combinatorics, modular forms, and discrete geometry...combinatorics, modular forms, and discrete...

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


“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.”


How Zeilberger tells the story of partition analysis (and more):


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




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




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


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)


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.




.... . .

...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










.... . .

...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







Combinatorics, Modular Forms, and Discrete Geometry / Omega and Mathematical Discovery11

Omega and Mathematical Discovery


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


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


∞∑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


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.


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?


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.


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.


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]


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]


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]


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)


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)


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.


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


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).


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)


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


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.


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].


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.


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 .


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]


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).


Reference

I Cristian-Silviu Radu: An Algorithmic Approach toRamanujan-Kolberg Identities. Journal of SymbolicComputation, 2014.

combinatorics, modular forms, and discrete geometry...combinatorics, modular forms, and discrete...

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