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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.
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