̄1111
4̄1̄
2̄1̄11
Abelian Group
A Borwein-Like Theorem for Overpartitions
Background
Borwein-Like Theorem
Proof
Final Thoughts
Proof: Jacobi Triple Product
Theorem (Jacobi Triple Product)
(−a; ab)∞ (−b; ab)∞ (ab; ab)∞ =
X
a
n(n+1)
2
b
n(n−1)
2
n∈Z
Abelian Group
A Borwein-Like Theorem for Overpartitions
Background
Borwein-Like Theorem
Proof
Final Thoughts
Proof: Jacobi Triple Product
Theorem (Jacobi Triple Product)
(−a; ab)∞ (−b; ab)∞ (ab; ab)∞ =
X
a
n(n+1)
2
b
n(n−1)
2
n∈Z
Sketch of Proof.
We deduce via the Jacobi triple product that
(q; q 3 )∞ (q 2 ; q 3 )∞
(q; q 3 )∞ (q 2 ; q 3 )∞ (q 3 ; q 3 )∞
=
(−q; q 3 )∞ (−q 2 ; q 3 )∞
(−q; q 3 )∞ (−q 2 ; q 3 )∞ (q 3 ; q 3 )∞
P
(−1)n q Pn
P
= n∈Z
,
Pn
n∈Z q
F (q) =
where Pn = 12 n(3n − 1) is the nth pentagonal number.
Abelian Group
A Borwein-Like Theorem for Overpartitions
Background
Borwein-Like Theorem
Proof
Final Thoughts
Proof: Sieve
Sketch of Proof, cont.
P
3n+2 as follows.
We sieve F (q) to get G (q) = ∞
n=0 b(3n + 2)q
Let ω be a primitive 3rd root of unity. Then
S(q) =F (q) + ωF (ωq) + ω 2 F (ω 2 q)
Abelian Group
A Borwein-Like Theorem for Overpartitions
Background
Borwein-Like Theorem
Proof
Final Thoughts
Proof: Sieve
Sketch of Proof, cont.
P
3n+2 as follows.
We sieve F (q) to get G (q) = ∞
n=0 b(3n + 2)q
Let ω be a primitive 3rd root of unity. Then
S(q) =F (q) + ωF (ωq) + ω 2 F (ω 2 q)
=b(0) + b(1)q + b(2)q 2 + b(3)q 3 + b(4)q 4 + b(5)q 5 + · · ·
+ωb(0) + ω 2 b(1)q + b(2)q 2 + ωb(3)q 3 + ω 2 b(4)q 4 + b(5)q 5 + · · ·
+ω 2 b(0) + ωb(1)q + b(2)q 2 + ω 2 b(3)q 3 + ωb(4)q 4 + b(5)q 5 + · · ·
Abelian Group
A Borwein-Like Theorem for Overpartitions
Background
Borwein-Like Theorem
Proof
Final Thoughts
Proof: Sieve
Sketch of Proof, cont.
P
3n+2 as follows.
We sieve F (q) to get G (q) = ∞
n=0 b(3n + 2)q
Let ω be a primitive 3rd root of unity. Then
S(q) =F (q) + ωF (ωq) + ω 2 F (ω 2 q)
=b(0) + b(1)q + b(2)q 2 + b(3)q 3 + b(4)q 4 + b(5)q 5 + · · ·
+ωb(0) + ω 2 b(1)q + b(2)q 2 + ωb(3)q 3 + ω 2 b(4)q 4 + b(5)q 5 + · · ·
+ω 2 b(0) + ωb(1)q + b(2)q 2 + ω 2 b(3)q 3 + ωb(4)q 4 + b(5)q 5 + · · ·
=0b(0) + 0b(1)q + 3b(2)q 2 + 0b(3)q 3 + 0b(4)q r + 3b(5)q 5 + · · ·
Abelian Group
A Borwein-Like Theorem for Overpartitions
Background
Borwein-Like Theorem
Proof
Final Thoughts
Proof: Sieve
Sketch of Proof, cont.
P
3n+2 as follows.
We sieve F (q) to get G (q) = ∞
n=0 b(3n + 2)q
Let ω be a primitive 3rd root of unity. Then
S(q) =F (q) + ωF (ωq) + ω 2 F (ω 2 q)
=b(0) + b(1)q + b(2)q 2 + b(3)q 3 + b(4)q 4 + b(5)q 5 + · · ·
+ωb(0) + ω 2 b(1)q + b(2)q 2 + ωb(3)q 3 + ω 2 b(4)q 4 + b(5)q 5 + · · ·
+ω 2 b(0) + ωb(1)q + b(2)q 2 + ω 2 b(3)q 3 + ωb(4)q 4 + b(5)q 5 + · · ·
=0b(0) + 0b(1)q + 3b(2)q 2 + 0b(3)q 3 + 0b(4)q r + 3b(5)q 5 + · · ·
=3G (q).
Abelian Group
A Borwein-Like Theorem for Overpartitions
Background
Borwein-Like Theorem
Proof
Final Thoughts
Proof: Denominator is Irrelevant
Sketch of Proof, cont.
Writing
P
F (q) =
n Pn
n∈Z (−1) q
P
Pn
n∈Z q
=
A(q)
,
B(q)
we get
G (q) =
A(q)B(ωq)B(ω 2 q) + ωA(ωq)B(ω 2 q)B(q) + ω 2 A(ω 2 q)B(q)B(ωq)
.
3B(q)B(ωq)B(ω 2 q)
Abelian Group
A Borwein-Like Theorem for Overpartitions
Background
Borwein-Like Theorem
Proof
Final Thoughts
Proof: Denominator is Irrelevant
Sketch of Proof, cont.
Writing
P
F (q) =
n Pn
n∈Z (−1) q
P
Pn
n∈Z q
=
A(q)
,
B(q)
we get
G (q) =
A(q)B(ωq)B(ω 2 q) + ωA(ωq)B(ω 2 q)B(q) + ω 2 A(ω 2 q)B(q)B(ωq)
.
3B(q)B(ωq)B(ω 2 q)
We see that the numerator is a q 3n+2 sieve of 3A(q)B(ωq)B(ω 2 q),
so the coefficients of q 3n and q 3n+1 will be 0. If we can show that
the numerator is in fact zero, then G (q) will be zero as well.
Abelian Group
A Borwein-Like Theorem for Overpartitions
Background
Borwein-Like Theorem
Proof
Final Thoughts
Proof: Closed Form of Coefficients
Sketch of Proof, cont.
P
N
We find that the numerator is equal to ∞
N=0 R(N)q , where
h
i
X
R(N) =
ω m−n (−1)k + (−1)m ω + (−1)n ω 2 .
k,m,n∈Z
Pk +Pm +Pn =N
We will define f (k, m, n) = ω m−n (−1)k + (−1)m ω + (−1)n ω 2
so that we may write
X
f (k, m, n).
R(N) =
k,m,n∈Z
Pk +Pm +Pn =N
Abelian Group
A Borwein-Like Theorem for Overpartitions
Background
Borwein-Like Theorem
Proof
Final Thoughts
Proof: Defining a Function φ
Sketch of Proof, cont.
Letting X = {(k, m, n) ∈ Z3 | k + m + n ≡ 2 (mod 3)}, we define
a function φ : X −→ X by
2s − 1
2s − 1
2s − 1
φ(k, m, n) = k −
,m −
,n −
= (k 0 , m0 , n0 ),
3
3
3
where s = k + m + n. This is a well-defined function, since
2s − 1 ≡ 2 · 2 − 1 ≡ 0 (mod 3) and
s 0 = k 0 + m0 + n0 = s − (2s − 1) = 1 − s ≡ 2
(mod 3).
We will also have Pk + Pm + Pn = Pk 0 + Pm0 + Pn0 , although we
will not show this here.
Abelian Group
A Borwein-Like Theorem for Overpartitions
Background
Borwein-Like Theorem
Proof
Final Thoughts
Proof: φ is an Involution
Sketch of Proof, cont.
We have
φ(k, m, n) =
2s − 1
2s − 1
2s − 1
k−
,m −
,n −
3
3
3
= (k 0 , m0 , n0 )
where s = k + m + n. We also see that φ is an involution, since
2s 0 − 1
0
0 0
0
φ(k , m , n ) = k −
,···
3
2s − 1 2(1 − s) − 1
= k−
−
,···
3
3
= (k, m, n),
with the second and third terms analogous to the first.
Abelian Group
A Borwein-Like Theorem for Overpartitions
Background
Borwein-Like Theorem
Proof
Final Thoughts
Proof: φ Changes Sign of f
Sketch of Proof, cont.
We see that
2s − 1
≡ 2s − 1 ≡ 1
3
(mod 2),
and
0
0
m −n =
2s − 1
m−
3
2s − 1
− n−
= m − n,
3
so
i
h
0
0
0
0
0
f (φ(k, m, n)) = ω m −n (−1)k + (−1)m ω + (−1)n ω 2
h
i
= −ω m−n (−1)k + (−1)m ω + (−1)n ω 2
= −f (k, m, n).
Abelian Group
A Borwein-Like Theorem for Overpartitions
Background
Borwein-Like Theorem
Proof
Final Thoughts
Proof: Conclusion
Sketch of Proof, cont.
Finally, we see that
X
R(N) =
f (k, m, n)
h
i
X
X
1
=
f (k, m, n) +
f (φ(k, m, n))
2
i
X
1 hX
f (k, m, n) +
[−f (k, m, n)]
=
2
= 0,
as desired.
Abelian Group
A Borwein-Like Theorem for Overpartitions
Background
Borwein-Like Theorem
Proof
Final Thoughts
General Theorem
Theorem (Overpartition Nonresidue)
Given a prime p, define the sequence bp (n) by
Fp (q) =
=
∞
X
n=0
∞
Y
k=1
=
bp (n)q n
(1 − q k )(1 + q kp )
(1 + q k )(1 − q kp )
(q; q)∞ (−q p ; q p )∞
.
(−q; q)∞ (q p ; q p )∞
Then when n is a quadratic nonresidue modulo p, we have
bp (n) = 0.
Abelian Group
A Borwein-Like Theorem for Overpartitions
Background
Borwein-Like Theorem
Proof
Final Thoughts
Additional Conjectures
Theorem (Overpartition Nonresidue, equivalent form)
Let n be a quadratic nonresidue modulo a prime p. Consider the
overpartitions of n into parts not divisible by p. Then the number
of such overpartitions into an even number of parts equals the
number of such overpartitions into an odd number of parts.
Abelian Group
A Borwein-Like Theorem for Overpartitions
Background
Borwein-Like Theorem
Proof
Final Thoughts
Proof (Andrews): An Identity
Sketch of Proof.
We have the identity
(q; q)∞
(q; q)∞ (q; q)∞
=
= (q; q 2 )∞ (q; q)∞
(−q; q)∞
(−q; q)∞ (q; q)∞
X
2
= (q; q 2 )2∞ (q 2 ; q 2 )∞ =
(−1)n q n
n∈Z
by the Jacobi triple product.
Abelian Group
A Borwein-Like Theorem for Overpartitions
Background
Borwein-Like Theorem
Proof
Final Thoughts
Proof (Andrews): Expression for bp (n)
Sketch of Proof, cont.
We may rewrite Fp (q) as
∞
X
n=0
(q; q)∞ (−q p ; q p )∞
(−q; q)∞ (q p ; q p )∞
! ∞
!
X
X
2
=
(−1)n q n
g (n)q pn
bp (n)q n =
n=0
n∈Z

∞ 
X

=


n=0

X
k
(−1) g
√
√
−b nc≤k≤b nc
2
p|n−k
Abelian Group
n − k2
p

 n
q .

A Borwein-Like Theorem for Overpartitions
Background
Borwein-Like Theorem
Proof
Final Thoughts
Proof (Andrews): Conclusion
Sketch of Proof, cont.
Thus
bp (n) =
X
√
k
(−1) g
√
−b nc≤k≤b nc
p|n−k 2
n − k2
p
.
When n is a quadratic nonresidue modulo p, there will be no k so
that p | n − k 2 , so the sum will be empty and bp (n) = 0.
Abelian Group
A Borwein-Like Theorem for Overpartitions
Background
Borwein-Like Theorem
Proof
Final Thoughts
References
I
The Theory of Partitions, G. E. Andrews, 1976.
I
G. E. Andrews private correspondence.
I
“Overpartitions”, S. Corteel and J. Lovejoy, 2004.
I
“Partition Theorems in Need of Bijections”, K. Garrett, 2003.
Abelian Group
A Borwein-Like Theorem for Overpartitions