Geometric and Enumerative Combinatorics, IMA University 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 Linz Research Institute for Symbolic Computation (RISC)
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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
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:
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