Counting partitions of a fixed genus G abor Hetyei Joint work with - - PowerPoint PPT Presentation

Outline Preliminaries and main result Proof and instances of our main result

Counting partitions of a fixed genus

G´ abor Hetyei Joint work with Robert Cori

Department of Mathematics and Statistics University of North Carolina at Charlotte http://www.math.uncc.edu/~ghetyei/

Monday, March 24, 2018

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result

1

Preliminaries and main result Where it all began Hypermaps Permutations of a fixed genus

2

Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Where it all began Hypermaps Permutations of a fixed genus

In the beginning . . .

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Where it all began Hypermaps Permutations of a fixed genus

In the beginning . . .

In the sixties Tutte wrote a series of papers on counting rooted planar maps, i.e., connected planar graphs with a distinguished edge.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Where it all began Hypermaps Permutations of a fixed genus

In the beginning . . .

In the sixties Tutte wrote a series of papers on counting rooted planar maps, i.e., connected planar graphs with a distinguished edge. His hope was to show that their number is (asymptotically) the same as the number of four-colorable rooted planar maps.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Where it all began Hypermaps Permutations of a fixed genus

In the beginning . . .

In the sixties Tutte wrote a series of papers on counting rooted planar maps, i.e., connected planar graphs with a distinguished edge. His hope was to show that their number is (asymptotically) the same as the number of four-colorable rooted planar maps. He found some explicit formulas and got stuck with some difficult equations.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Where it all began Hypermaps Permutations of a fixed genus

Visual definition of a hypermap

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Where it all began Hypermaps Permutations of a fixed genus

Visual definition of a hypermap

α 2 4 13 11 7 8 1 3 5 6 9 10 12 14 α α α α

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Where it all began Hypermaps Permutations of a fixed genus

Visual definition of a hypermap

α 2 4 13 11 7 8 1 3 5 6 9 10 12 14 α α α α

σ = (1, 2)(3, 4, 5, 6)(7, 8, 9)(10, 11, 12, 13)(14) (counterclockwise)

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Where it all began Hypermaps Permutations of a fixed genus

Visual definition of a hypermap

α 2 4 13 11 7 8 1 3 5 6 9 10 12 14 α α α α

σ = (1, 2)(3, 4, 5, 6)(7, 8, 9)(10, 11, 12, 13)(14) (counterclockwise) α = (1, 4)(2, 5, 7, 10)(3, 14)(6, 8, 9)(11, 12, 13) (clockwise)

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Where it all began Hypermaps Permutations of a fixed genus

A few remarks on hypermaps

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Where it all began Hypermaps Permutations of a fixed genus

A few remarks on hypermaps

We can think of the cycles of σ as “vertices”, and of the cycles of α as “hyperedges”.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Where it all began Hypermaps Permutations of a fixed genus

A few remarks on hypermaps

We can think of the cycles of σ as “vertices”, and of the cycles of α as “hyperedges”. If α is an involution then the hypermap is a map.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Where it all began Hypermaps Permutations of a fixed genus

A few remarks on hypermaps

We can think of the cycles of σ as “vertices”, and of the cycles of α as “hyperedges”. If α is an involution then the hypermap is a map. We can represent every hypermap by a map by replacing each i ∈ {1, 2, . . . , n} with four copies i1, . . . , i4 and by setting ˜ σ = (11, 12, 13, 14) · · · (n1, n2, n3, n4) and ˜ α = · · · (i1, α(i)4) · · · (j2, σ(j)3) · · · .

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Where it all began Hypermaps Permutations of a fixed genus

A few remarks on hypermaps

We can think of the cycles of σ as “vertices”, and of the cycles of α as “hyperedges”. If α is an involution then the hypermap is a map. We can represent every hypermap by a map by replacing each i ∈ {1, 2, . . . , n} with four copies i1, . . . , i4 and by setting ˜ σ = (11, 12, 13, 14) · · · (n1, n2, n3, n4) and ˜ α = · · · (i1, α(i)4) · · · (j2, σ(j)3) · · · . A hypermap is connected if and only if the permutations σ and α generate a transitive permutation group.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Where it all began Hypermaps Permutations of a fixed genus

The genus of a hypermap

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Where it all began Hypermaps Permutations of a fixed genus

The genus of a hypermap

The genus is given by g(σ, α) = 1 + (n − z(α) − z(σ) − z(α−1σ))/2,

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Where it all began Hypermaps Permutations of a fixed genus

The genus of a hypermap

The genus is given by g(σ, α) = 1 + (n − z(α) − z(σ) − z(α−1σ))/2, where z(π) is the number of cycles of π.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Where it all began Hypermaps Permutations of a fixed genus

The genus of a hypermap

The genus is given by g(σ, α) = 1 + (n − z(α) − z(σ) − z(α−1σ))/2, where z(π) is the number of cycles of π. An important special case is the hypermonopole where σ is the cycle ζn = (1, 2, . . . , n).

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Where it all began Hypermaps Permutations of a fixed genus

The genus of a hypermap

The genus is given by g(σ, α) = 1 + (n − z(α) − z(σ) − z(α−1σ))/2, where z(π) is the number of cycles of π. An important special case is the hypermonopole where σ is the cycle ζn = (1, 2, . . . , n). The genus of the permutation α is defined as the genus of the hypermonopole (ζn, α):

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Where it all began Hypermaps Permutations of a fixed genus

The genus of a hypermap

The genus is given by g(σ, α) = 1 + (n − z(α) − z(σ) − z(α−1σ))/2, where z(π) is the number of cycles of π. An important special case is the hypermonopole where σ is the cycle ζn = (1, 2, . . . , n). The genus of the permutation α is defined as the genus of the hypermonopole (ζn, α): g(α) = (n + 1 − z(α) − z(α−1σ))/2.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Where it all began Hypermaps Permutations of a fixed genus

The genus of a hypermap

The genus is given by g(σ, α) = 1 + (n − z(α) − z(σ) − z(α−1σ))/2, where z(π) is the number of cycles of π. An important special case is the hypermonopole where σ is the cycle ζn = (1, 2, . . . , n). The genus of the permutation α is defined as the genus of the hypermonopole (ζn, α): g(α) = (n + 1 − z(α) − z(α−1σ))/2. An element i is a back point of the permutation α if α(i) < i and α(i) is not the smallest element in its cycle.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Where it all began Hypermaps Permutations of a fixed genus

The genus of a hypermap

The genus is given by g(σ, α) = 1 + (n − z(α) − z(σ) − z(α−1σ))/2, where z(π) is the number of cycles of π. An important special case is the hypermonopole where σ is the cycle ζn = (1, 2, . . . , n). The genus of the permutation α is defined as the genus of the hypermonopole (ζn, α): g(α) = (n + 1 − z(α) − z(α−1σ))/2. An element i is a back point of the permutation α if α(i) < i and α(i) is not the smallest element in its cycle. Lemma The sum of the number of back points of the permutation α and the number of those of α−1ζn is equal to 2g(α).

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Where it all began Hypermaps Permutations of a fixed genus

Permutations of genus zero

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Where it all began Hypermaps Permutations of a fixed genus

Permutations of genus zero

A permutation is a partition if in every cycle the elements are listed in increasing order from the smallest to the largest element: (13)(245) is a partition, (132) is not a partition.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Where it all began Hypermaps Permutations of a fixed genus

Permutations of genus zero

A permutation is a partition if in every cycle the elements are listed in increasing order from the smallest to the largest element: (13)(245) is a partition, (132) is not a partition. Theorem (Cori) A permutation has genus zero, if and only if it is a noncrossing partition

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Where it all began Hypermaps Permutations of a fixed genus

Permutations of genus zero

A permutation is a partition if in every cycle the elements are listed in increasing order from the smallest to the largest element: (13)(245) is a partition, (132) is not a partition. Theorem (Cori) A permutation has genus zero, if and only if it is a noncrossing partition As a consequence, the number of genus zero permutations on n elements with k cycles is the Narayana number n

k

n

k−1

• /n, and

the number of all partitions of genus zero is the Catalan number Cn.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Where it all began Hypermaps Permutations of a fixed genus

Permutations and partitions of higher genus

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Where it all began Hypermaps Permutations of a fixed genus

Permutations and partitions of higher genus

Goupil and Schaeffer have computed the number of permutations

• f a fixed genus, on n elements with k cycles. Their computation

follows from character theoretic results.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Where it all began Hypermaps Permutations of a fixed genus

Permutations and partitions of higher genus

Goupil and Schaeffer have computed the number of permutations

• f a fixed genus, on n elements with k cycles. Their computation

follows from character theoretic results. Martha Yip experimentally

• bserved and conjectured that the number of genus one partitions
• n n elements with k parts is the same as the number of genus one

permutations on n − 1 elements with k − 1 cycles. This formula was proved by us.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Where it all began Hypermaps Permutations of a fixed genus

Permutations and partitions of higher genus

Goupil and Schaeffer have computed the number of permutations

• f a fixed genus, on n elements with k cycles. Their computation

follows from character theoretic results. Martha Yip experimentally

• bserved and conjectured that the number of genus one partitions
• n n elements with k parts is the same as the number of genus one

permutations on n − 1 elements with k − 1 cycles. This formula was proved by us. Theorem (Cori-H) The number of genus one partitions on n elements with k parts is

1 6

n

2

n−2

k

n−2

k−2

• .
• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Where it all began Hypermaps Permutations of a fixed genus

Permutations and partitions of higher genus

Goupil and Schaeffer have computed the number of permutations

• f a fixed genus, on n elements with k cycles. Their computation

follows from character theoretic results. Martha Yip experimentally

• bserved and conjectured that the number of genus one partitions
• n n elements with k parts is the same as the number of genus one

permutations on n − 1 elements with k − 1 cycles. This formula was proved by us. Theorem (Cori-H) The number of genus one partitions on n elements with k parts is

1 6

n

2

n−2

k

n−2

k−2

• .

There is not even a conjecture regarding the number of partitions

• f higher genus.
• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Where it all began Hypermaps Permutations of a fixed genus

Our main result

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Where it all began Hypermaps Permutations of a fixed genus

Our main result

Although we have no idea what the number of partitions of a fixed genus is, we can show the following.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Where it all began Hypermaps Permutations of a fixed genus

Our main result

Although we have no idea what the number of partitions of a fixed genus is, we can show the following. Theorem For a fixed g the generating function P(x, y) =

n,k p(n, k)xnyk

• f genus g partitions of n elements with k parts is algebraic. More

precisely, it may be obtained by substituting x, y and

• (x + xy − 1)2 − 4x2y into a rational expression.
• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Where it all began Hypermaps Permutations of a fixed genus

Our main result

Although we have no idea what the number of partitions of a fixed genus is, we can show the following. Theorem For a fixed g the generating function P(x, y) =

n,k p(n, k)xnyk

• f genus g partitions of n elements with k parts is algebraic. More

precisely, it may be obtained by substituting x, y and

• (x + xy − 1)2 − 4x2y into a rational expression.

Note that the generating function of genus zero (noncrossing) partitions is 1 − x − xy −

• (x + xy − 1)2 − 4x2y

2 · x + 1

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

A general genus 2 partition α

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

A general genus 2 partition α

1 2 3 4 5 6 10 11 12 13 15 14 9 8 7 17 16

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

A general genus 2 partition α

1 2 3 4 5 6 10 11 12 13 15 14 9 8 7 17 16

5 is a fixed point of the partition α.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

A general genus 2 partition α

1 2 3 4 5 6 10 11 12 13 15 14 9 8 7 17 16

5 is a fixed point of the partition α. 6 is a dual fixed point, i.e., a fixed point of α−1ζ17.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Elementary reductions and extensions

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Elementary reductions and extensions

Definition Let α be any permutation of {1, . . . , n}. An elementary reduction

• f the first kind is the removal of a fixed point of α: given an i

such that α(i) = i, we remove the cycle (i) from the cycle decomposition of α and replace all j > i by j − 1, thus obtaining the cycle decomposition of a permutation α′ of {1, . . . , n − 1}. We call the inverse of this operation, assigning α to α′, an elementary extension of the first kind.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Elementary reductions and extensions

Definition Let α be any permutation of {1, . . . , n}. An elementary reduction

• f the second kind is the removal of a dual fixed point of α as

follows: given an i < n such that α(i) = i + 1, in the decomposition of α we remove i from the cycle containing it, and replace all j > i by j − 1, thus obtaining the cycle decomposition

• f a permutation α′ of {1, . . . , n − 1}. We call the inverse of this
• peration, assigning α to α′, an elementary extension of the

second kind.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Elementary reductions and extensions

Definition Let α be any permutation of {1, . . . , n}. An elementary reduction

• f the second kind is the removal of a dual fixed point of α as

follows: given an i < n such that α(i) = i + 1, in the decomposition of α we remove i from the cycle containing it, and replace all j > i by j − 1, thus obtaining the cycle decomposition

• f a permutation α′ of {1, . . . , n − 1}. We call the inverse of this
• peration, assigning α to α′, an elementary extension of the

second kind. Elementary reductions and extensions do not change the genus.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Reducing the counting problem to reduced permutations

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Reducing the counting problem to reduced permutations

Theorem Let α be any permutation on {1, . . . , n}. Let us keep performing elementary reductions until we arrive at a reduced permutation. The resulting permutation is unique, regardless of the order in which the elementary reductions were performed.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Reducing the counting problem to reduced permutations

Theorem Consider a class C of permutations that is closed under elementary reductions and extensions. Let p(n, k) and r(n, k) respectively be the number of all, respectively all reduced permutations of {1, . . . , n} in the class having k cycles. Then the generating functions P(x, y) :=

n,k p(n, k)xnyk and

R(x, y) :=

n,k r(n, k)xnyk satisfy the equation

P(x, y) = R D(x, y) − 1 y , y

• ·

1

• (x + xy − 1)2 − 4x2y
• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Reducing the counting problem to reduced permutations

A key idea behind the proof to represent each permutation by a bicolored matching.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Reducing the counting problem to reduced permutations

A key idea behind the proof to represent each permutation by a bicolored matching. For example, the bicolored matching associated to α = (1, 5, 3, 4, 8)(2, 7)(6) is

−3 3 −4 −5 −2 2 4 −1 1 7 −7 −8 8 6 −6 5

Elementary reductions of both kinds correspond to removing arcs connecting consecutive points.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Reduced partitions with parallel edges

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Reduced partitions with parallel edges

After removing all dual fixpoints from this partition

1 2 3 4 5 6 10 11 12 13 15 14 9 8 7 17 16

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Reduced partitions with parallel edges

we obtain

1 2 3 6 7 8 9 10 5 4 12 11

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Reduced partitions with parallel edges

we obtain

1 2 3 6 7 8 9 10 5 4 12 11

it has several parallel edges.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Parallel edges

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Parallel edges

j i + 1 i j + 1

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Parallel edges

j i + 1 i j + 1

Definition Given a reduced permutation α of {1, . . . , n} and a pair of numbers {i, j} ⊆ {1, . . . , n} such that α(i) = j + 1 and α(j) = i + 1, we say that the ordered pairs (i, α(i)) and (j, α(j)) are parallel edges.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Parallel edges

j i + 1 i j + 1

Lemma For any reduced permutation α of {1, . . . , n}, the ordered directed edges i → α(i) and j → α(j) are parallel if and only if (i, j) is a 2-cycle of α−1ζn.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Removing parallel edges

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Removing parallel edges

j i + 1 i j + 1

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Removing parallel edges

j i + 1 i j + 1 j i + 1 i j + 1

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Removing parallel edges

j i + 1 i j + 1 j − 1 i

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Removing parallel edges

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Removing parallel edges

Proposition Let α be a reduced partition on {1, . . . , n} and let {i → α(i), j → α(j)} be a pair of parallel edges. Then i and j belong to different cycles of α and the permutation γi,j[α], given by γi,j[α](k) =      i + 1 if k = i; j + 1 if k = j; α(k) if k / ∈ {i, j} is also a partition. Furthermore, γi,j[α] has the same genus as α.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Removing parallel edges

Proposition Let α be a reduced partition on {1, . . . , n} and let {i → α(i), j → α(j)} be a pair of parallel edges. Then i and j belong to different cycles of α and the permutation γi,j[α], given by γi,j[α](k) =      i + 1 if k = i; j + 1 if k = j; α(k) if k / ∈ {i, j} is also a partition. Furthermore, γi,j[α] has the same genus as α. Note that γi,j[α] is not reduced, but we can make it reduced by removing the dual fixed points i and j.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Removing parallel edges

Lemma The effect on α−1ζn of the removal of the pair of parallel edges {i → α(i), j → α(j)} is the following. The 2-cycle (i, j) is deleted and each label k in the remaining cycles is decreased by the number of elements in {1, . . . , k} ∩ {i, j}.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Primitive and semiprimitive partitions

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Primitive and semiprimitive partitions

Definition We call a partition α semiprimitive if it has no pair of parallel edges {i → α(i), j → α(j)} such that (i, α(i)) is a 2-cycle. We call a partition α primitive if it contains no pairs of parallel edges at all.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Primitive and semiprimitive partitions

Definition We call a partition α semiprimitive if it has no pair of parallel edges {i → α(i), j → α(j)} such that (i, α(i)) is a 2-cycle. We call a partition α primitive if it contains no pairs of parallel edges at all. If i → α(i) is part of a 2-cycle, then this 2-cycle with another polygon and then removing the arising dual fixed points has the same pictorial effect as simply removing this 2-cycle.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Primitive and semiprimitive partitions

Definition We call a partition α semiprimitive if it has no pair of parallel edges {i → α(i), j → α(j)} such that (i, α(i)) is a 2-cycle. We call a partition α primitive if it contains no pairs of parallel edges at all. If i → α(i) is part of a 2-cycle, then this 2-cycle with another polygon and then removing the arising dual fixed points has the same pictorial effect as simply removing this 2-cycle. Counting the ways of reinserting parallel 2-cycles is much easier than counting all reduced permutations that can be simplified to the same primitive partition.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Our example

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Our example

Original partition:

1 2 3 4 5 6 10 11 12 13 15 14 9 8 7 17 16

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Our example

Resulting reduced partition:

1 2 3 6 7 8 9 10 5 4 12 11

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Our example

Resulting semiprimitive partition:

10 2 3 1 4 5 6 7 8 9

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Our example

Resulting primitive partition:

8 2 3 1 4 5 6 7

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

The breakthrough

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

The breakthrough

Theorem A primitive partition α of genus g is a partition of a set with at most 6(2g − 1) elements. Moreover for any g there is a finite number of semiprimitive partitions of genus g, hence also a finite number of primitive ones.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

The breakthrough

Theorem A primitive partition α of genus g is a partition of a set with at most 6(2g − 1) elements. Moreover for any g there is a finite number of semiprimitive partitions of genus g, hence also a finite number of primitive ones. n + 1 − 2g(α) = z(α) + z(α−1ζn).

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

The breakthrough

Theorem A primitive partition α of genus g is a partition of a set with at most 6(2g − 1) elements. Moreover for any g there is a finite number of semiprimitive partitions of genus g, hence also a finite number of primitive ones. n + 1 − 2g(α) = z(α) + z(α−1ζn). Since α has no fixed points, every cycle of α has length at least 2 and z(α) ≤ n/2. Similarly α−1ζn has no fixed point, by the primitivity of α, there are no 2-cycles in α−1ζn either. Thus z(α−1ζn) ≤ n/3. Hence we get

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

The breakthrough

Theorem A primitive partition α of genus g is a partition of a set with at most 6(2g − 1) elements. Moreover for any g there is a finite number of semiprimitive partitions of genus g, hence also a finite number of primitive ones. n + 1 − 2g(α) = z(α) + z(α−1ζn). n + 1 − 2g = z(α) + z(α−1ζn) ≤ n 2 + n 3.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

The breakthrough

Theorem A primitive partition α of genus g is a partition of a set with at most 6(2g − 1) elements. Moreover for any g there is a finite number of semiprimitive partitions of genus g, hence also a finite number of primitive ones. n + 1 − 2g(α) = z(α) + z(α−1ζn). n + 1 − 2g = z(α) + z(α−1ζn) ≤ n 2 + n 3. Solving for n yields n ≤ 6(2g − 1).

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Selecting parallel class representatives

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Selecting parallel class representatives

Definition In a semiprimitive partition we select each edge of its diagram that is not part of a parallel pair as a parallel class representative and from each parallel pair of edges we select exactly one as a parallel class representative. Subject to this selection we say that a point has type 0, 1, or 2, respectively if the number of edges incident to it in the diagram that are parallel class representatives is 0, 1, or 2, respectively.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Selecting parallel class representatives

10 2 3 1 4 5 6 7 8 9

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Selecting parallel class representatives

10 2 3 1 4 5 6 7 8 9

Select 9 → 5. as a parallel class Type 1 points: 1, 3, 4, 6, 8, 10. The type 2 points: 2, 5, 7, 9. There are no type 0 points.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Counting modulo cyclic relabeling

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Counting modulo cyclic relabeling

Definition Let β be a semiprimitive partition of {1, . . . , m} . We call the average contribution of β to R(x, y) modulo cyclic relabeling the generating function Rβ(x, y) = 1 m ·

m−1

• j=0

Rζj

mβζ−j m (x, y).

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Counting modulo cyclic relabeling

Definition Let β be a semiprimitive partition of {1, . . . , m} . We call the average contribution of β to R(x, y) modulo cyclic relabeling the generating function Rβ(x, y) = 1 m ·

m−1

• j=0

Rζj

mβζ−j m (x, y).

We over-count the contribution of each equivalent semiprimitive partition m times, but then we divide by m. Thus R(x, y) =

• β

Rβ(x, y).

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Completing the proof of the main result

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Completing the proof of the main result

Theorem Let β be a semiprimitive partition on m points with c cycles, whose diagram has p parallel classes. Suppose we have selected parallel class representatives and, subject to this selection, β has mi points of type i for i = 0, 1, 2. Then the average contribution

• f β to R(x, y) modulo cyclic relabeling is given by

Rβ(x, y) = xmyc · m0 · (1 − x2y) + m1 + m2 · (1 + x2y) m · (1 − x2y)p+1 = xmyc (1 − x2y)p+1 + (m2 − m0) · xm+2yc+1 m · (1 − x2y)p+1

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Completing the proof of the main result

Corollary For a fixed genus g, the generating function R(x, y) of reduced partitions of genus g is a rational function of x and y. Moreover, the denominator of R(x, y) is a power of 1 − x2y.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Completing the proof of the main result

Corollary For a fixed genus g, the generating function R(x, y) of reduced partitions of genus g is a rational function of x and y. Moreover, the denominator of R(x, y) is a power of 1 − x2y.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Genus one partitions

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Genus one partitions

n ≤ 6(2g − 1) gives n ≤ 6.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Genus one partitions

n ≤ 6(2g − 1) gives n ≤ 6. The only primitive partitions are β1 = (1, 3)(2, 4) and β2 = (1, 4)(2, 5)(3, 6), these are also the only semiprimitive partitions and all points have type 1.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Genus one partitions

n ≤ 6(2g − 1) gives n ≤ 6. The only primitive partitions are β1 = (1, 3)(2, 4) and β2 = (1, 4)(2, 5)(3, 6), these are also the only semiprimitive partitions and all points have type 1. R(x, y) = Rβ1(x, y) + Rβ2(x, y) = x4y2 · 4 4 · (1 − x2y)3 + x6y3 · 6 6 · (1 − x2y)4 = x4y2 (1 − x2y)4 .

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Genus one partitions

n ≤ 6(2g − 1) gives n ≤ 6. The only primitive partitions are β1 = (1, 3)(2, 4) and β2 = (1, 4)(2, 5)(3, 6), these are also the only semiprimitive partitions and all points have type 1. R(x, y) = Rβ1(x, y) + Rβ2(x, y) = x4y2 · 4 4 · (1 − x2y)3 + x6y3 · 6 6 · (1 − x2y)4 = x4y2 (1 − x2y)4 . The generating of all genus one partitions is P(x, y) = x4y2 (1 − 2(1 + y)x + x2(1 − y)2)5/2 .

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Genus two partitions-overview

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Genus two partitions-overview

n ≤ 6(2g − 1) gives n ≤ 18.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Genus two partitions-overview

n Transpositions only

• ne 3-cycle

two 3-cycles

• ne 4-cycle

6 1 7 14 8 21 20 6 9 141 10 168 65 15 11 407 12 483 52 9 13 455 14 651 15 16 420 17 18 105

Table: Numbers of primitive partitions of genus 2

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Genus two partitions-overview

n Transpositions only

• ne 3-cycle

two 3-cycles

• ne 4-cycle

6 1 7 14 8 21 20 6 9 141 10 168 65 15 11 407 12 483 52 9 13 455 14 651 15 16 420 17 18 105

Table: Numbers of primitive partitions of genus 2

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Reduced matchings of genus 2

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Reduced matchings of genus 2

RM(x, y) = 21 · (x2y)4 · 1+3·x2y+(x2y)2

(1−x2y)10

.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Reduced matchings of genus 2

RM(x, y) = 21 · (x2y)4 · 1+3·x2y+(x2y)2

(1−x2y)10

. The generating function

• n≥0 m2(n) · tn for all matchings of genus 2 with n edges is

RM(C(t), t), where C(t) = (1 − √1 − 4t)/(2t) is a generating function of the Catalan numbers (the coefficient of tn counts the number of noncrossing partitions of 2n such that each part has two elements).

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Reduced matchings of genus 2

RM(x, y) = 21 · (x2y)4 · 1+3·x2y+(x2y)2

(1−x2y)10

. The generating function

• n≥0 m2(n) · tn for all matchings of genus 2 with n edges is

RM(C(t), t), where C(t) = (1 − √1 − 4t)/(2t) is a generating function of the Catalan numbers (the coefficient of tn counts the number of noncrossing partitions of 2n such that each part has two elements). Evaluating the Taylor series of RM(C(t), t) gives RM(C(t), t) = t4 + 483 · t5 + 6468 · t6 + 66066 · t7 + 570570 · t8 + 4390386 · t9 + 31039008 · t10 + · · · The coefficients are listed as sequence A006298 in OEIS, as counting “genus 2 rooted maps with 1 face with n points”, the main reference being the work of Walsh and Lehman.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

The contribution of the other primitive partitions

❈

✷

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

The contribution of the other primitive partitions

R△(x, y) = 7 · x7y3(2 + 13x2y + 13(x2y)2 + 2(x2y)3) (1 − x2y)10 . ❈

✷

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

The contribution of the other primitive partitions

R△(x, y) = 7 · x7y3(2 + 13x2y + 13(x2y)2 + 2(x2y)3) (1 − x2y)10 . R❈(x, y) = x6y2(1 + 18x2y + 55(x2y)2 + 30(x2y)3 + (x2y)4) (1 − x2y)10 .

✷

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

The contribution of the other primitive partitions

R△(x, y) = 7 · x7y3(2 + 13x2y + 13(x2y)2 + 2(x2y)3) (1 − x2y)10 . R❈(x, y) = x6y2(1 + 18x2y + 55(x2y)2 + 30(x2y)3 + (x2y)4) (1 − x2y)10 . R✷(x, y) = 6x8y3(1 + x2y) (1 − x2y)9 .

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Semiprimitive partitions of genus 2 that are not primitive

✜

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Semiprimitive partitions of genus 2 that are not primitive

The primitive partitions listed below all contain a 4-cycle and have several rotational equivalents. These are the ones which may be

• btained by removing a pair of parallel edges in semiprimitive

partitions containing 2 three-cycles.

✜

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Semiprimitive partitions of genus 2 that are not primitive

n Primitive partition c Semiprimitive partition c 10 (1, 3, 5, 7)(2, 8)(4, 6) 4 (1, 3, 5)(6, 8, 10)(2, 9)(4, 7) 5 (1, 3, 9)(4, 6, 8)(2, 10)(5, 7) 5 (1, 3, 5, 7)(2, 6)(4, 8) 2 (1, 3, 5)(6, 8, 10)(2, 7)(4, 9) 5 12 (1, 4, 7, 9)(2, 5) 10 (1, 4, 7)(8, 10, 12)(2, 5)(3, 6)(9, 11) 12 (3, 6)(8, 10) (1, 4, 11)(5, 8, 10)(2, 6)(3, 7)(9, 12) 12 (1, 4, 6, 9)(2, 7) 5 (1, 4, 6)(7, 10, 12)(2, 8)(3, 9)(5, 11) 6 (3, 8)(5, 10) (1, 3, 6)(7, 9, 12)(2, 8)(4, 10)(5, 11) 6 14 (1, 4, 7, 10)(2, 5) 6 (1, 4, 7)(8, 11, 14)(2, 5) 7 (3, 6)(8, 11)(9, 12) (3, 6)(9, 12)(10, 14) (1, 4, 12)(5, 8, 11)(2, 6) 7 (3, 7)(9, 13)(10, 14) (1, 4, 7, 10)(2, 8) 3 (1, 4, 7)(8, 11, 14)(2, 9) 7 (3, 9)(5, 11)(6, 12) (3, 10)(5, 12)(6, 13)

✜

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Semiprimitive partitions of genus 2 that are not primitive

R✜(x, y) = 3x10y4 ·

5+4x2y (1−x2y)10 .

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Adding up the contributions

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Adding up the contributions

Theorem The generating function R(x, y) =

n,k r(n, k)xnyk of all reduced

genus 2 partitions of n elements with k parts is given by R(x, y) = x6y2 · r(x, y) (1 − x2y)10 for r(x, y) = 1 + 14xy + 24x2y + 21x2y2 + 91x3y2 + 55x4y2 + 63x4y3 + 91x5y3 + 21x6y4 + 24x6y3 + 14x7y4 + x8y4.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Adding up the contributions

The rest is pain . . .

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Adding up the contributions

Theorem The generating function P(x, y) =

n,k p(n, k) · xnyk of the

numbers p(n, k) of all genus 2 partitions of n elements with k parts is given by P(x, y) = x6y2 · p(x, y) (x2 − 2x2y − 2x + x2y2 − 2xy + 1)11/2 Here p(x, y) = x4 · (8y4 − 4y3 − 15y2 + 10y + 1) + x3 · (−4y3 + 39y2 − 10y − 4) + x2 · (−15y2 − 10y + 6) + x · (10y − 4) + 1

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Adding up the contributions

Don’t trust me, trust Maple. . .

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Adding up the contributions

The number p(n, k) of genus 2 partitions of n elements with k parts is given by p(n, k) = 8 · γ(n − 10, k − 6) − 4 · γ(n − 10, k − 5) − 15 · γ(n − 10, k − 4) + 10 · γ(n − 10, k − 3) + γ(n − 10, k − 2) − 4 · γ(n − 9, k − 5) + 39 · γ(n − 9, k − 4) − 10 · γ(n − 9, k − 3) − 4 · γ(n − 9, k − 2) − 15 · γ(n − 8, k − 4) − 10 · γ(n − 8, k − 3) + 6 · γ(n − 8, k − 2) − 4 · γ(n − 7, k − 2) + 10 · γ(n − 7, k − 3) + γ(n − 6, k − 2). Here γ(n, k) = n+10

5

n+5

k

n+5

n−k

• /

10

5

• .
• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

A table

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

A table

❅ ❅ ❅

n k 2 3 4 5 6 7 8 6 1 7 7 21 8 28 210 161 9 84 1134 2184 777 10 210 4410 15330 13713 2835 11 462 13860 75075 121275 63063 8547 12 924 37422 289905 729960 685608 233772 22407

Table: Numbers of partitions of genus 2 of n elements with k parts

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

A table

These numbers agree with the numbers published by M. Yip except for the values of p(11, 4) and p(11, 5), where M. Yip’s computer search has found p(11, 4) = 75675 and p(11, 5) = 110880. Our exhaustive search for partitions of genus 2 has found the same numbers as published in our table.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

A table

These numbers agree with the numbers published by M. Yip except for the values of p(11, 4) and p(11, 5), where M. Yip’s computer search has found p(11, 4) = 75675 and p(11, 5) = 110880. Our exhaustive search for partitions of genus 2 has found the same numbers as published in our table.

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Thank you!

• G. Hetyei and R. Cori

Partitions of a fixed genus

Outline Preliminaries and main result Proof and instances of our main result Elementary reductions and reduced permutations Primitive and semiprimitive partitions The cases of genus one and two

Thank you!

Please read: [1] S´ eminaire Lotharingien de Combinatoire, B70e (2014), 28 pp. [2] arXiv:1710.09992 [math.CO]

• G. Hetyei and R. Cori

Partitions of a fixed genus