Numerical Semigroups and theirCorresponding Core Partitions

Benjamin Houston-EdwardsJoint with Hannah Constantin

Yale University

August 7, 2014

Background and Review

DefinitionA set S is a numerical semigroup if

S ⊆N0 ∈ S

S is closed under addition

N\ S is finite

Example

S = ⟨3,8⟩ = {0,3,6,8,9,11,12,14,15,16, . . .}

Background and Review

DefinitionA set S is a numerical semigroup if

S ⊆N0 ∈ S

S is closed under addition

N\ S is finite

Example

S = ⟨3,8⟩

= {0,3,6,8,9,11,12,14,15,16, . . .}

Background and Review

DefinitionA set S is a numerical semigroup if

S ⊆N0 ∈ S

S is closed under addition

N\ S is finite

Example

S = ⟨3,8⟩ = {0,3,6,8,9,11,12,14,15,16, . . .}

Background and ReviewThere is an injective map ϕ from numerical semigroups to integerpartitions

Example

S = ⟨3,8⟩ = {0,3,6,8,9,11,12,14, . . .}

0

1

23

4

56

78 9

1011 12

13

ϕ(S) = (7,5,3,2,2,1,1)

Background and ReviewThere is an injective map ϕ from numerical semigroups to integerpartitions

Example

S = ⟨3,8⟩ = {0,3,6,8,9,11,12,14, . . .}

0

1

23

4

56

78 9

1011 12

13

ϕ(S) = (7,5,3,2,2,1,1)

Background and ReviewThere is an injective map ϕ from numerical semigroups to integerpartitions

Example

S = ⟨3,8⟩ = {0,3,6,8,9,11,12,14, . . .}

0

1

23

4

56

78 9

1011 12

13

ϕ(S) = (7,5,3,2,2,1,1)

Background and ReviewThere is an injective map ϕ from numerical semigroups to integerpartitions

Example

S = ⟨3,8⟩ = {0,3,6,8,9,11,12,14, . . .}

0

1

23

4

56

78 9

1011 12

13

ϕ(S) = (7,5,3,2,2,1,1)

Background and ReviewThere is an injective map ϕ from numerical semigroups to integerpartitions

Example

S = ⟨3,8⟩ = {0,3,6,8,9,11,12,14, . . .}

0

1

23

4

56

78 9

1011 12

13

ϕ(S) = (7,5,3,2,2,1,1)

Background and ReviewThere is an injective map ϕ from numerical semigroups to integerpartitions

Example

S = ⟨3,8⟩ = {0,3,6,8,9,11,12,14, . . .}

0

1

23

4

56

78 9

1011 12

13

ϕ(S) = (7,5,3,2,2,1,1)

Background and ReviewThere is an injective map ϕ from numerical semigroups to integerpartitions

Example

S = ⟨3,8⟩ = {0,3,6,8,9,11,12,14, . . .}

0

1

23

4

56

78 9

1011 12

13

ϕ(S) = (7,5,3,2,2,1,1)

Background and ReviewThere is an injective map ϕ from numerical semigroups to integerpartitions

Example

S = ⟨3,8⟩ = {0,3,6,8,9,11,12,14, . . .}

0

1

23

4

56

78 9

1011 12

13

ϕ(S) = (7,5,3,2,2,1,1)

Background and Review

We can also assign a set of hook lengths to each partition:

Example

0

123

456

78 9

1011 12

13

7

13 10 7 5 4 2 1

10 7 4 2 1

7 4 1

5 2

4 1

2

1

ϕ(⟨3,8⟩)

Background and Review

We can also assign a set of hook lengths to each partition:

Example

0

123

456

78 9

1011 12

13

7

13 10 7 5 4 2 1

10 7 4 2 1

7 4 1

5 2

4 1

2

1

ϕ(⟨3,8⟩)

Background and Review

We can also assign a set of hook lengths to each partition:

Example

0

123

456

78 9

1011 12

13

7

13 10 7 5 4 2 1

10 7 4 2 1

7 4 1

5 2

4 1

2

1

ϕ(⟨3,8⟩)

Background and Review

We can also assign a set of hook lengths to each partition:

Example

0

123

456

78 9

1011 12

13

7

13 10 7 5 4 2 1

10 7 4 2 1

7 4 1

5 2

4 1

2

1

ϕ(⟨3,8⟩)

Background and Review

We can also assign a set of hook lengths to each partition:

Example

0

123

456

78 9

1011 12

13

7

13 10 7 5 4 2 1

10 7 4 2 1

7 4 1

5 2

4 1

2

1

ϕ(⟨3,8⟩)

Background and Review

We can also assign a set of hook lengths to each partition:

Example

0

123

456

78 9

1011 12

13

7

13 10 7 5 4 2 1

10 7 4 2 1

7 4 1

5 2

4 1

2

1

ϕ(⟨3,8⟩)

Background and Review

We can also assign a set of hook lengths to each partition:

Example

0

123

456

78 9

1011 12

13

7

13 10 7 5 4 2 1

10 7 4 2 1

7 4 1

5 2

4 1

2

1

ϕ(⟨3,8⟩)

Background

DefinitionA partition λ is an a–core partition if a does not divide any of thehook lengths of λ. An (a,b)–core partition is both an a−core and ab−core.

Example

λ= (7,5,3,2,2,1,1) is a (3,8)−core

13 10 7 5 4 2 1

10 7 4 2 1

7 4 1

5 2

4 1

2

1

Background

DefinitionA partition λ is an a–core partition if a does not divide any of thehook lengths of λ. An (a,b)–core partition is both an a−core and ab−core.

Example

λ= (7,5,3,2,2,1,1) is a (3,8)−core

13 10 7 5 4 2 1

10 7 4 2 1

7 4 1

5 2

4 1

2

1

Background

Theorem (Anderson)For coprime a and b, the total number of (a,b)−core partitions is

1

a+b

(a+b

a

).

We are interested in counting the subset of (a,b)−cores thatcome from numerical semigroups via the map ϕ.

Background

Theorem (Anderson)For coprime a and b, the total number of (a,b)−core partitions is

1

a+b

(a+b

a

).

We are interested in counting the subset of (a,b)−cores thatcome from numerical semigroups via the map ϕ.

Background

Proposition

Suppose λ=ϕ(S) for some semigroup S. Then λ is an (a,b)−core ifand only if a,b ∈ S.

Example

λ= (7,5,3,2,2,1,1) is a (3,8)−core and λ=ϕ(S) whereS = ⟨3,8⟩ = {0,3,6,8,9,11,12,14,15,16, . . .}

0

3

6

8 9

11 12

13 10 7 5 4 2 1

10 7 4 2 1

7 4 1

5 2

4 1

2

1

Background

Proposition

Suppose λ=ϕ(S) for some semigroup S. Then λ is an (a,b)−core ifand only if a,b ∈ S.

Example

λ= (7,5,3,2,2,1,1) is a (3,8)−core and λ=ϕ(S) whereS = ⟨3,8⟩ = {0,3,6,8,9,11,12,14,15,16, . . .}

0

3

6

8 9

11 12

13 10 7 5 4 2 1

10 7 4 2 1

7 4 1

5 2

4 1

2

1

Background

Proposition

Suppose λ=ϕ(S) for some semigroup S. Then λ is an (a,b)−core ifand only if a,b ∈ S.

DefinitionGiven a numerical semigroup S, the set of oversemigroups of S is

{T ⊇ S : T is a numerical semigroup}.

The cardinality of this set is denoted O(S).

The number of (a,b)−core partitions from numerical semi-groups is exactly O(⟨a,b⟩).

Background

Proposition

Suppose λ=ϕ(S) for some semigroup S. Then λ is an (a,b)−core ifand only if a,b ∈ S.

DefinitionGiven a numerical semigroup S, the set of oversemigroups of S is

{T ⊇ S : T is a numerical semigroup}.

The cardinality of this set is denoted O(S).

The number of (a,b)−core partitions from numerical semi-groups is exactly O(⟨a,b⟩).

Background

Proposition

Suppose λ=ϕ(S) for some semigroup S. Then λ is an (a,b)−core ifand only if a,b ∈ S.

DefinitionGiven a numerical semigroup S, the set of oversemigroups of S is

{T ⊇ S : T is a numerical semigroup}.

The cardinality of this set is denoted O(S).

The number of (a,b)−core partitions from numerical semi-groups is exactly O(⟨a,b⟩).

Apéry Tuples

DefinitionIf S is a numerical semigroup, then the Apéry tuple of S with respectto some n ∈ S is the tuple (k1,k2, . . . ,kn−1) such that nki + i is thesmallest element of S in its residue class (mod n) for each i.

This tuple is denoted Ap′(S,n).

Example

If S = ⟨3,8⟩ = {0,3,6,8,9,11,12,14, . . .}, then 16 and 8 are the smallestelements of S in their residue classes mod 3, so Ap′(S,3) = (5,2).

Apéry Tuples

DefinitionIf S is a numerical semigroup, then the Apéry tuple of S with respectto some n ∈ S is the tuple (k1,k2, . . . ,kn−1) such that nki + i is thesmallest element of S in its residue class (mod n) for each i.

This tuple is denoted Ap′(S,n).

Example

If S = ⟨3,8⟩ = {0,3,6,8,9,11,12,14, . . .}, then 16 and 8 are the smallestelements of S in their residue classes mod 3, so Ap′(S,3) = (5,2).

Apéry Tuples

Suppose S is a numerical semigroup with

Ap′(S,n) = (k1, . . . ,kn−1).

A tuple (`1,`2, . . . ,`n−1) is an Apéry tuple of some numericalsemigroup T ⊇ S if and only if the following inequalities aresatisfied:

`i ≥ 0, ∀1 ≤ i ≤ n−1

`i +`j ≥ `i+j, i+ j < n

`i +`j +1 ≥ `n−i−j, i+ j > n

`i ≤ ki for all i

RemarkThese inequalities define an n−1 dimensional polytope in whichthe integer lattice points correspond exactly with theoversemigroups of S.

Apéry Tuples

Suppose S is a numerical semigroup with

Ap′(S,n) = (k1, . . . ,kn−1).

A tuple (`1,`2, . . . ,`n−1) is an Apéry tuple of some numericalsemigroup T ⊇ S if and only if the following inequalities aresatisfied:

`i ≥ 0, ∀1 ≤ i ≤ n−1

`i +`j ≥ `i+j, i+ j < n

`i +`j +1 ≥ `n−i−j, i+ j > n

`i ≤ ki for all i

RemarkThese inequalities define an n−1 dimensional polytope in whichthe integer lattice points correspond exactly with theoversemigroups of S.

Apéry Tuples

Suppose S is a numerical semigroup with

Ap′(S,n) = (k1, . . . ,kn−1).

A tuple (`1,`2, . . . ,`n−1) is an Apéry tuple of some numericalsemigroup T ⊇ S if and only if the following inequalities aresatisfied:

`i ≥ 0, ∀1 ≤ i ≤ n−1

`i +`j ≥ `i+j, i+ j < n

`i +`j +1 ≥ `n−i−j, i+ j > n

`i ≤ ki for all i

RemarkThese inequalities define an n−1 dimensional polytope in whichthe integer lattice points correspond exactly with theoversemigroups of S.

Apéry Tuples and Polytopes

Example

S = ⟨3,8⟩ and Ap′(S,3) = (5,2). The relevant polytope is defined byx ≤ 5, y ≤ 2, 2x ≥ y, and 2y+1 ≥ x:

x

y

There are 10 integer lattice points in this polytope, so O(⟨3,8⟩) = 10.

Apéry Tuples and Polytopes

Example

S = ⟨3,8⟩ and Ap′(S,3) = (5,2). The relevant polytope is defined byx ≤ 5, y ≤ 2, 2x ≥ y, and 2y+1 ≥ x:

x

y

There are 10 integer lattice points in this polytope, so O(⟨3,8⟩) = 10.

Apéry Tuples and Polytopes

Example

S = ⟨3,8⟩ and Ap′(S,3) = (5,2). The relevant polytope is defined byx ≤ 5, y ≤ 2, 2x ≥ y, and 2y+1 ≥ x:

x

y

There are 10 integer lattice points in this polytope, so O(⟨3,8⟩) = 10.

The case of a = 3

x

y

1 1 1

2

2

1

1

1

The case of a = 3

x

y

1 1 1

2

2

1

1

1

The case of a = 3

x

y

1 1 1

2

2

1

1

1

The case of a = 3

x

y

1 1 1

2

2

1

1

1

The case of a = 3

x

y

1 1 1

2

2

1

1

1

The case of a = 3

x

y

Theorem (Constantin – H.E.)If S = ⟨3,6k+`⟩ then O(S) = (3k+`)(k+1).

Example

O(⟨3,8⟩) = O(⟨3,6 ·1+2⟩) = (3+2)(1+1) = 10

The case of a = 3

x

y

Theorem (Constantin – H.E.)If S = ⟨3,6k+`⟩ then O(S) = (3k+`)(k+1).

Example

O(⟨3,8⟩) = O(⟨3,6 ·1+2⟩) = (3+2)(1+1) = 10

The case of a = 3

x

y

Theorem (Constantin – H.E.)If S = ⟨3,6k+`⟩ then O(S) = (3k+`)(k+1).

Example

O(⟨3,8⟩) = O(⟨3,6 ·1+2⟩) = (3+2)(1+1) = 10

The case of a = 4

Theorem (Constantin – H.E.)If S = ⟨4,12k+`⟩ then O(S) ∼ 24k3.

In fact, we can find the explicit formula for each `:

` O(S)1 24k3 +30k2 +11k+13 24k3 +42k2 +23k+45 24k3 +54k2 +39k+97 24k3 +66k2 +59k+179 24k3 +78k2 +83k+29

11 24k3 +90k2 +111k+45

The case of a = 4

Theorem (Constantin – H.E.)If S = ⟨4,12k+`⟩ then O(S) ∼ 24k3.In fact, we can find the explicit formula for each `:

` O(S)1 24k3 +30k2 +11k+13 24k3 +42k2 +23k+45 24k3 +54k2 +39k+97 24k3 +66k2 +59k+179 24k3 +78k2 +83k+29

11 24k3 +90k2 +111k+45

Asymptotic Behavior

Let A(a,b) = (a+ba

)/(a+b), the total number of (a,b)−core partitions

by Anderson’s theorem.

Comparing O(⟨a,b⟩) with A(a,b) in the limit:

a limb→∞ O(⟨a,b⟩)/A(a,b)2 13 1/24 1/3

Asymptotic Behavior

Let A(a,b) = (a+ba

)/(a+b), the total number of (a,b)−core partitions

by Anderson’s theorem.

Comparing O(⟨a,b⟩) with A(a,b) in the limit:

a limb→∞ O(⟨a,b⟩)/A(a,b)

2 13 1/24 1/3

Asymptotic Behavior

Let A(a,b) = (a+ba

)/(a+b), the total number of (a,b)−core partitions

by Anderson’s theorem.

Comparing O(⟨a,b⟩) with A(a,b) in the limit:

a limb→∞ O(⟨a,b⟩)/A(a,b)2 1

3 1/24 1/3

Asymptotic Behavior

Let A(a,b) = (a+ba

)/(a+b), the total number of (a,b)−core partitions

by Anderson’s theorem.

Comparing O(⟨a,b⟩) with A(a,b) in the limit:

a limb→∞ O(⟨a,b⟩)/A(a,b)2 13 1/2

4 1/3

Asymptotic Behavior

Let A(a,b) = (a+ba

)/(a+b), the total number of (a,b)−core partitions

by Anderson’s theorem.

Comparing O(⟨a,b⟩) with A(a,b) in the limit:

a limb→∞ O(⟨a,b⟩)/A(a,b)2 13 1/24 1/3

Future work

a limb→∞ O(⟨a,b⟩)/A(a,b)2 13 1/24 1/3

In the future we would like to look at

limb→∞

O(⟨a,b⟩)A(a,b)

for general values of a.

We suspect that as a →∞, this fraction will decrease to 0, meaningthat almost no (a,b)−cores come from semigroups in the limit.

Future work

a limb→∞ O(⟨a,b⟩)/A(a,b)2 13 1/24 1/3

In the future we would like to look at

limb→∞

O(⟨a,b⟩)A(a,b)

for general values of a.

We suspect that as a →∞, this fraction will decrease to 0, meaningthat almost no (a,b)−cores come from semigroups in the limit.

Future work

a limb→∞ O(⟨a,b⟩)/A(a,b)2 13 1/24 1/3

In the future we would like to look at

limb→∞

O(⟨a,b⟩)A(a,b)

for general values of a.

We suspect that as a →∞, this fraction will decrease to 0, meaningthat almost no (a,b)−cores come from semigroups in the limit.

Acknowledgments

We would like the thank . . .

Nathan Kaplan for guiding our research

Flor Orosz Hunziker and Dan Corey for all their help asmentors

Kyle Luh for helping us understand polytopes

The rest of the SUMRY staff and students for creating such agreat program

References

J. Anderson, Partitions which are simultaneously t1− and t2−core.Discrete Math. 248 (2002), no. 1–3, 237–243.

P.A. García-Sánchez and J.C. Rosales, Numerical semigroups. NewYork: Springer, 2009.

N. Kaplan, Counting numerical semigroups by genus and some casesof a question of Wilf, J. Pure Appl. Algebra 216 (2012), no. 5,1016–1032.