combinatorics, modular forms, and discrete geometry...combinatorics, modular forms, and discrete...
TRANSCRIPT
Combinatorics, Modular Forms, and Discrete Geometry / 1
Geometric and Enumerative Combinatorics, IMAUniversity of Minnesota, Nov 10–14, 2014
Combinatorics, Modular Forms,
and Discrete Geometry
Peter Paule(joint work with: G.E. Andrews, S. Radu)
Johannes Kepler University LinzResearch Institute for Symbolic Computation (RISC)
Combinatorics, Modular Forms, and Discrete Geometry / Partition Analysis 2
Partition Analysis
Combinatorics, Modular Forms, and Discrete Geometry / Partition Analysis 3
“The no. of partitions of N of the formN = b1 + · · ·+ bn satisfying
bnn≥ bn−1n− 1
≥ · · · ≥ b22≥ b1
1≥ 0
equals the no. of partitions of N into odd parts each ≤ 2n− 1.
This problem cried out for MacMahon’s Partition Analysis, . . .
Given that Partition Analysis is an algorithm for producingpartition generating functions, I was able to convince Peter Pauleand Axel Riese to join an effort to automate this algorithm.”
Combinatorics, Modular Forms, and Discrete Geometry / Partition Analysis 3
“The no. of partitions of N of the formN = b1 + · · ·+ bn satisfying
bnn≥ bn−1n− 1
≥ · · · ≥ b22≥ b1
1≥ 0
equals the no. of partitions of N into odd parts each ≤ 2n− 1.
This problem cried out for MacMahon’s Partition Analysis, . . .
Given that Partition Analysis is an algorithm for producingpartition generating functions, I was able to convince Peter Pauleand Axel Riese to join an effort to automate this algorithm.”
Combinatorics, Modular Forms, and Discrete Geometry / Partition Analysis 7
How Zeilberger tells the story of partition analysis (and more):
Combinatorics, Modular Forms, and Discrete Geometry / Partition Analysis 8
Example (PA and the Omega package)
Find a suitable closed form of
L(x1, x2, x3) :=∑
b1,b2,b3∈N s.t. 2b3 − 3b2 ≥ 0, b2 − 2b1 ≥ 0
xb11 xb22 x
b33
= Ω=
∑b1,b2,b3≥0 λ
2b3−3b21 λb2−2b12 xb11 x
b22 x
b33
Combinatorics, Modular Forms, and Discrete Geometry / Partition Analysis 8
Example (PA and the Omega package)
Find a suitable closed form of
L(x1, x2, x3) :=∑
b1,b2,b3∈N s.t. 2b3 − 3b2 ≥ 0, b2 − 2b1 ≥ 0
xb11 xb22 x
b33
= Ω=
∑b1,b2,b3≥0 λ
2b3−3b21 λb2−2b12 xb11 x
b22 x
b33
Combinatorics, Modular Forms, and Discrete Geometry / Partition Analysis 8
Example (PA and the Omega package)
Find a suitable closed form of
L(x1, x2, x3) :=∑
b1,b2,b3∈N s.t. 2b3 − 3b2 ≥ 0, b2 − 2b1 ≥ 0
xb11 xb22 x
b33
= Ω=
∑b1,b2,b3≥0 λ
2b3−3b21 λb2−2b12 xb11 x
b22 x
b33
= Ω=
1
1− x1λ22
1
1− λ2x2λ31
1
1− λ21x3
Combinatorics, Modular Forms, and Discrete Geometry / Partition Analysis 8
Example (PA and the Omega package)
Find a suitable closed form of
L(x1, x2, x3) :=∑
b1,b2,b3∈N s.t. 2b3 − 3b2 ≥ 0, b2 − 2b1 ≥ 0
xb11 xb22 x
b33
= Ω=
∑b1,b2,b3≥0 λ
2b3−3b21 λb2−2b12 xb11 x
b22 x
b33
In[1]:= << Omega2.m
Omega Package by Axel Riese (in cooperation with George E. Andrewsand Peter Paule) - c©RISC, JKU Linz - V 2.47
Combinatorics, Modular Forms, and Discrete Geometry / Partition Analysis 9
In[2]:= LCrude = OSum[ x1b1 x2b2 x3b3,
2 b3 - 3 b2 ≥ 0, b2 - 2 b1 ≥ 0 , b1 ≥ 0, λ]
Out[2]= Ω≥
λ1, λ2
1(1− x1
λ22
)(1−λ2 x2
λ31
)(1−λ21 x3)
In[3]:= L=OR[LCrude]
Out[3]= 1+x2 x32
(1−x3)(1−x22 x33)(1−x1 x22 x33)
In[4]:= L /. x1->q, x2->q, x3->q
Out[4]= 1+q3
(1−q)(1−q5)(1−q6)
Combinatorics, Modular Forms, and Discrete Geometry / Partition Analysis 9
In[2]:= LCrude = OSum[ x1b1 x2b2 x3b3,
2 b3 - 3 b2 ≥ 0, b2 - 2 b1 ≥ 0 , b1 ≥ 0, λ]
Out[2]= Ω≥
λ1, λ2
1(1− x1
λ22
)(1−λ2 x2
λ31
)(1−λ21 x3)
In[3]:= L=OR[LCrude]
Out[3]= 1+x2 x32
(1−x3)(1−x22 x33)(1−x1 x22 x33)
In[4]:= L /. x1->q, x2->q, x3->q
Out[4]= 1+q3
(1−q)(1−q5)(1−q6)
Combinatorics, Modular Forms, and Discrete Geometry / Partition Analysis 9
In[2]:= LCrude = OSum[ x1b1 x2b2 x3b3,
2 b3 - 3 b2 ≥ 0, b2 - 2 b1 ≥ 0 , b1 ≥ 0, λ]
Out[2]= Ω≥
λ1, λ2
1(1− x1
λ22
)(1−λ2 x2
λ31
)(1−λ21 x3)
In[3]:= L=OR[LCrude]
Out[3]= 1+x2 x32
(1−x3)(1−x22 x33)(1−x1 x22 x33)
In[4]:= L /. x1->q, x2->q, x3->q
Out[4]= 1+q3
(1−q)(1−q5)(1−q6)
Combinatorics, Modular Forms, and Discrete Geometry / Partition Analysis 9
In[2]:= LCrude = OSum[ x1b1 x2b2 x3b3,
2 b3 - 3 b2 ≥ 0, b2 - 2 b1 ≥ 0 , b1 ≥ 0, λ]
Out[2]= Ω≥
λ1, λ2
1(1− x1
λ22
)(1−λ2 x2
λ31
)(1−λ21 x3)
In[3]:= L=OR[LCrude]
Out[3]= 1+x2 x32
(1−x3)(1−x22 x33)(1−x1 x22 x33)
In[4]:= L /. x1->q, x2->q, x3->q
Out[4]= 1+q3
(1−q)(1−q5)(1−q6)
Combinatorics, Modular Forms, and Discrete Geometry / Partition Analysis 10
GENERAL THEME: linear Diophantine constraints
I Find b1, . . . bn ∈ N such thatc1,1 · · · c1,nc2,1 · · · c2,n
.... . .
...cm,1 · · · cm,n
b1...bn
≥c1c2...cm
I New algorithm by F. Breuer & Z. Zafeirakopolous [poster:“A Linear Diophantine System Solver”, Lind Hall 400, 4 pm]Polyhedral Omega is a new algorithm for solving linear Diophantine systems (LDS), i.e., for computing a
multivariate rational function representation of the set of all non-negative integer solutions to a system of
linear equations and inequalities. Polyhedral Omega combines methods from partition analysis with
methods from polyhedral geometry. In particular, we combine MacMahon’s iterative approach based on the
Omega operator and explicit formulas for its evaluation with geometric tools such as Brion decompositions
and Barvinok’s short rational function representations. In this way, we connect two recent branches of
research that have so far remained separate, unified by the concept of symbolic cones which we introduce.
Combinatorics, Modular Forms, and Discrete Geometry / Partition Analysis 10
GENERAL THEME: linear Diophantine constraints
I Find b1, . . . bn ∈ N such thatc1,1 · · · c1,nc2,1 · · · c2,n
.... . .
...cm,1 · · · cm,n
b1...bn
=
c1c2...cm
I New algorithm by F. Breuer & Z. Zafeirakopolous [poster:“A Linear Diophantine System Solver”, Lind Hall 400, 4 pm]Polyhedral Omega is a new algorithm for solving linear Diophantine systems (LDS), i.e., for computing a
multivariate rational function representation of the set of all non-negative integer solutions to a system of
linear equations and inequalities. Polyhedral Omega combines methods from partition analysis with
methods from polyhedral geometry. In particular, we combine MacMahon’s iterative approach based on the
Omega operator and explicit formulas for its evaluation with geometric tools such as Brion decompositions
and Barvinok’s short rational function representations. In this way, we connect two recent branches of
research that have so far remained separate, unified by the concept of symbolic cones which we introduce.
Combinatorics, Modular Forms, and Discrete Geometry / Partition Analysis 10
GENERAL THEME: linear Diophantine constraints
I Find b1, . . . bn ∈ N such thatc1,1 · · · c1,nc2,1 · · · c2,n
.... . .
...cm,1 · · · cm,n
b1...bn
=
c1c2...cm
I New algorithm by F. Breuer & Z. Zafeirakopolous [poster:
“A Linear Diophantine System Solver”, Lind Hall 400, 4 pm]Polyhedral Omega is a new algorithm for solving linear Diophantine systems (LDS), i.e., for computing a
multivariate rational function representation of the set of all non-negative integer solutions to a system of
linear equations and inequalities. Polyhedral Omega combines methods from partition analysis with
methods from polyhedral geometry. In particular, we combine MacMahon’s iterative approach based on the
Omega operator and explicit formulas for its evaluation with geometric tools such as Brion decompositions
and Barvinok’s short rational function representations. In this way, we connect two recent branches of
research that have so far remained separate, unified by the concept of symbolic cones which we introduce.
Combinatorics, Modular Forms, and Discrete Geometry / Omega and Mathematical Discovery11
Omega and Mathematical Discovery
Combinatorics, Modular Forms, and Discrete Geometry / Omega and Mathematical Discovery12
ca1
AAA
U ca2
-
ca3AAAAAA
-
U
ca4
-
ca5AAAAAA
-
U
ca6
. . . . . . . . . .
ca7 . . . . . . . . . . ca2k−1
AAAAAA
-
U
ca2k−2
-
ca2k+1
AAAU
ca2k
c a2k+2
A k-elongated partition diamond of length 1
ca1
AAA
U ca2
. . . . . . . .ca3
. . . . . . . . ca2k
AAA
ca2k+1
U ca2k+2 cAAA
U ca2k+3
. . . . . . . .ca2k+4
. . . . . . . . ca4k+1
AAA
ca4k+2
U ca4k+3 . . . . . . cAAA
U c
. . . . . . . .c
. . . . . . . . ca(2k+1)n−1
AAA
ca(2k+1)n
U ca(2k+1)n+1
A k-elongated partition diamond of length n
1
Combinatorics, Modular Forms, and Discrete Geometry / Omega and Mathematical Discovery13
Generating function for k-elongated diamonds of length n:
hn,k(q) =
∏n−1j=0 (1 + q(2k+1)j+2)(1 + q(2k+1)j+4) · · · (1 + q(2k+1)j+2k)∏(2k+1)n+1
j=1 (1− qj)
Andrews’ great idea: delete the source:
h∗n,k(q) =
∏n−1j=0 (1 + q(2k+1)j+1)(1 + q(2k+1)j+3) · · · (1 + q(2k+1)j+2k−1)∏(2k+1)n
j=1 (1− qj)and glue the diamonds together:
cb(2k+1)n+1
AAAc
b(2k+1)n−1
. . . . . . . .cb(2k+1)n
. . . . . . . . cAAA
c
c
K
K
. . . cAAA
c. . . . . . . .
c. . . . . . . .K
cb7AAAAAA
cb6
K
cb5
Kc
b4
AAAAAA
cb3
cb2
b2k+2
b2k+1
b2k
ca1
AAA
U ca2
. . . . . . . .ca3
. . . . . . . . ca2k
AAA
ca2k+1
U ca2k+2 . . . cAAA
U c
. . . . . . . .c
. . . . . . . . ca(2k+1)n−1
AAA
ca(2k+1)n
U ca(2k+1)n+1
A broken k-diamond of length 2n
1
Combinatorics, Modular Forms, and Discrete Geometry / Omega and Mathematical Discovery13
Generating function for k-elongated diamonds of length n:
hn,k(q) =
∏n−1j=0 (1 + q(2k+1)j+2)(1 + q(2k+1)j+4) · · · (1 + q(2k+1)j+2k)∏(2k+1)n+1
j=1 (1− qj)
Andrews’ great idea: delete the source:
h∗n,k(q) =
∏n−1j=0 (1 + q(2k+1)j+1)(1 + q(2k+1)j+3) · · · (1 + q(2k+1)j+2k−1)∏(2k+1)n
j=1 (1− qj)and glue the diamonds together:
cb(2k+1)n+1
AAAc
b(2k+1)n−1
. . . . . . . .cb(2k+1)n
. . . . . . . . cAAA
c
c
K
K
. . . cAAA
c. . . . . . . .
c. . . . . . . .K
cb7AAAAAA
cb6
K
cb5
Kc
b4
AAAAAA
cb3
cb2
b2k+2
b2k+1
b2k
ca1
AAA
U ca2
. . . . . . . .ca3
. . . . . . . . ca2k
AAA
ca2k+1
U ca2k+2 . . . cAAA
U c
. . . . . . . .c
. . . . . . . . ca(2k+1)n−1
AAA
ca(2k+1)n
U ca(2k+1)n+1
A broken k-diamond of length 2n
1
Combinatorics, Modular Forms, and Discrete Geometry / Omega and Mathematical Discovery13
Generating function for k-elongated diamonds of length n:
hn,k(q) =
∏n−1j=0 (1 + q(2k+1)j+2)(1 + q(2k+1)j+4) · · · (1 + q(2k+1)j+2k)∏(2k+1)n+1
j=1 (1− qj)
Andrews’ great idea: delete the source:
h∗n,k(q) =
∏n−1j=0 (1 + q(2k+1)j+1)(1 + q(2k+1)j+3) · · · (1 + q(2k+1)j+2k−1)∏(2k+1)n
j=1 (1− qj)and glue the diamonds together:
cb(2k+1)n+1
AAAc
b(2k+1)n−1
. . . . . . . .cb(2k+1)n
. . . . . . . . cAAA
c
c
K
K
. . . cAAA
c. . . . . . . .
c. . . . . . . .K
cb7AAAAAA
cb6
K
cb5
Kc
b4
AAAAAA
cb3
cb2
b2k+2
b2k+1
b2k
ca1
AAA
U ca2
. . . . . . . .ca3
. . . . . . . . ca2k
AAA
ca2k+1
U ca2k+2 . . . cAAA
U c
. . . . . . . .c
. . . . . . . . ca(2k+1)n−1
AAA
ca(2k+1)n
U ca(2k+1)n+1
A broken k-diamond of length 2n
1
Combinatorics, Modular Forms, and Discrete Geometry / Omega and Mathematical Discovery14
∞∑m=0
∆k(m)qm:= limn→∞
hn,k(q)h∗n,k(q)
=
∏∞j=1(1 + qj)∏∞
j=1(1− qj)2∏∞j=1(1 + q(2k+1)j)
=
∏∞j=1(1 + qj)(1− qj)∏∞
j=1(1− qj)3∏∞j=1(1 + q(2k+1)j)
=∞∏j=1
(1− q2j)(1− q(2k+1)j)
(1− qj)3(1− q(4k+2)j)
cb(2k+1)n+1
AAAc
b(2k+1)n−1
. . . . . . . .cb(2k+1)n
. . . . . . . . cAAA
c
c
K
K
. . . cAAA
c. . . . . . . .
c. . . . . . . .K
cb7AAAAAA
cb6
K
cb5
Kc
b4
AAAAAA
cb3
cb2
b2k+2
b2k+1
b2k
ca1
AAA
U ca2
. . . . . . . .ca3
. . . . . . . . ca2k
AAA
ca2k+1
U ca2k+2 . . . cAAA
U c
. . . . . . . .c
. . . . . . . . ca(2k+1)n−1
AAA
ca(2k+1)n
U ca(2k+1)n+1
A broken k-diamond of length 2n
1
Combinatorics, Modular Forms, and Discrete Geometry / Omega and Mathematical Discovery15
Consequently,
∞∑m=0
∆k(m)qm = limn→∞
hn,k(q)h∗n,k(q)
=
∞∏j=1
(1− q2j)(1− q(2k+1)j)
(1− qj)3(1− q(4k+2)j)
= q(k+1)/12 η(2τ)η((2k + 1)τ)
η(τ)3η((4k + 2)τ)
with η the Dedekind eta function:
η(τ) := q124
∞∏n=1
(1− qn) (q = e2πiτ )
NOTE. η24 is a modular form of weight 12 for SL2(Z), because of
η
(aτ + b
cτ + d
)= ε(a, b, c, d)
√−i(cτ + d)η(τ)
where a d− b c = 1 and c > 0.
Combinatorics, Modular Forms, and Discrete Geometry / Omega and Mathematical Discovery15
Consequently,
∞∑m=0
∆k(m)qm = limn→∞
hn,k(q)h∗n,k(q)
=
∞∏j=1
(1− q2j)(1− q(2k+1)j)
(1− qj)3(1− q(4k+2)j)
= q(k+1)/12 η(2τ)η((2k + 1)τ)
η(τ)3η((4k + 2)τ)
with η the Dedekind eta function:
η(τ) := q124
∞∏n=1
(1− qn) (q = e2πiτ )
NOTE. η24 is a modular form of weight 12 for SL2(Z), because of
η
(aτ + b
cτ + d
)= ε(a, b, c, d)
√−i(cτ + d)η(τ)
where a d− b c = 1 and c > 0.
Combinatorics, Modular Forms, and Discrete Geometry / Omega and Mathematical Discovery16
Recall
η
(aτ + b
cτ + d
)= ε(a, b, c, d)
√−i(cτ + d)η(τ).
Hence, for τ ∈ H (upper half complex plane):
η(τ + 1)24 = η
(1 τ + 1
0 τ + 1
)24
= ε(1, 1, 0, 1)24√−i(0 τ + 1)
24η(τ)24
= η(τ)24
The Fourier series expansion (“q-series expansion”, q = e2πiτ ) is
η(τ) = q
∞∏n=1
(1− qn)24.
WHY η-QUOTIENTS?
Combinatorics, Modular Forms, and Discrete Geometry / Omega and Mathematical Discovery17
Combinatorics, Modular Forms, and Discrete Geometry / Omega and Mathematical Discovery18
Congruences for ∆k(n)
Theorem [Andrews & P, Partition Analysis XI]. For all n ∈ N,
∆1(2n+ 1) ≡ 0 (mod 3).
Proof.
Because of (1− qj)3 ≡ 1− q3j (mod 3),
∞∑m=0
∆1(m)qm =∞∏j=1
(1− q2j)(1− q3j)(1− qj)3(1− q6j)
≡∞∏j=1
(1− q2j)(1− q3j)(1− q3j)(1− q6j)
(mod 3).
Hence the coefficients of odd powers of q have to be zero.
Combinatorics, Modular Forms, and Discrete Geometry / Omega and Mathematical Discovery18
Congruences for ∆k(n)
Theorem [Andrews & P, Partition Analysis XI]. For all n ∈ N,
∆1(2n+ 1) ≡ 0 (mod 3).
Proof. Because of (1− qj)3 ≡ 1− q3j (mod 3),
∞∑m=0
∆1(m)qm =
∞∏j=1
(1− q2j)(1− q3j)(1− qj)3(1− q6j)
≡∞∏j=1
(1− q2j)(1− q3j)(1− q3j)(1− q6j)
(mod 3).
Hence the coefficients of odd powers of q have to be zero.
Combinatorics, Modular Forms, and Discrete Geometry / Omega and Mathematical Discovery19
Recall:
Theorem. For all n ∈ N,
∆1(2n+ 1) ≡ 0 (mod 3).
Algorithmic Proof [Radu 2014]:
∞∑n=0
∆1(2n+ 1)qn = 3∞∏j=1
(1− q2j)2(1− q6j)2
(1− qj)6
NOTE. Human proof [Hirschhorn & Sellers, 2007]
Combinatorics, Modular Forms, and Discrete Geometry / Omega and Mathematical Discovery19
Recall:
Theorem. For all n ∈ N,
∆1(2n+ 1) ≡ 0 (mod 3).
Algorithmic Proof [Radu 2014]:
∞∑n=0
∆1(2n+ 1)qn = 3
∞∏j=1
(1− q2j)2(1− q6j)2
(1− qj)6
NOTE. Human proof [Hirschhorn & Sellers, 2007]
Combinatorics, Modular Forms, and Discrete Geometry / Omega and Mathematical Discovery19
Recall:
Theorem. For all n ∈ N,
∆1(2n+ 1) ≡ 0 (mod 3).
Algorithmic Proof [Radu 2014]:
∞∑n=0
∆1(2n+ 1)qn = 3
∞∏j=1
(1− q2j)2(1− q6j)2
(1− qj)6
NOTE. Human proof [Hirschhorn & Sellers, 2007]
Combinatorics, Modular Forms, and Discrete Geometry / Omega and Mathematical Discovery20
Some conjectures [Andrews & P, PA XI]: For all n ∈ N,
∆2(10n+ 2) ≡ 0 (mod 2)
and∆2(25n+ 14) ≡ 0 (mod 5).
S.H. Chan [2008] proved this and also
∆2(10n+ 6) ≡ 0 (mod 2)
and∆2(25n+ 24) ≡ 0 (mod 5).
NOTE. First proof of the 10-case [Hirschhorn & Sellers, 2007]
Combinatorics, Modular Forms, and Discrete Geometry / Omega and Mathematical Discovery20
Some conjectures [Andrews & P, PA XI]: For all n ∈ N,
∆2(10n+ 2) ≡ 0 (mod 2)
and∆2(25n+ 14) ≡ 0 (mod 5).
S.H. Chan [2008] proved this and also
∆2(10n+ 6) ≡ 0 (mod 2)
and∆2(25n+ 24) ≡ 0 (mod 5).
NOTE. First proof of the 10-case [Hirschhorn & Sellers, 2007]
Combinatorics, Modular Forms, and Discrete Geometry / Omega and Mathematical Discovery20
Some conjectures [Andrews & P, PA XI]: For all n ∈ N,
∆2(10n+ 2) ≡ 0 (mod 2)
and∆2(25n+ 14) ≡ 0 (mod 5).
S.H. Chan [2008] proved this and also
∆2(10n+ 6) ≡ 0 (mod 2)
and∆2(25n+ 24) ≡ 0 (mod 5).
NOTE. First proof of the 10-case [Hirschhorn & Sellers, 2007]
Combinatorics, Modular Forms, and Discrete Geometry / Omega and Mathematical Discovery21
Recall [S.H. Chan, 2008]:
∆2(25n+ 14) ≡ ∆2(25n+ 24) ≡ 0 (mod 5).
Algorithmic Proof [Radu 2014].
Human preprocessing:since (1− qj)5 ≡ 1− q5j (mod 5),
∆2(n) ≡ d(n) (mod 5),
where∞∑m=0
∆2(n)qn =∞∏j=1
(1− q2j)(1− q5j)(1− qj)3(1− q10j)
and∞∑m=0
d(n)qn:=∞∏j=1
(1− q2j)(1− qj)2
(1− q10j)
Combinatorics, Modular Forms, and Discrete Geometry / Omega and Mathematical Discovery21
Recall [S.H. Chan, 2008]:
∆2(25n+ 14) ≡ ∆2(25n+ 24) ≡ 0 (mod 5).
Algorithmic Proof [Radu 2014]. Human preprocessing:since (1− qj)5 ≡ 1− q5j (mod 5),
∆2(n) ≡ d(n) (mod 5),
where∞∑m=0
∆2(n)qn =
∞∏j=1
(1− q2j)(1− q5j)(1− qj)3(1− q10j)
and∞∑m=0
d(n)qn:=
∞∏j=1
(1− q2j)(1− qj)2
(1− q10j)
Combinatorics, Modular Forms, and Discrete Geometry / Omega and Mathematical Discovery22
Radu’s program “Ramanujan-Kolberg” delivers:
q32η(2τ)η(5τ)10
η(τ)6η(10τ)20
( ∞∑m=0
d(25n+ 14)qn
)( ∞∑m=0
d(25n+ 24)qn
)= 25(2t4 + 28t3 + 155t2 + 400t+ 400)
where
t =η(τ)3η(5τ)
η(2τ)η(10τ)3∈M(Γ0(10)).
NOTE 1.
The program computes a similar identity also for ∆2(n)instead of d(n), but the output is much bigger.
NOTE 2. There are numerous other congruences for brokendiamonds and generalizations.
Combinatorics, Modular Forms, and Discrete Geometry / Omega and Mathematical Discovery22
Radu’s program “Ramanujan-Kolberg” delivers:
q32η(2τ)η(5τ)10
η(τ)6η(10τ)20
( ∞∑m=0
d(25n+ 14)qn
)( ∞∑m=0
d(25n+ 24)qn
)= 25(2t4 + 28t3 + 155t2 + 400t+ 400)
where
t =η(τ)3η(5τ)
η(2τ)η(10τ)3∈M(Γ0(10)).
NOTE 1.The program computes a similar identity also for ∆2(n)instead of d(n), but the output is much bigger.
NOTE 2. There are numerous other congruences for brokendiamonds and generalizations.
Combinatorics, Modular Forms, and Discrete Geometry / Omega and Mathematical Discovery22
Radu’s program “Ramanujan-Kolberg” delivers:
q32η(2τ)η(5τ)10
η(τ)6η(10τ)20
( ∞∑m=0
d(25n+ 14)qn
)( ∞∑m=0
d(25n+ 24)qn
)= 25(2t4 + 28t3 + 155t2 + 400t+ 400)
where
t =η(τ)3η(5τ)
η(2τ)η(10τ)3∈M(Γ0(10)).
NOTE 1.The program computes a similar identity also for ∆2(n)instead of d(n), but the output is much bigger.
NOTE 2. There are numerous other congruences for brokendiamonds and generalizations.
Combinatorics, Modular Forms, and Discrete Geometry / Radu’s Ramanujan-Kolberg Package24
Radu’s Ramanujan-Kolberg Package
Combinatorics, Modular Forms, and Discrete Geometry / Radu’s Ramanujan-Kolberg Package25
Back to Euler (= limit of Lecture Hall)
Define∞∑n=0
Q(n)qn:=
∞∏j=1
1
1− q2j−1:
Radu’s “Ramanujan-Kolberg” package delivers (computing overE∞(14)):
∞∑n=0
Q(7n+ 3)qn ·∞∑n=0
Q(7n+ 4)qn ·∞∑n=0
Q(7n+ 6)qn
= 8 q5∞∏j=1
(1− q2j)5(1− q14j)16
(1− qj)13(1− q7j)8(−16E1 + 9E2
1 + 2E1E4)
NOTE. This implies:
Q(7n+ 3) ≡ Q(7n+ 4) ≡ Q(7n+ 6) ≡ 0 (mod 2).
Combinatorics, Modular Forms, and Discrete Geometry / Radu’s Ramanujan-Kolberg Package25
Back to Euler (= limit of Lecture Hall)
Define∞∑n=0
Q(n)qn:=
∞∏j=1
1
1− q2j−1:
Radu’s “Ramanujan-Kolberg” package delivers (computing overE∞(14)):
∞∑n=0
Q(7n+ 3)qn ·∞∑n=0
Q(7n+ 4)qn ·∞∑n=0
Q(7n+ 6)qn
= 8 q5∞∏j=1
(1− q2j)5(1− q14j)16
(1− qj)13(1− q7j)8(−16E1 + 9E2
1 + 2E1E4)
NOTE. This implies:
Q(7n+ 3) ≡ Q(7n+ 4) ≡ Q(7n+ 6) ≡ 0 (mod 2).
Combinatorics, Modular Forms, and Discrete Geometry / Radu’s Ramanujan-Kolberg Package26
Radu’s “Ramanujan-Kolberg” package also delivers:
∞∑n=0
Q(7n)qn ·∞∑n=0
Q(7n+ 1)qn ·∞∑n=0
Q(7n+ 5)qn
= q6∞∏j=1
(1− q2j)5(1− q14j)16
(1− qj)13(1− q7j)8(3E3
1 + 24E21 + 64E1)
and
∞∑n=0
Q(7n+ 2)qn
= q3∞∏j=1
(1− q14j)8
(1− qj)3(1− q2j)(1− q7j)4(8E1 + E4 − 8)
Combinatorics, Modular Forms, and Discrete Geometry / Radu’s Ramanujan-Kolberg Package27
I STEP 1. Find generators of the multiplicativemonoid E∞(14):
Solving a problem for nonnegative integers with linear Diophantineconstraints, we obtain the generators
E1 =
(η(2τ)
η(τ)
)1(η(7τ)
η(τ)
)7(η(14τ)
η(τ)
)−7∈ E∞(14),
E2 =
(η(2τ)
η(τ)
)8(η(7τ)
η(τ)
)4(η(14τ)
η(τ)
)−8∈ E∞(14),
E3 =
(η(2τ)
η(τ)
)−5(η(7τ)
η(τ)
)5(η(14τ)
η(τ)
)−13∈ E∞(14),
and
Combinatorics, Modular Forms, and Discrete Geometry / Radu’s Ramanujan-Kolberg Package28
E4 =
(η(2τ)
η(τ)
)1(η(7τ)
η(τ)
)3(η(14τ)
η(τ)
)−7∈ E∞(14),
E5 =
(η(2τ)
η(τ)
)5(η(7τ)
η(τ)
)7(η(14τ)
η(τ)
)−11∈ E∞(14),
and
E6 =
(η(2τ)
η(τ)
)−2(η(7τ)
η(τ)
)6(η(14τ)
η(τ)
)−10∈ E∞(14).
Summary: STEP 1 computes generators E1, . . . , E6 of themultiplicative monoid E∞(14) consisting of eta quotients whichare modular functions with poles only at infinity.
Combinatorics, Modular Forms, and Discrete Geometry / Radu’s Ramanujan-Kolberg Package29
A crucial FINITE representation
I GOAL: We want to represent our object as an element in theinfinite dimensional vectorspace
〈E∞(14)〉Q = c1 e1 + · · ·+ ck ek : ci ∈ Q, ej ∈ E∞(14)= Q[E1, . . . , E6].
I STEP 2. Represent E∞(14) as a Q[E1]-module which isfreely generated by 1 and E4; i.e.,
〈E∞(14)〉Q = 〈1, E4〉Q[E1].
Combinatorics, Modular Forms, and Discrete Geometry / Radu’s Ramanujan-Kolberg Package30
NOTE 1. Ramanujan [1919] proved for
∞∑n=0
p(n)qn:=
∞∏j=1
1
1− qj:
∞∑n=0
p(5n+ 4)qn = 5
∞∏j=1
(1− q5j)5
(1− qj)6
and
∞∑n=0
p(7n+ 5)qn
= 7
∞∏j=1
(1− q7j)3
(1− qj)4+ 49q
∞∏j=1
(1− q7j)7
(1− qj)8.
Combinatorics, Modular Forms, and Discrete Geometry / Radu’s Ramanujan-Kolberg Package31
NOTE 2. An alternative formulation in terms of
z5:=q
∞∏j=1
(1− q5j)6
(1− qj)6=
(η(5τ)
η(τ)
)6
and
z7:=q
∞∏j=1
(1− q7j)4
(1− qj)4=
(η(7τ)
η(τ)
)4
:
q
∞∏j=1
(1− q5j)∞∑n=0
p(5n+ 4)qn = 5 z5
and
q
∞∏j=1
(1− q7j)∞∑n=0
p(7n+ 5)qn = 7 z7 + 49 q z27 .
Combinatorics, Modular Forms, and Discrete Geometry / Radu’s Ramanujan-Kolberg Package32
NOTE 3.
∞∑n=0
p(5n+ 4)qn = 5
∞∏j=1
(1− q5j)5
(1− qj)6
“It would be difficult to find more beautiful formulaethan the ‘Rogers-Ramanujan’ identities . . . ; but hereRamanujan must take second place to Prof. Rogers;and, if I had to select one formula from allRamanujan’s work, I would agree with MajorMacMahon in selecting . . . ” [G.H. Hardy]
Combinatorics, Modular Forms, and Discrete Geometry / Radu’s Ramanujan-Kolberg Package33
NOTE. The “Ramanujan-Kolberg” package computes in E∞(22):∞∑n=0
p(11n+ 6)qn = q14∞∏j=1
(1− q22j)22
(1− qj)10(1− q2j)2(1− q11j)11
×(1078t4 + 13893t3 + 31647t2 + 11209t− 21967
+z1(187t3 + 5390t2 + 594t− 9581)
+z2(11t3 + 2761t2 + 5368t− 6754)
with
t:=3
88w1 +
1
11w2 −
1
8w3, z1:=−
5
88w1 +
2
11w2 −
1
8w3 − 3,
z2:=1
44w1 −
3
11w2 +
5
4w3,
where
w1:=[−3, 3,−7], w2:=[8, 4,−8], w3:=[1, 11,−11] ∈ E∞(22)
and
[r2, r11, r22]:=∏δ|22
(η(δτ
η(τ)
)rδ∈ E∞(22).
Combinatorics, Modular Forms, and Discrete Geometry / Radu’s Ramanujan-Kolberg Package33
NOTE. The “Ramanujan-Kolberg” package computes in E∞(22):∞∑n=0
p(11n+ 6)qn = q14∞∏j=1
(1− q22j)22
(1− qj)10(1− q2j)2(1− q11j)11
×(1078t4 + 13893t3 + 31647t2 + 11209t− 21967
+z1(187t3 + 5390t2 + 594t− 9581)
+z2(11t3 + 2761t2 + 5368t− 6754)
with
t:=3
88w1 +
1
11w2 −
1
8w3, z1:=−
5
88w1 +
2
11w2 −
1
8w3 − 3,
z2:=1
44w1 −
3
11w2 +
5
4w3,
where
w1:=[−3, 3,−7], w2:=[8, 4,−8], w3:=[1, 11,−11] ∈ E∞(22)
and
[r2, r11, r22]:=∏δ|22
(η(δτ
η(τ)
)rδ∈ E∞(22).
Combinatorics, Modular Forms, and Discrete Geometry / Radu’s Ramanujan-Kolberg Package33
NOTE. The “Ramanujan-Kolberg” package computes in E∞(22):∞∑n=0
p(11n+ 6)qn = q14∞∏j=1
(1− q22j)22
(1− qj)10(1− q2j)2(1− q11j)11
×(1078t4 + 13893t3 + 31647t2 + 11209t− 21967
+z1(187t3 + 5390t2 + 594t− 9581)
+z2(11t3 + 2761t2 + 5368t− 6754)
with
t:=3
88w1 +
1
11w2 −
1
8w3, z1:=−
5
88w1 +
2
11w2 −
1
8w3 − 3,
z2:=1
44w1 −
3
11w2 +
5
4w3,
where
w1:=[−3, 3,−7], w2:=[8, 4,−8], w3:=[1, 11,−11] ∈ E∞(22)
and
[r2, r11, r22]:=∏δ|22
(η(δτ
η(τ)
)rδ∈ E∞(22).
Combinatorics, Modular Forms, and Discrete Geometry / Radu’s Ramanujan-Kolberg Package33
NOTE. The “Ramanujan-Kolberg” package computes in E∞(22):∞∑n=0
p(11n+ 6)qn = q14∞∏j=1
(1− q22j)22
(1− qj)10(1− q2j)2(1− q11j)11
×(1078t4 + 13893t3 + 31647t2 + 11209t− 21967
+z1(187t3 + 5390t2 + 594t− 9581)
+z2(11t3 + 2761t2 + 5368t− 6754)
with
t:=3
88w1 +
1
11w2 −
1
8w3, z1:=−
5
88w1 +
2
11w2 −
1
8w3 − 3,
z2:=1
44w1 −
3
11w2 +
5
4w3,
where
w1:=[−3, 3,−7], w2:=[8, 4,−8], w3:=[1, 11,−11] ∈ E∞(22)
and
[r2, r11, r22]:=∏δ|22
(η(δτ
η(τ)
)rδ∈ E∞(22).
Combinatorics, Modular Forms, and Discrete Geometry / Radu’s Ramanujan-Kolberg Package34
Reference
I Cristian-Silviu Radu: An Algorithmic Approach toRamanujan-Kolberg Identities. Journal of SymbolicComputation, 2014.