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Nilpotent ideals of upper triangular matrices andVariations of the Catalan numbers
Jia Huang
University of Nebraska at KearneyE-mail address: huangj2@unk.edu
October 19, 2017
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 1 / 25
Ideals of upper triangular matrices
Definition
Let Un be the algebra of all n-by-n upper triangular matrices∗ ∗ ∗ · · · ∗0 ∗ ∗ · · · ∗0 0 ∗ · · · ∗...
.... . .
. . ....
0 0 · · · 0 ∗
where a star ∗ is an arbitrary entry from a fixed field F (e.g., R).
A (two-sided) ideal I of Un is a vector subspace of Un such thatXI ⊆ I and IX ⊆ I for all X ∈ Un.
A ideal I is nilpotent if I k = 0 for some k ≥ 1. The smallest k suchthat I k = 0 is the (nilpotent) order of I .
A ideal I of Un is commutative if AB = BA for all A,B ∈ I .
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 2 / 25
Ideals of upper triangular matrices
Definition
Let Un be the algebra of all n-by-n upper triangular matrices∗ ∗ ∗ · · · ∗0 ∗ ∗ · · · ∗0 0 ∗ · · · ∗...
.... . .
. . ....
0 0 · · · 0 ∗
where a star ∗ is an arbitrary entry from a fixed field F (e.g., R).
A (two-sided) ideal I of Un is a vector subspace of Un such thatXI ⊆ I and IX ⊆ I for all X ∈ Un.
A ideal I is nilpotent if I k = 0 for some k ≥ 1. The smallest k suchthat I k = 0 is the (nilpotent) order of I .
A ideal I of Un is commutative if AB = BA for all A,B ∈ I .
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 2 / 25
Ideals of upper triangular matrices
Definition
Let Un be the algebra of all n-by-n upper triangular matrices∗ ∗ ∗ · · · ∗0 ∗ ∗ · · · ∗0 0 ∗ · · · ∗...
.... . .
. . ....
0 0 · · · 0 ∗
where a star ∗ is an arbitrary entry from a fixed field F (e.g., R).
A (two-sided) ideal I of Un is a vector subspace of Un such thatXI ⊆ I and IX ⊆ I for all X ∈ Un.
A ideal I is nilpotent if I k = 0 for some k ≥ 1. The smallest k suchthat I k = 0 is the (nilpotent) order of I .
A ideal I of Un is commutative if AB = BA for all A,B ∈ I .
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 2 / 25
Ideals of upper triangular matrices
Definition
Let Un be the algebra of all n-by-n upper triangular matrices∗ ∗ ∗ · · · ∗0 ∗ ∗ · · · ∗0 0 ∗ · · · ∗...
.... . .
. . ....
0 0 · · · 0 ∗
where a star ∗ is an arbitrary entry from a fixed field F (e.g., R).
A (two-sided) ideal I of Un is a vector subspace of Un such thatXI ⊆ I and IX ⊆ I for all X ∈ Un.
A ideal I is nilpotent if I k = 0 for some k ≥ 1. The smallest k suchthat I k = 0 is the (nilpotent) order of I .
A ideal I of Un is commutative if AB = BA for all A,B ∈ I .
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 2 / 25
Nilpotent ideals
Example (A nilpotent ideal of U6 and its corresponding Dyck path)
I =
0 0 ∗ ∗ ∗ ∗0 0 0 0 ∗ ∗0 0 0 0 ∗ ∗0 0 0 0 0 ∗0 0 0 0 0 ∗0 0 0 0 0 0
height = 3
Observation
An nilpotent ideal of Un is represented by a matrix of 0’s and ∗’sseparated by a Dyck path of length 2n.
The number of such ideals is the Catalan number Cn := 1n+1
(2nn
).
The number of all ideals of Un is the Catalan number Cn+1.
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 3 / 25
Nilpotent ideals
Example (A nilpotent ideal of U6 and its corresponding Dyck path)
I =
0 0 ∗ ∗ ∗ ∗0 0 0 0 ∗ ∗0 0 0 0 ∗ ∗0 0 0 0 0 ∗0 0 0 0 0 ∗0 0 0 0 0 0
height = 3
Observation
An nilpotent ideal of Un is represented by a matrix of 0’s and ∗’sseparated by a Dyck path of length 2n.
The number of such ideals is the Catalan number Cn := 1n+1
(2nn
).
The number of all ideals of Un is the Catalan number Cn+1.
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 3 / 25
Nilpotent ideals
Example (A nilpotent ideal of U6 and its corresponding Dyck path)
I =
0 0 ∗ ∗ ∗ ∗0 0 0 0 ∗ ∗0 0 0 0 ∗ ∗0 0 0 0 0 ∗0 0 0 0 0 ∗0 0 0 0 0 0
height = 3
Observation
An nilpotent ideal of Un is represented by a matrix of 0’s and ∗’sseparated by a Dyck path of length 2n.
The number of such ideals is the Catalan number Cn := 1n+1
(2nn
).
The number of all ideals of Un is the Catalan number Cn+1.
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 3 / 25
Nilpotent ideals
Example (A nilpotent ideal of U6 and its corresponding Dyck path)
I =
0 0 ∗ ∗ ∗ ∗0 0 0 0 ∗ ∗0 0 0 0 ∗ ∗0 0 0 0 0 ∗0 0 0 0 0 ∗0 0 0 0 0 0
height = 3
Observation
An nilpotent ideal of Un is represented by a matrix of 0’s and ∗’sseparated by a Dyck path of length 2n.
The number of such ideals is the Catalan number Cn := 1n+1
(2nn
).
The number of all ideals of Un is the Catalan number Cn+1.
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 3 / 25
Nilpotent ideals
Example (A nilpotent ideal of U6 and its corresponding Dyck path)
I =
0 0 ∗ ∗ ∗ ∗0 0 0 0 ∗ ∗0 0 0 0 ∗ ∗0 0 0 0 0 ∗0 0 0 0 0 ∗0 0 0 0 0 0
height = 3
Observation
An nilpotent ideal of Un is represented by a matrix of 0’s and ∗’sseparated by a Dyck path of length 2n.
The number of such ideals is the Catalan number Cn := 1n+1
(2nn
).
The number of all ideals of Un is the Catalan number Cn+1.
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 3 / 25
Commutative ideals
Proposition (L. Shapiro, 1975)
The number of commutative ideals of Un is 2n−1.
Problem
Find a direct proof of the above result.
Example
The number of subsets of {1, 2, . . . , n} is(n0
)+(n1
)+(n2
)+ · · ·+
(nn
)= 2n.
This can be proved by considering whether a subset contains i for each i .
Observation
An ideal of Un is commutative if and only if it has nilpotent order 1 or 2.
Definition
Let Cdn be the number of nilpotent ideals of Un with order at most d .
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 4 / 25
Commutative ideals
Proposition (L. Shapiro, 1975)
The number of commutative ideals of Un is 2n−1.
Problem
Find a direct proof of the above result.
Example
The number of subsets of {1, 2, . . . , n} is(n0
)+(n1
)+(n2
)+ · · ·+
(nn
)= 2n.
This can be proved by considering whether a subset contains i for each i .
Observation
An ideal of Un is commutative if and only if it has nilpotent order 1 or 2.
Definition
Let Cdn be the number of nilpotent ideals of Un with order at most d .
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 4 / 25
Commutative ideals
Proposition (L. Shapiro, 1975)
The number of commutative ideals of Un is 2n−1.
Problem
Find a direct proof of the above result.
Example
The number of subsets of {1, 2, . . . , n} is(n0
)+(n1
)+(n2
)+ · · ·+
(nn
)= 2n.
This can be proved by considering whether a subset contains i for each i .
Observation
An ideal of Un is commutative if and only if it has nilpotent order 1 or 2.
Definition
Let Cdn be the number of nilpotent ideals of Un with order at most d .
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 4 / 25
Commutative ideals
Proposition (L. Shapiro, 1975)
The number of commutative ideals of Un is 2n−1.
Problem
Find a direct proof of the above result.
Example
The number of subsets of {1, 2, . . . , n} is(n0
)+(n1
)+(n2
)+ · · ·+
(nn
)= 2n.
This can be proved by considering whether a subset contains i for each i .
Observation
An ideal of Un is commutative if and only if it has nilpotent order 1 or 2.
Definition
Let Cdn be the number of nilpotent ideals of Un with order at most d .
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 4 / 25
Commutative ideals
Proposition (L. Shapiro, 1975)
The number of commutative ideals of Un is 2n−1.
Problem
Find a direct proof of the above result.
Example
The number of subsets of {1, 2, . . . , n} is(n0
)+(n1
)+(n2
)+ · · ·+
(nn
)= 2n.
This can be proved by considering whether a subset contains i for each i .
Observation
An ideal of Un is commutative if and only if it has nilpotent order 1 or 2.
Definition
Let Cdn be the number of nilpotent ideals of Un with order at most d .
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 4 / 25
Nilpotent order
Observation
The order of a nilpotent ideal I of Un is the largest possible length d of anadmissible sequence, that is, a sequence (i1, i2, . . . , id) such that the entry(ij , ij+1) is a star ∗ in the matrix form of I for all j = 1, 2, . . . , d − 1.
Example
The following ideal has nilpotent order is 4 since the sequence (1, 3, 5, 6) isadmissible and there is no longer admissible sequence.
I =
0 0 ∗ ∗ ∗ ∗0 0 0 0 ∗ ∗0 0 0 0 ∗ ∗0 0 0 0 0 ∗0 0 0 0 0 ∗0 0 0 0 0 0
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 5 / 25
Nilpotent order
Observation
The order of a nilpotent ideal I of Un is the largest possible length d of anadmissible sequence, that is, a sequence (i1, i2, . . . , id) such that the entry(ij , ij+1) is a star ∗ in the matrix form of I for all j = 1, 2, . . . , d − 1.
Example
The following ideal has nilpotent order is 4 since the sequence (1, 3, 5, 6) isadmissible and there is no longer admissible sequence.
I =
0 0 ∗ ∗ ∗ ∗0 0 0 0 ∗ ∗0 0 0 0 ∗ ∗0 0 0 0 0 ∗0 0 0 0 0 ∗0 0 0 0 0 0
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 5 / 25
Bounce Paths
Observation
Let I be an ideal of Un corresponding to a Dyck path D. Then thenilpotent order of I is the number of times the bounce path of D bouncesoff the main diagonal.
Example (Bounce Path)
The bounce path has 4 bounces.
The Dyck path D has height 3.
Fact (Andrews–Krattenthaler–Orsina–Papi 2002, Haglund 2008)
Bijection ζ : Dyck paths with height d ↔ Dyck paths with d bounces.
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 6 / 25
Bounce Paths
Observation
Let I be an ideal of Un corresponding to a Dyck path D. Then thenilpotent order of I is the number of times the bounce path of D bouncesoff the main diagonal.
Example (Bounce Path)
The bounce path has 4 bounces.
The Dyck path D has height 3.
Fact (Andrews–Krattenthaler–Orsina–Papi 2002, Haglund 2008)
Bijection ζ : Dyck paths with height d ↔ Dyck paths with d bounces.
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 6 / 25
Bounce Paths
Observation
Let I be an ideal of Un corresponding to a Dyck path D. Then thenilpotent order of I is the number of times the bounce path of D bouncesoff the main diagonal.
Example (Bounce Path)
The bounce path has 4 bounces.
The Dyck path D has height 3.
Fact (Andrews–Krattenthaler–Orsina–Papi 2002, Haglund 2008)
Bijection ζ : Dyck paths with height d ↔ Dyck paths with d bounces.
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 6 / 25
Bounce Paths
Observation
Let I be an ideal of Un corresponding to a Dyck path D. Then thenilpotent order of I is the number of times the bounce path of D bouncesoff the main diagonal.
Example (Bounce Path)
The bounce path has 4 bounces.
The Dyck path D has height 3.
Fact (Andrews–Krattenthaler–Orsina–Papi 2002, Haglund 2008)
Bijection ζ : Dyck paths with height d ↔ Dyck paths with d bounces.
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 6 / 25
Generalization of Commutative Ideals
Theorem (H.-Rhoades)
Dyck paths of length 2n with height at most d are counted by Cdn . Hence
Cdn is the sequence A080934 in OEIS and interpolates between 1 and Cn.
Example
n 1 2 3 4 5 6 7 n
C 1n 1 1 1 1 1 1 1 1
C 2n 1 2 4 8 16 32 64 2n−1
C 3n 1 2 5 13 34 89 233 F2n−1
C 4n 1 2 5 14 41 122 365 1
2(1 + 3n−1)
Cn 1 2 5 14 42 132 429 1n+1
(2nn
)Problem
Is there a nice (q, t)-analogue of the number Cdn ?
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 7 / 25
Generalization of Commutative Ideals
Theorem (H.-Rhoades)
Dyck paths of length 2n with height at most d are counted by Cdn . Hence
Cdn is the sequence A080934 in OEIS and interpolates between 1 and Cn.
Example
n 1 2 3 4 5 6 7 n
C 1n 1 1 1 1 1 1 1 1
C 2n 1 2 4 8 16 32 64 2n−1
C 3n 1 2 5 13 34 89 233 F2n−1
C 4n 1 2 5 14 41 122 365 1
2(1 + 3n−1)
Cn 1 2 5 14 42 132 429 1n+1
(2nn
)
Problem
Is there a nice (q, t)-analogue of the number Cdn ?
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 7 / 25
Generalization of Commutative Ideals
Theorem (H.-Rhoades)
Dyck paths of length 2n with height at most d are counted by Cdn . Hence
Cdn is the sequence A080934 in OEIS and interpolates between 1 and Cn.
Example
n 1 2 3 4 5 6 7 n
C 1n 1 1 1 1 1 1 1 1
C 2n 1 2 4 8 16 32 64 2n−1
C 3n 1 2 5 13 34 89 233 F2n−1
C 4n 1 2 5 14 41 122 365 1
2(1 + 3n−1)
Cn 1 2 5 14 42 132 429 1n+1
(2nn
)Problem
Is there a nice (q, t)-analogue of the number Cdn ?
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 7 / 25
Ideals of Lie Algebras
Definition
Let sln(C) be the (type A semisimple) Lie algebra of all n× n complexmatrices with zero trace under the Lie bracket [X ,Y ] := XY − YX .
Let b be the Borel subalgebra of upper triangular matrices of sln(C).
Theorem (Andrews–Krattenthaler–Orsina–Papi 2002)
The number of ad-nilpotent ideals of b with order at most d − 1 is Cdn .
This gives a nonsymmetric (q, t)-analogue of Cn using (#bounces, area).
Problem
Find a natural order-preserving bijection between nilpotent ideals ofUn and ad-nilpotent ideals of b. (The exponential map?)
The above theorem has been generalized from type A to other types[Krattenthaler–Orsina–Papi 2002]. Is there a similar generalization fornilpotent ideals of Un?
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 8 / 25
Ideals of Lie Algebras
Definition
Let sln(C) be the (type A semisimple) Lie algebra of all n× n complexmatrices with zero trace under the Lie bracket [X ,Y ] := XY − YX .
Let b be the Borel subalgebra of upper triangular matrices of sln(C).
Theorem (Andrews–Krattenthaler–Orsina–Papi 2002)
The number of ad-nilpotent ideals of b with order at most d − 1 is Cdn .
This gives a nonsymmetric (q, t)-analogue of Cn using (#bounces, area).
Problem
Find a natural order-preserving bijection between nilpotent ideals ofUn and ad-nilpotent ideals of b. (The exponential map?)
The above theorem has been generalized from type A to other types[Krattenthaler–Orsina–Papi 2002]. Is there a similar generalization fornilpotent ideals of Un?
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 8 / 25
Ideals of Lie Algebras
Definition
Let sln(C) be the (type A semisimple) Lie algebra of all n× n complexmatrices with zero trace under the Lie bracket [X ,Y ] := XY − YX .
Let b be the Borel subalgebra of upper triangular matrices of sln(C).
Theorem (Andrews–Krattenthaler–Orsina–Papi 2002)
The number of ad-nilpotent ideals of b with order at most d − 1 is Cdn .
This gives a nonsymmetric (q, t)-analogue of Cn using (#bounces, area).
Problem
Find a natural order-preserving bijection between nilpotent ideals ofUn and ad-nilpotent ideals of b. (The exponential map?)
The above theorem has been generalized from type A to other types[Krattenthaler–Orsina–Papi 2002]. Is there a similar generalization fornilpotent ideals of Un?
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 8 / 25
Ideals of Lie Algebras
Definition
Let sln(C) be the (type A semisimple) Lie algebra of all n× n complexmatrices with zero trace under the Lie bracket [X ,Y ] := XY − YX .
Let b be the Borel subalgebra of upper triangular matrices of sln(C).
Theorem (Andrews–Krattenthaler–Orsina–Papi 2002)
The number of ad-nilpotent ideals of b with order at most d − 1 is Cdn .
This gives a nonsymmetric (q, t)-analogue of Cn using (#bounces, area).
Problem
Find a natural order-preserving bijection between nilpotent ideals ofUn and ad-nilpotent ideals of b. (The exponential map?)
The above theorem has been generalized from type A to other types[Krattenthaler–Orsina–Papi 2002]. Is there a similar generalization fornilpotent ideals of Un?
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 8 / 25
Ideals of Lie Algebras
Definition
Let sln(C) be the (type A semisimple) Lie algebra of all n× n complexmatrices with zero trace under the Lie bracket [X ,Y ] := XY − YX .
Let b be the Borel subalgebra of upper triangular matrices of sln(C).
Theorem (Andrews–Krattenthaler–Orsina–Papi 2002)
The number of ad-nilpotent ideals of b with order at most d − 1 is Cdn .
This gives a nonsymmetric (q, t)-analogue of Cn using (#bounces, area).
Problem
Find a natural order-preserving bijection between nilpotent ideals ofUn and ad-nilpotent ideals of b. (The exponential map?)
The above theorem has been generalized from type A to other types[Krattenthaler–Orsina–Papi 2002]. Is there a similar generalization fornilpotent ideals of Un?
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 8 / 25
Generating function
Definition
Let Cd(x) :=∑
n≥0 Cdn x
n+1 for d ≥ 1, and let C 0(x) := x .
Let Fi (x) := i for i = 0, 1, and Fn(x) := Fn−1(x)− xFn−2(x), n ≥ 2.
Proposition (de Bruijn–Knuth–Rice 1972)
For n ≥ 1 we have Fn(x) =∑
0≤i≤(n−1)/2(n−1−i
i
)(−x)i .
Proposition (Kreweras 1970)
For d ≥ 1 we have Cd(x) =x
1− Cd−1(x)=
xFd+1(x)
Fd+2(x).
Example
C 1(x) = x1−x , C 2(x) = x
1− x1−x
= x(1−x)1−2x , C 3(x) = x
1− x1− x
1−x
= x(1−2x)1−3x+x2
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 9 / 25
Generating function
Definition
Let Cd(x) :=∑
n≥0 Cdn x
n+1 for d ≥ 1, and let C 0(x) := x .
Let Fi (x) := i for i = 0, 1, and Fn(x) := Fn−1(x)− xFn−2(x), n ≥ 2.
Proposition (de Bruijn–Knuth–Rice 1972)
For n ≥ 1 we have Fn(x) =∑
0≤i≤(n−1)/2(n−1−i
i
)(−x)i .
Proposition (Kreweras 1970)
For d ≥ 1 we have Cd(x) =x
1− Cd−1(x)=
xFd+1(x)
Fd+2(x).
Example
C 1(x) = x1−x , C 2(x) = x
1− x1−x
= x(1−x)1−2x , C 3(x) = x
1− x1− x
1−x
= x(1−2x)1−3x+x2
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 9 / 25
Generating function
Definition
Let Cd(x) :=∑
n≥0 Cdn x
n+1 for d ≥ 1, and let C 0(x) := x .
Let Fi (x) := i for i = 0, 1, and Fn(x) := Fn−1(x)− xFn−2(x), n ≥ 2.
Proposition (de Bruijn–Knuth–Rice 1972)
For n ≥ 1 we have Fn(x) =∑
0≤i≤(n−1)/2(n−1−i
i
)(−x)i .
Proposition (Kreweras 1970)
For d ≥ 1 we have Cd(x) =x
1− Cd−1(x)=
xFd+1(x)
Fd+2(x).
Example
C 1(x) = x1−x , C 2(x) = x
1− x1−x
= x(1−x)1−2x , C 3(x) = x
1− x1− x
1−x
= x(1−2x)1−3x+x2
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 9 / 25
Generating function
Definition
Let Cd(x) :=∑
n≥0 Cdn x
n+1 for d ≥ 1, and let C 0(x) := x .
Let Fi (x) := i for i = 0, 1, and Fn(x) := Fn−1(x)− xFn−2(x), n ≥ 2.
Proposition (de Bruijn–Knuth–Rice 1972)
For n ≥ 1 we have Fn(x) =∑
0≤i≤(n−1)/2(n−1−i
i
)(−x)i .
Proposition (Kreweras 1970)
For d ≥ 1 we have Cd(x) =x
1− Cd−1(x)=
xFd+1(x)
Fd+2(x).
Example
C 1(x) = x1−x , C 2(x) = x
1− x1−x
= x(1−x)1−2x , C 3(x) = x
1− x1− x
1−x
= x(1−2x)1−3x+x2
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 9 / 25
Generating function
Definition
Let Cd(x) :=∑
n≥0 Cdn x
n+1 for d ≥ 1, and let C 0(x) := x .
Let Fi (x) := i for i = 0, 1, and Fn(x) := Fn−1(x)− xFn−2(x), n ≥ 2.
Proposition (de Bruijn–Knuth–Rice 1972)
For n ≥ 1 we have Fn(x) =∑
0≤i≤(n−1)/2(n−1−i
i
)(−x)i .
Proposition (Kreweras 1970)
For d ≥ 1 we have Cd(x) =x
1− Cd−1(x)=
xFd+1(x)
Fd+2(x).
Example
C 1(x) = x1−x , C 2(x) = x
1− x1−x
= x(1−x)1−2x , C 3(x) = x
1− x1− x
1−x
= x(1−2x)1−3x+x2
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 9 / 25
Closed Formulas for C dn
Theorem (Andrews–Krattenthaler–Orsina–Papi 2002)
The number Cdn has the following closed formulas:
Cdn =
∑i∈Z
2i(d + 2) + 1
2n + 1
(2n + 1
n − i(d + 2)
)= det
[(i − j + d
j − i + 1
)]n−1i ,j=1
=∑
0=i0≤i1≤···≤id−1≤id=n
∏0≤j≤d−2
(ij+2 − ij − 1
ij+1 − ij
).
Theorem (de Bruijn–Knuth–Rice 1972)
The number of plane trees with n + 1 nodes of depth at most d is
Cdn =
22n+1
d + 2
∑1≤j≤d+1
sin2(jπ/(d + 2)) cos2n(jπ/(d + 2)).
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 10 / 25
Closed Formulas for C dn
Theorem (Andrews–Krattenthaler–Orsina–Papi 2002)
The number Cdn has the following closed formulas:
Cdn =
∑i∈Z
2i(d + 2) + 1
2n + 1
(2n + 1
n − i(d + 2)
)= det
[(i − j + d
j − i + 1
)]n−1i ,j=1
=∑
0=i0≤i1≤···≤id−1≤id=n
∏0≤j≤d−2
(ij+2 − ij − 1
ij+1 − ij
).
Theorem (de Bruijn–Knuth–Rice 1972)
The number of plane trees with n + 1 nodes of depth at most d is
Cdn =
22n+1
d + 2
∑1≤j≤d+1
sin2(jπ/(d + 2)) cos2n(jπ/(d + 2)).
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 10 / 25
Nonassociativity of binary operations
Fact
Let ∗ be a binary operation on a set X , and let x0, x1, . . . , xn beX -valued indeterminates.
There are Cn ways to parenthesize x0 ∗ x1 ∗ · · · ∗ xn.
The number C∗,n of distinct results from parenthesizations ofx0 ∗ x1 ∗ · · · ∗ xn satisfies 1 ≤ C∗,n ≤ Cn.
We have C∗,n = 1 for all n ≥ 0 if and only if ∗ is associative. So ingeneral, C∗,n measures the failure of ∗ to be associative.
Example (Subtraction, n = 3)
((x0−x1)−x2)−x3
= x0 − x1 − x2 − x3
(x0−(x1−x2))−x3
= x0 − x1 + x2 − x3
(x0−x1)−(x2−x3)
= x0 − x1 − x2 + x3
x0−((x1−x2)−x3)
= x0 − x1 + x2 + x3
x0−(x1−(x2−x3))
= x0 − x1 + x2 − x3
⇒{
C3 = 5
C−,3 = 4
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 11 / 25
Nonassociativity of binary operations
Fact
Let ∗ be a binary operation on a set X , and let x0, x1, . . . , xn beX -valued indeterminates.
There are Cn ways to parenthesize x0 ∗ x1 ∗ · · · ∗ xn.
The number C∗,n of distinct results from parenthesizations ofx0 ∗ x1 ∗ · · · ∗ xn satisfies 1 ≤ C∗,n ≤ Cn.
We have C∗,n = 1 for all n ≥ 0 if and only if ∗ is associative. So ingeneral, C∗,n measures the failure of ∗ to be associative.
Example (Subtraction, n = 3)
((x0−x1)−x2)−x3
= x0 − x1 − x2 − x3
(x0−(x1−x2))−x3
= x0 − x1 + x2 − x3
(x0−x1)−(x2−x3)
= x0 − x1 − x2 + x3
x0−((x1−x2)−x3)
= x0 − x1 + x2 + x3
x0−(x1−(x2−x3))
= x0 − x1 + x2 − x3
⇒{
C3 = 5
C−,3 = 4
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 11 / 25
Nonassociativity of binary operations
Fact
Let ∗ be a binary operation on a set X , and let x0, x1, . . . , xn beX -valued indeterminates.
There are Cn ways to parenthesize x0 ∗ x1 ∗ · · · ∗ xn.
The number C∗,n of distinct results from parenthesizations ofx0 ∗ x1 ∗ · · · ∗ xn satisfies 1 ≤ C∗,n ≤ Cn.
We have C∗,n = 1 for all n ≥ 0 if and only if ∗ is associative. So ingeneral, C∗,n measures the failure of ∗ to be associative.
Example (Subtraction, n = 3)
((x0−x1)−x2)−x3
= x0 − x1 − x2 − x3
(x0−(x1−x2))−x3
= x0 − x1 + x2 − x3
(x0−x1)−(x2−x3)
= x0 − x1 − x2 + x3
x0−((x1−x2)−x3)
= x0 − x1 + x2 + x3
x0−(x1−(x2−x3))
= x0 − x1 + x2 − x3
⇒{
C3 = 5
C−,3 = 4
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 11 / 25
Nonassociativity of binary operations
Fact
Let ∗ be a binary operation on a set X , and let x0, x1, . . . , xn beX -valued indeterminates.
There are Cn ways to parenthesize x0 ∗ x1 ∗ · · · ∗ xn.
The number C∗,n of distinct results from parenthesizations ofx0 ∗ x1 ∗ · · · ∗ xn satisfies 1 ≤ C∗,n ≤ Cn.
We have C∗,n = 1 for all n ≥ 0 if and only if ∗ is associative. So ingeneral, C∗,n measures the failure of ∗ to be associative.
Example (Subtraction, n = 3)
((x0−x1)−x2)−x3
= x0 − x1 − x2 − x3
(x0−(x1−x2))−x3
= x0 − x1 + x2 − x3
(x0−x1)−(x2−x3)
= x0 − x1 − x2 + x3
x0−((x1−x2)−x3)
= x0 − x1 + x2 + x3
x0−(x1−(x2−x3))
= x0 − x1 + x2 − x3
⇒{
C3 = 5
C−,3 = 4
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 11 / 25
Nonassociativity of binary operations
Fact
Let ∗ be a binary operation on a set X , and let x0, x1, . . . , xn beX -valued indeterminates.
There are Cn ways to parenthesize x0 ∗ x1 ∗ · · · ∗ xn.
The number C∗,n of distinct results from parenthesizations ofx0 ∗ x1 ∗ · · · ∗ xn satisfies 1 ≤ C∗,n ≤ Cn.
We have C∗,n = 1 for all n ≥ 0 if and only if ∗ is associative. So ingeneral, C∗,n measures the failure of ∗ to be associative.
Example (Subtraction, n = 3)
((x0−x1)−x2)−x3
= x0 − x1 − x2 − x3
(x0−(x1−x2))−x3
= x0 − x1 + x2 − x3
(x0−x1)−(x2−x3)
= x0 − x1 − x2 + x3
x0−((x1−x2)−x3)
= x0 − x1 + x2 + x3
x0−(x1−(x2−x3))
= x0 − x1 + x2 − x3
⇒{
C3 = 5
C−,3 = 4
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 11 / 25
Nonassociativity of binary operations
Fact
Let ∗ be a binary operation on a set X , and let x0, x1, . . . , xn beX -valued indeterminates.
There are Cn ways to parenthesize x0 ∗ x1 ∗ · · · ∗ xn.
The number C∗,n of distinct results from parenthesizations ofx0 ∗ x1 ∗ · · · ∗ xn satisfies 1 ≤ C∗,n ≤ Cn.
We have C∗,n = 1 for all n ≥ 0 if and only if ∗ is associative. So ingeneral, C∗,n measures the failure of ∗ to be associative.
Example (Subtraction, n = 3)
((x0−x1)−x2)−x3 = x0 − x1 − x2 − x3(x0−(x1−x2))−x3 = x0 − x1 + x2 − x3(x0−x1)−(x2−x3) = x0 − x1 − x2 + x3x0−((x1−x2)−x3) = x0 − x1 + x2 + x3x0−(x1−(x2−x3)) = x0 − x1 + x2 − x3
⇒{
C3 = 5
C−,3 = 4
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 11 / 25
Nonassociativity of binary operations
Fact
Let ∗ be a binary operation on a set X , and let x0, x1, . . . , xn beX -valued indeterminates.
There are Cn ways to parenthesize x0 ∗ x1 ∗ · · · ∗ xn.
The number C∗,n of distinct results from parenthesizations ofx0 ∗ x1 ∗ · · · ∗ xn satisfies 1 ≤ C∗,n ≤ Cn.
We have C∗,n = 1 for all n ≥ 0 if and only if ∗ is associative. So ingeneral, C∗,n measures the failure of ∗ to be associative.
Example (Subtraction, n = 3)
((x0−x1)−x2)−x3 = x0 − x1 − x2 − x3(x0−(x1−x2))−x3 = x0 − x1 + x2 − x3(x0−x1)−(x2−x3) = x0 − x1 − x2 + x3x0−((x1−x2)−x3) = x0 − x1 + x2 + x3x0−(x1−(x2−x3)) = x0 − x1 + x2 − x3
⇒{
C3 = 5
C−,3 = 4
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 11 / 25
A four-parameter generalization of C dn
Definition
Define a binary operation ⊕ on R := C[x , y ]/(xd+k − xd , y e+` − y e):
f ⊕ g := xf + yg ∀f , g ∈ R.
Let Cd ,ek,`,n := C⊕,n be the number of distinct results obtained by
inserting parentheses to the expression z0 ⊕ z1 ⊕ · · · ⊕ zn, wherez0, z1, . . . , zn are indeterminates taking values in R.
Example (n = 3, d = 2, e = k = ` = 1)
For R := C[x , y ]/(x2+1 − x2, y1+1 − y1) we have
((z0 ⊕ z1)⊕ z2)⊕ z3
= x3z0 + x2yz1 + xyz2 + yz3
(z0 ⊕ (z1 ⊕ z2))⊕ z3
= x2z0 + x2yz1 + xy2z2 + yz3
(z0 ⊕ z1)⊕ (z2 ⊕ z3)
= x2z0 + xyz1 + xyz2 + y2z3
z0 ⊕ ((z1 ⊕ z2)⊕ z3)
= xz0 + x2yz1 + xy2z2 + y2z3
z0 ⊕ (z1 ⊕ (z2 ⊕ z3))
= xz0 + xyz1 + xy2z2 + y3z3
⇒C5 = 5 = 1
n+1
(2nn
)C 2,11,1,3 = 4 = 2n−1
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 12 / 25
A four-parameter generalization of C dn
Definition
Define a binary operation ⊕ on R := C[x , y ]/(xd+k − xd , y e+` − y e):
f ⊕ g := xf + yg ∀f , g ∈ R.
Let Cd ,ek,`,n := C⊕,n be the number of distinct results obtained by
inserting parentheses to the expression z0 ⊕ z1 ⊕ · · · ⊕ zn, wherez0, z1, . . . , zn are indeterminates taking values in R.
Example (n = 3, d = 2, e = k = ` = 1)
For R := C[x , y ]/(x2+1 − x2, y1+1 − y1) we have
((z0 ⊕ z1)⊕ z2)⊕ z3
= x3z0 + x2yz1 + xyz2 + yz3
(z0 ⊕ (z1 ⊕ z2))⊕ z3
= x2z0 + x2yz1 + xy2z2 + yz3
(z0 ⊕ z1)⊕ (z2 ⊕ z3)
= x2z0 + xyz1 + xyz2 + y2z3
z0 ⊕ ((z1 ⊕ z2)⊕ z3)
= xz0 + x2yz1 + xy2z2 + y2z3
z0 ⊕ (z1 ⊕ (z2 ⊕ z3))
= xz0 + xyz1 + xy2z2 + y3z3
⇒C5 = 5 = 1
n+1
(2nn
)C 2,11,1,3 = 4 = 2n−1
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 12 / 25
A four-parameter generalization of C dn
Definition
Define a binary operation ⊕ on R := C[x , y ]/(xd+k − xd , y e+` − y e):
f ⊕ g := xf + yg ∀f , g ∈ R.
Let Cd ,ek,`,n := C⊕,n be the number of distinct results obtained by
inserting parentheses to the expression z0 ⊕ z1 ⊕ · · · ⊕ zn, wherez0, z1, . . . , zn are indeterminates taking values in R.
Example (n = 3, d = 2, e = k = ` = 1)
For R := C[x , y ]/(x2+1 − x2, y1+1 − y1) we have
((z0 ⊕ z1)⊕ z2)⊕ z3
= x3z0 + x2yz1 + xyz2 + yz3
(z0 ⊕ (z1 ⊕ z2))⊕ z3
= x2z0 + x2yz1 + xy2z2 + yz3
(z0 ⊕ z1)⊕ (z2 ⊕ z3)
= x2z0 + xyz1 + xyz2 + y2z3
z0 ⊕ ((z1 ⊕ z2)⊕ z3)
= xz0 + x2yz1 + xy2z2 + y2z3
z0 ⊕ (z1 ⊕ (z2 ⊕ z3))
= xz0 + xyz1 + xy2z2 + y3z3
⇒C5 = 5 = 1
n+1
(2nn
)C 2,11,1,3 = 4 = 2n−1
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 12 / 25
A four-parameter generalization of C dn
Definition
Define a binary operation ⊕ on R := C[x , y ]/(xd+k − xd , y e+` − y e):
f ⊕ g := xf + yg ∀f , g ∈ R.
Let Cd ,ek,`,n := C⊕,n be the number of distinct results obtained by
inserting parentheses to the expression z0 ⊕ z1 ⊕ · · · ⊕ zn, wherez0, z1, . . . , zn are indeterminates taking values in R.
Example (n = 3, d = 2, e = k = ` = 1)
For R := C[x , y ]/(x2+1 − x2, y1+1 − y1) we have
((z0 ⊕ z1)⊕ z2)⊕ z3 = x3z0 + x2yz1 + xyz2 + yz3(z0 ⊕ (z1 ⊕ z2))⊕ z3 = x2z0 + x2yz1 + xy2z2 + yz3(z0 ⊕ z1)⊕ (z2 ⊕ z3) = x2z0 + xyz1 + xyz2 + y2z3z0 ⊕ ((z1 ⊕ z2)⊕ z3) = xz0 + x2yz1 + xy2z2 + y2z3z0 ⊕ (z1 ⊕ (z2 ⊕ z3)) = xz0 + xyz1 + xy2z2 + y3z3
⇒C5 = 5 = 1
n+1
(2nn
)C 2,11,1,3 = 4 = 2n−1
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 12 / 25
A four-parameter generalization of C dn
Definition
Define a binary operation ⊕ on R := C[x , y ]/(xd+k − xd , y e+` − y e):
f ⊕ g := xf + yg ∀f , g ∈ R.
Let Cd ,ek,`,n := C⊕,n be the number of distinct results obtained by
inserting parentheses to the expression z0 ⊕ z1 ⊕ · · · ⊕ zn, wherez0, z1, . . . , zn are indeterminates taking values in R.
Example (n = 3, d = 2, e = k = ` = 1)
For R := C[x , y ]/(x2+1 − x2, y1+1 − y1) we have
((z0 ⊕ z1)⊕ z2)⊕ z3 = x3z0 + x2yz1 + xyz2 + yz3(z0 ⊕ (z1 ⊕ z2))⊕ z3 = x2z0 + x2yz1 + xy2z2 + yz3(z0 ⊕ z1)⊕ (z2 ⊕ z3) = x2z0 + xyz1 + xyz2 + y2z3z0 ⊕ ((z1 ⊕ z2)⊕ z3) = xz0 + x2yz1 + xy2z2 + y2z3z0 ⊕ (z1 ⊕ (z2 ⊕ z3)) = xz0 + xyz1 + xy2z2 + y3z3
⇒C5 = 5 = 1
n+1
(2nn
)C 2,11,1,3 = 4 = 2n−1
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 12 / 25
Binary trees
Fact
Parenthesizations of z0 ⊕ · · · ⊕ zn ↔ (full) binary trees with n + 1 leaves.
Example
0 12
3
01 2
3 0 1 2 3 0
1 23
01
2 3
l l l l l((z0⊕z1)⊕z2)⊕z3 (z0⊕(z1⊕z2))⊕z3 (z0⊕z1)⊕(z2⊕z3) z0⊕((z1⊕z2)⊕z3) z0⊕(z1⊕(z2⊕z3))
δ = (3, 2, 1, 0) δ = (2, 2, 1, 0) δ = (2, 1, 1, 0) δ = (1, 2, 1, 0) δ = (1, 1, 1, 0)ρ = (0, 1, 1, 1) ρ = (0, 1, 2, 1) ρ = (0, 1, 1, 2) ρ = (0, 1, 2, 2) ρ = (0, 1, 2, 3)
Observation
A parenthesization of z0 ⊕ · · · ⊕ zn corresponding to t ∈ Tn equals
xδ0(t)yρ0(t)z0 + · · ·+ xδn(t)yρn(t)znwhere the left depth δi (t) (or right depth ρi (t), resp.) of leaf i in t ∈ Tn isthe number of edges to the left (or right, resp.) in the unique path fromthe root of t down to i .
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 13 / 25
Binary trees
Fact
Parenthesizations of z0 ⊕ · · · ⊕ zn ↔ (full) binary trees with n + 1 leaves.
Example
0 12
3
01 2
3 0 1 2 3 0
1 23
01
2 3
l l l l l((z0⊕z1)⊕z2)⊕z3 (z0⊕(z1⊕z2))⊕z3 (z0⊕z1)⊕(z2⊕z3) z0⊕((z1⊕z2)⊕z3) z0⊕(z1⊕(z2⊕z3))
δ = (3, 2, 1, 0) δ = (2, 2, 1, 0) δ = (2, 1, 1, 0) δ = (1, 2, 1, 0) δ = (1, 1, 1, 0)ρ = (0, 1, 1, 1) ρ = (0, 1, 2, 1) ρ = (0, 1, 1, 2) ρ = (0, 1, 2, 2) ρ = (0, 1, 2, 3)
Observation
A parenthesization of z0 ⊕ · · · ⊕ zn corresponding to t ∈ Tn equals
xδ0(t)yρ0(t)z0 + · · ·+ xδn(t)yρn(t)znwhere the left depth δi (t) (or right depth ρi (t), resp.) of leaf i in t ∈ Tn isthe number of edges to the left (or right, resp.) in the unique path fromthe root of t down to i .
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 13 / 25
Binary trees
Fact
Parenthesizations of z0 ⊕ · · · ⊕ zn ↔ (full) binary trees with n + 1 leaves.
Example
0 12
3
01 2
3 0 1 2 3 0
1 23
01
2 3
l l l l l((z0⊕z1)⊕z2)⊕z3 (z0⊕(z1⊕z2))⊕z3 (z0⊕z1)⊕(z2⊕z3) z0⊕((z1⊕z2)⊕z3) z0⊕(z1⊕(z2⊕z3))
δ = (3, 2, 1, 0) δ = (2, 2, 1, 0) δ = (2, 1, 1, 0) δ = (1, 2, 1, 0) δ = (1, 1, 1, 0)ρ = (0, 1, 1, 1) ρ = (0, 1, 2, 1) ρ = (0, 1, 1, 2) ρ = (0, 1, 2, 2) ρ = (0, 1, 2, 3)
Observation
A parenthesization of z0 ⊕ · · · ⊕ zn corresponding to t ∈ Tn equals
xδ0(t)yρ0(t)z0 + · · ·+ xδn(t)yρn(t)znwhere the left depth δi (t) (or right depth ρi (t), resp.) of leaf i in t ∈ Tn isthe number of edges to the left (or right, resp.) in the unique path fromthe root of t down to i .
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 13 / 25
Associativity at left depth d
Definition (Associativity and Rotation)
A binary operation ∗ is associative if (a ∗ b) ∗ c = a ∗ (b ∗ c) alwaysholds.
The rotation for binary trees is given byr s t
↔r s t
.
Observation
If e = k = ` = 1 then two parenthesizations of z0 ⊕ · · · ⊕ zn are equal ifand only if their corresponding binary trees can be obtained from eachother by applying rotations at nodes of left depth at least d − 1.
Example
The first two binary trees are equivalent for ⊕ with d = 2, e = k = ` = 1.
0 12
3
01 2
3 0 1 2 3 0
1 23
01
2 3
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 14 / 25
Associativity at left depth d
Definition (Associativity and Rotation)
A binary operation ∗ is associative if (a ∗ b) ∗ c = a ∗ (b ∗ c) alwaysholds.
The rotation for binary trees is given byr s t
↔r s t
.
Observation
If e = k = ` = 1 then two parenthesizations of z0 ⊕ · · · ⊕ zn are equal ifand only if their corresponding binary trees can be obtained from eachother by applying rotations at nodes of left depth at least d − 1.
Example
The first two binary trees are equivalent for ⊕ with d = 2, e = k = ` = 1.
0 12
3
01 2
3 0 1 2 3 0
1 23
01
2 3
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 14 / 25
Associativity at left depth d
Definition (Associativity and Rotation)
A binary operation ∗ is associative if (a ∗ b) ∗ c = a ∗ (b ∗ c) alwaysholds.
The rotation for binary trees is given byr s t
↔r s t
.
Observation
If e = k = ` = 1 then two parenthesizations of z0 ⊕ · · · ⊕ zn are equal ifand only if their corresponding binary trees can be obtained from eachother by applying rotations at nodes of left depth at least d − 1.
Example
The first two binary trees are equivalent for ⊕ with d = 2, e = k = ` = 1.
0 12
3
01 2
3 0 1 2 3 0
1 23
01
2 3
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 14 / 25
Associativity at left depth d
Definition (Associativity and Rotation)
A binary operation ∗ is associative if (a ∗ b) ∗ c = a ∗ (b ∗ c) alwaysholds.
The rotation for binary trees is given byr s t
↔r s t
.
Observation
If e = k = ` = 1 then two parenthesizations of z0 ⊕ · · · ⊕ zn are equal ifand only if their corresponding binary trees can be obtained from eachother by applying rotations at nodes of left depth at least d − 1.
Example
The first two binary trees are equivalent for ⊕ with d = 2, e = k = ` = 1.
0 12
3
01 2
3 0 1 2 3 0
1 23
01
2 3
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 14 / 25
More on the number C dn
Theorem (Hein and H.)
For n, d ≥ 1 we have Cd ,11,1,n = Cd
n .
Definition
A composition of n is a sequence α = (α1, . . . , α`) of positive integers suchthat α1 + · · ·+ α` = n. Let max(α) := max{α1, . . . , α`} and `(α) = `.
Proposition (Hein and H.)
For n, d ≥ 1 we have
Cdn =
∑α|=n
max(α)≤(d+1)/2
(−1)n−`(α)(d − α1
α1 − 1
) ∏2≤i≤`(α)
(d + 1− αi
αi
)
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 15 / 25
More on the number C dn
Theorem (Hein and H.)
For n, d ≥ 1 we have Cd ,11,1,n = Cd
n .
Definition
A composition of n is a sequence α = (α1, . . . , α`) of positive integers suchthat α1 + · · ·+ α` = n. Let max(α) := max{α1, . . . , α`} and `(α) = `.
Proposition (Hein and H.)
For n, d ≥ 1 we have
Cdn =
∑α|=n
max(α)≤(d+1)/2
(−1)n−`(α)(d − α1
α1 − 1
) ∏2≤i≤`(α)
(d + 1− αi
αi
)
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 15 / 25
More on the number C dn
Theorem (Hein and H.)
For n, d ≥ 1 we have Cd ,11,1,n = Cd
n .
Definition
A composition of n is a sequence α = (α1, . . . , α`) of positive integers suchthat α1 + · · ·+ α` = n. Let max(α) := max{α1, . . . , α`} and `(α) = `.
Proposition (Hein and H.)
For n, d ≥ 1 we have
Cdn =
∑α|=n
max(α)≤(d+1)/2
(−1)n−`(α)(d − α1
α1 − 1
) ∏2≤i≤`(α)
(d + 1− αi
αi
)
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 15 / 25
Closed Formulas for C dn
Theorem (Andrews–Krattenthaler–Orsina–Papi 2002)
The number Cdn has the following closed formulas:
Cdn =
∑i∈Z
2i(d + 2) + 1
2n + 1
(2n + 1
n − i(d + 2)
)= det
[(i − j + d
j − i + 1
)]n−1i ,j=1
=∑
0=i0≤i1≤···≤id−1≤id=n
∏0≤j≤d−2
(ij+2 − ij − 1
ij+1 − ij
).
Theorem (de Bruijn–Knuth–Rice 1972)
The number of plane trees with n + 1 nodes of depth at most d is
Cdn =
22n+1
d + 2
∑1≤j≤d+1
sin2(jπ/(d + 2)) cos2n(jπ/(d + 2)).
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 16 / 25
Associativity at left depth d and right depth e
Observation
The number Cd ,en := Cd ,e
1,1,n enumerates equivalence classes of binarytrees with n + 1 leaves under rotations at left depth at least d − 1and right depth at least e − 1.
We have a symmetry in d and e: Cd ,en = C e,d
n .
Definition
Let Cd ,e(x) :=∑
n≥0 Cd ,en xn+1 for d , e ≥ 1. Let C 0,e(x) := C 1,e(x) and
Cd ,0(x) := Cd ,1(x).
Proposition (Hein and H.)
For d , e ≥ 1 we have
Cd ,e(x) = x + Cd−1,e(x)Cd ,e−1(x).
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 17 / 25
Associativity at left depth d and right depth e
Observation
The number Cd ,en := Cd ,e
1,1,n enumerates equivalence classes of binarytrees with n + 1 leaves under rotations at left depth at least d − 1and right depth at least e − 1.
We have a symmetry in d and e: Cd ,en = C e,d
n .
Definition
Let Cd ,e(x) :=∑
n≥0 Cd ,en xn+1 for d , e ≥ 1. Let C 0,e(x) := C 1,e(x) and
Cd ,0(x) := Cd ,1(x).
Proposition (Hein and H.)
For d , e ≥ 1 we have
Cd ,e(x) = x + Cd−1,e(x)Cd ,e−1(x).
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 17 / 25
Associativity at left depth d and right depth e
Observation
The number Cd ,en := Cd ,e
1,1,n enumerates equivalence classes of binarytrees with n + 1 leaves under rotations at left depth at least d − 1and right depth at least e − 1.
We have a symmetry in d and e: Cd ,en = C e,d
n .
Definition
Let Cd ,e(x) :=∑
n≥0 Cd ,en xn+1 for d , e ≥ 1. Let C 0,e(x) := C 1,e(x) and
Cd ,0(x) := Cd ,1(x).
Proposition (Hein and H.)
For d , e ≥ 1 we have
Cd ,e(x) = x + Cd−1,e(x)Cd ,e−1(x).
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 17 / 25
Associativity at left depth d and right depth e
Observation
The number Cd ,en := Cd ,e
1,1,n enumerates equivalence classes of binarytrees with n + 1 leaves under rotations at left depth at least d − 1and right depth at least e − 1.
We have a symmetry in d and e: Cd ,en = C e,d
n .
Definition
Let Cd ,e(x) :=∑
n≥0 Cd ,en xn+1 for d , e ≥ 1. Let C 0,e(x) := C 1,e(x) and
Cd ,0(x) := Cd ,1(x).
Proposition (Hein and H.)
For d , e ≥ 1 we have
Cd ,e(x) = x + Cd−1,e(x)Cd ,e−1(x).
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 17 / 25
The special case e = 2
Theorem (Hein and H.)
For d , n ≥ 2 we have
Cd ,2(x) = Cd(x) +xd+2
(1− 2x)Fd+2(x)and
C d,2n = C d
n +∑
1≤i≤n−d
2i−1∑
α|=n−d−imax(α)≤(d+1)/2
(−1)n−d−i−`(α)∏
1≤j≤`(α)
(d + 1− αj
αj
).
Corollary
For n ≥ 2 we have C 2,2n = (n + 2)2n−3 [OEIS sequence A045623].
For n ≥ 2 we have C 3,2n =
(1+√5
2
)2n−2+(1−√5
2
)2n−2− 2n−2 [OEIS
A142586].
For n ≥ 3 we have C 4,2n = 1 + 5 · 3n−3 − 2n−3 (not found in OEIS).
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 18 / 25
The special case e = 2
Theorem (Hein and H.)
For d , n ≥ 2 we have
Cd ,2(x) = Cd(x) +xd+2
(1− 2x)Fd+2(x)and
C d,2n = C d
n +∑
1≤i≤n−d
2i−1∑
α|=n−d−imax(α)≤(d+1)/2
(−1)n−d−i−`(α)∏
1≤j≤`(α)
(d + 1− αj
αj
).
Corollary
For n ≥ 2 we have C 2,2n = (n + 2)2n−3 [OEIS sequence A045623].
For n ≥ 2 we have C 3,2n =
(1+√5
2
)2n−2+(1−√5
2
)2n−2− 2n−2 [OEIS
A142586].
For n ≥ 3 we have C 4,2n = 1 + 5 · 3n−3 − 2n−3 (not found in OEIS).
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 18 / 25
The special case e = 2
Theorem (Hein and H.)
For d , n ≥ 2 we have
Cd ,2(x) = Cd(x) +xd+2
(1− 2x)Fd+2(x)and
C d,2n = C d
n +∑
1≤i≤n−d
2i−1∑
α|=n−d−imax(α)≤(d+1)/2
(−1)n−d−i−`(α)∏
1≤j≤`(α)
(d + 1− αj
αj
).
Corollary
For n ≥ 2 we have C 2,2n = (n + 2)2n−3 [OEIS sequence A045623].
For n ≥ 2 we have C 3,2n =
(1+√5
2
)2n−2+(1−√5
2
)2n−2− 2n−2 [OEIS
A142586].
For n ≥ 3 we have C 4,2n = 1 + 5 · 3n−3 − 2n−3 (not found in OEIS).
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 18 / 25
The special case e = 2
Theorem (Hein and H.)
For d , n ≥ 2 we have
Cd ,2(x) = Cd(x) +xd+2
(1− 2x)Fd+2(x)and
C d,2n = C d
n +∑
1≤i≤n−d
2i−1∑
α|=n−d−imax(α)≤(d+1)/2
(−1)n−d−i−`(α)∏
1≤j≤`(α)
(d + 1− αj
αj
).
Corollary
For n ≥ 2 we have C 2,2n = (n + 2)2n−3 [OEIS sequence A045623].
For n ≥ 2 we have C 3,2n =
(1+√5
2
)2n−2+(1−√5
2
)2n−2− 2n−2 [OEIS
A142586].
For n ≥ 3 we have C 4,2n = 1 + 5 · 3n−3 − 2n−3 (not found in OEIS).
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 18 / 25
The special case e = 2
Theorem (Hein and H.)
For d , n ≥ 2 we have
Cd ,2(x) = Cd(x) +xd+2
(1− 2x)Fd+2(x)and
C d,2n = C d
n +∑
1≤i≤n−d
2i−1∑
α|=n−d−imax(α)≤(d+1)/2
(−1)n−d−i−`(α)∏
1≤j≤`(α)
(d + 1− αj
αj
).
Corollary
For n ≥ 2 we have C 2,2n = (n + 2)2n−3 [OEIS sequence A045623].
For n ≥ 2 we have C 3,2n =
(1+√5
2
)2n−2+(1−√5
2
)2n−2− 2n−2 [OEIS
A142586].
For n ≥ 3 we have C 4,2n = 1 + 5 · 3n−3 − 2n−3 (not found in OEIS).
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 18 / 25
The k-associativity
Definition
We call Ck,n := C 1,1k,1,n the k-modular Catalan number.
k-associativity: (z0 ∗ · · · ∗ zk) ∗ zk+1 = z0 ∗ (z1 ∗ · · · ∗ zk+1)
k-rotation: (t0 ∧ · · · ∧ tk) ∧ tk+1 ↔ t0 ∧ (t1 ∧ · · · ∧ tk+1).
Proposition (Hein and H. 2017)
For d = e = ` = 1, the following statements are equivalent.
Two parenthesizations of z0 ⊕ z1 ⊕ · · · ⊕ zn are equal.
The corresponding binary trees can be obtained from each other byk-rotations.
The left depth sequences of the two trees are congruent modulo k .
Remark
We have two closed formulas for Ck,n and several restricted families ofCatalan objects enumerated by Ck,n [Hein and H. 2007].
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 19 / 25
The k-associativity
Definition
We call Ck,n := C 1,1k,1,n the k-modular Catalan number.
k-associativity: (z0 ∗ · · · ∗ zk) ∗ zk+1 = z0 ∗ (z1 ∗ · · · ∗ zk+1)
k-rotation: (t0 ∧ · · · ∧ tk) ∧ tk+1 ↔ t0 ∧ (t1 ∧ · · · ∧ tk+1).
Proposition (Hein and H. 2017)
For d = e = ` = 1, the following statements are equivalent.
Two parenthesizations of z0 ⊕ z1 ⊕ · · · ⊕ zn are equal.
The corresponding binary trees can be obtained from each other byk-rotations.
The left depth sequences of the two trees are congruent modulo k .
Remark
We have two closed formulas for Ck,n and several restricted families ofCatalan objects enumerated by Ck,n [Hein and H. 2007].
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 19 / 25
The k-associativity
Definition
We call Ck,n := C 1,1k,1,n the k-modular Catalan number.
k-associativity: (z0 ∗ · · · ∗ zk) ∗ zk+1 = z0 ∗ (z1 ∗ · · · ∗ zk+1)
k-rotation: (t0 ∧ · · · ∧ tk) ∧ tk+1 ↔ t0 ∧ (t1 ∧ · · · ∧ tk+1).
Proposition (Hein and H. 2017)
For d = e = ` = 1, the following statements are equivalent.
Two parenthesizations of z0 ⊕ z1 ⊕ · · · ⊕ zn are equal.
The corresponding binary trees can be obtained from each other byk-rotations.
The left depth sequences of the two trees are congruent modulo k .
Remark
We have two closed formulas for Ck,n and several restricted families ofCatalan objects enumerated by Ck,n [Hein and H. 2007].
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 19 / 25
The k-associativity
Definition
We call Ck,n := C 1,1k,1,n the k-modular Catalan number.
k-associativity: (z0 ∗ · · · ∗ zk) ∗ zk+1 = z0 ∗ (z1 ∗ · · · ∗ zk+1)
k-rotation: (t0 ∧ · · · ∧ tk) ∧ tk+1 ↔ t0 ∧ (t1 ∧ · · · ∧ tk+1).
Proposition (Hein and H. 2017)
For d = e = ` = 1, the following statements are equivalent.
Two parenthesizations of z0 ⊕ z1 ⊕ · · · ⊕ zn are equal.
The corresponding binary trees can be obtained from each other byk-rotations.
The left depth sequences of the two trees are congruent modulo k .
Remark
We have two closed formulas for Ck,n and several restricted families ofCatalan objects enumerated by Ck,n [Hein and H. 2007].
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 19 / 25
The k-associativity
Definition
We call Ck,n := C 1,1k,1,n the k-modular Catalan number.
k-associativity: (z0 ∗ · · · ∗ zk) ∗ zk+1 = z0 ∗ (z1 ∗ · · · ∗ zk+1)
k-rotation: (t0 ∧ · · · ∧ tk) ∧ tk+1 ↔ t0 ∧ (t1 ∧ · · · ∧ tk+1).
Proposition (Hein and H. 2017)
For d = e = ` = 1, the following statements are equivalent.
Two parenthesizations of z0 ⊕ z1 ⊕ · · · ⊕ zn are equal.
The corresponding binary trees can be obtained from each other byk-rotations.
The left depth sequences of the two trees are congruent modulo k .
Remark
We have two closed formulas for Ck,n and several restricted families ofCatalan objects enumerated by Ck,n [Hein and H. 2007].
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 19 / 25
The k-associativity
Definition
We call Ck,n := C 1,1k,1,n the k-modular Catalan number.
k-associativity: (z0 ∗ · · · ∗ zk) ∗ zk+1 = z0 ∗ (z1 ∗ · · · ∗ zk+1)
k-rotation: (t0 ∧ · · · ∧ tk) ∧ tk+1 ↔ t0 ∧ (t1 ∧ · · · ∧ tk+1).
Proposition (Hein and H. 2017)
For d = e = ` = 1, the following statements are equivalent.
Two parenthesizations of z0 ⊕ z1 ⊕ · · · ⊕ zn are equal.
The corresponding binary trees can be obtained from each other byk-rotations.
The left depth sequences of the two trees are congruent modulo k .
Remark
We have two closed formulas for Ck,n and several restricted families ofCatalan objects enumerated by Ck,n [Hein and H. 2007].
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 19 / 25
The k-associativity
Definition
We call Ck,n := C 1,1k,1,n the k-modular Catalan number.
k-associativity: (z0 ∗ · · · ∗ zk) ∗ zk+1 = z0 ∗ (z1 ∗ · · · ∗ zk+1)
k-rotation: (t0 ∧ · · · ∧ tk) ∧ tk+1 ↔ t0 ∧ (t1 ∧ · · · ∧ tk+1).
Proposition (Hein and H. 2017)
For d = e = ` = 1, the following statements are equivalent.
Two parenthesizations of z0 ⊕ z1 ⊕ · · · ⊕ zn are equal.
The corresponding binary trees can be obtained from each other byk-rotations.
The left depth sequences of the two trees are congruent modulo k .
Remark
We have two closed formulas for Ck,n and several restricted families ofCatalan objects enumerated by Ck,n [Hein and H. 2007].
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 19 / 25
Rotation and 2-rotation
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 20 / 25
The k-associativity at left depth d
Definition
Let Cdk,n := Cd ,1
k,1,n and Cdk (x) :=
∑n≥0 C
dk,nx
n+1.
Observation
If e = ` = 1 then two parenthesizations of z0 ⊕ · · · ⊕ zn are equal if andonly if their corresponding binary trees can be obtained from each other byapplying k-rotations at nodes of left depth at least d − 1.
Proposition (Hein and H.)
We have Cd+1k (x) = x
/ (1− Cd
k (x)).
Proposition (Hein and H.)
We have Cd2 (x) = Cd+1
1 (x) and Cd2,n = Cd+1
1,n .
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 21 / 25
The k-associativity at left depth d
Definition
Let Cdk,n := Cd ,1
k,1,n and Cdk (x) :=
∑n≥0 C
dk,nx
n+1.
Observation
If e = ` = 1 then two parenthesizations of z0 ⊕ · · · ⊕ zn are equal if andonly if their corresponding binary trees can be obtained from each other byapplying k-rotations at nodes of left depth at least d − 1.
Proposition (Hein and H.)
We have Cd+1k (x) = x
/ (1− Cd
k (x)).
Proposition (Hein and H.)
We have Cd2 (x) = Cd+1
1 (x) and Cd2,n = Cd+1
1,n .
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 21 / 25
The k-associativity at left depth d
Definition
Let Cdk,n := Cd ,1
k,1,n and Cdk (x) :=
∑n≥0 C
dk,nx
n+1.
Observation
If e = ` = 1 then two parenthesizations of z0 ⊕ · · · ⊕ zn are equal if andonly if their corresponding binary trees can be obtained from each other byapplying k-rotations at nodes of left depth at least d − 1.
Proposition (Hein and H.)
We have Cd+1k (x) = x
/ (1− Cd
k (x)).
Proposition (Hein and H.)
We have Cd2 (x) = Cd+1
1 (x) and Cd2,n = Cd+1
1,n .
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 21 / 25
The k-associativity at left depth d
Definition
Let Cdk,n := Cd ,1
k,1,n and Cdk (x) :=
∑n≥0 C
dk,nx
n+1.
Observation
If e = ` = 1 then two parenthesizations of z0 ⊕ · · · ⊕ zn are equal if andonly if their corresponding binary trees can be obtained from each other byapplying k-rotations at nodes of left depth at least d − 1.
Proposition (Hein and H.)
We have Cd+1k (x) = x
/ (1− Cd
k (x)).
Proposition (Hein and H.)
We have Cd2 (x) = Cd+1
1 (x) and Cd2,n = Cd+1
1,n .
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 21 / 25
The special case k = 3
Proposition (Hein and H.)
For d , n ≥ 0 the number Cd3,n enumerates (a) permutations of 1, 2, . . . , n
avoiding 321 and (d + 3)1̄(d + 4)2 · · · (d + 2) [Barcucci–DelLungo–Pergola–Pinzani 2000], and (b) certain lattice paths [Flajolet 1980].
Theorem (Hein and H.)
C d3 (x) =
2xFd+1(x)Fd+2(x)− xd − xd+1 + xd√
1− 2x − 3x2
2(Fd+2(x)2 − xd − xd+1)and
Cd3,n =
∑α|=n+1
h>1⇒αh≤d+1
− δα1,d
2+ (−1)α1−1
∑i+j=α1−1
(d − i
i
)(d + 1− j
j
)+
∑i+j=α1−d
(−3)i
2
( 12
i
)( 12
j
)
·∏h≥2
δαh,d+ (−1)αh−1
∑i+j=αh
(d + 1− i
i
)(d + 1− j
j
)
where δm,d := 1 if m ∈ {d , d + 1} or δm,d := 0 otherwise.
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 22 / 25
The special case k = 3
Proposition (Hein and H.)
For d , n ≥ 0 the number Cd3,n enumerates (a) permutations of 1, 2, . . . , n
avoiding 321 and (d + 3)1̄(d + 4)2 · · · (d + 2) [Barcucci–DelLungo–Pergola–Pinzani 2000], and (b) certain lattice paths [Flajolet 1980].
Theorem (Hein and H.)
C d3 (x) =
2xFd+1(x)Fd+2(x)− xd − xd+1 + xd√
1− 2x − 3x2
2(Fd+2(x)2 − xd − xd+1)and
Cd3,n =
∑α|=n+1
h>1⇒αh≤d+1
− δα1,d
2+ (−1)α1−1
∑i+j=α1−1
(d − i
i
)(d + 1− j
j
)+
∑i+j=α1−d
(−3)i
2
( 12
i
)( 12
j
)
·∏h≥2
δαh,d+ (−1)αh−1
∑i+j=αh
(d + 1− i
i
)(d + 1− j
j
)
where δm,d := 1 if m ∈ {d , d + 1} or δm,d := 0 otherwise.
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 22 / 25
The special case k = 3
Proposition (Hein and H.)
For d , n ≥ 0 the number Cd3,n enumerates (a) permutations of 1, 2, . . . , n
avoiding 321 and (d + 3)1̄(d + 4)2 · · · (d + 2) [Barcucci–DelLungo–Pergola–Pinzani 2000], and (b) certain lattice paths [Flajolet 1980].
Theorem (Hein and H.)
C d3 (x) =
2xFd+1(x)Fd+2(x)− xd − xd+1 + xd√
1− 2x − 3x2
2(Fd+2(x)2 − xd − xd+1)and
Cd3,n =
∑α|=n+1
h>1⇒αh≤d+1
− δα1,d
2+ (−1)α1−1
∑i+j=α1−1
(d − i
i
)(d + 1− j
j
)+
∑i+j=α1−d
(−3)i
2
( 12
i
)( 12
j
)
·∏h≥2
δαh,d+ (−1)αh−1
∑i+j=αh
(d + 1− i
i
)(d + 1− j
j
)
where δm,d := 1 if m ∈ {d , d + 1} or δm,d := 0 otherwise.
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 22 / 25
The special case d = 2
Theorem (Hein and H.)
For n ≥ 0 and k ≥ 1 we have
C 2k,n(x) = 1 +
∑1≤i≤n−1
i
n − i
∑0≤j≤(n−i−1)/k
(−1)j(n − i
j
)(2n − i − jk − 1
n
)
= 1 +∑
1≤i≤n−1
∑λ⊆(k−1)n−i
n − i − |λ|n − i
(n − |λ| − 1
n − |λ| − i
)mλ(1n−i ).
Proposition (Hein and H.)
For n ≥ 0 we have
C 22,n =
∑0≤j≤n
(n + j − 1
2j
)= F2n−1
(= C 3
1,n = C 3n
).
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 23 / 25
The special case d = 2
Theorem (Hein and H.)
For n ≥ 0 and k ≥ 1 we have
C 2k,n(x) = 1 +
∑1≤i≤n−1
i
n − i
∑0≤j≤(n−i−1)/k
(−1)j(n − i
j
)(2n − i − jk − 1
n
)
= 1 +∑
1≤i≤n−1
∑λ⊆(k−1)n−i
n − i − |λ|n − i
(n − |λ| − 1
n − |λ| − i
)mλ(1n−i ).
Proposition (Hein and H.)
For n ≥ 0 we have
C 22,n =
∑0≤j≤n
(n + j − 1
2j
)= F2n−1
(= C 3
1,n = C 3n
).
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 23 / 25
Questions
Conjecture
For all k , ` ≥ 1 and n ≥ 0, C 1,1k,`,n = C 1,1
k+`−1,1,n.
For all d , ` ≥ 1 and n ≥ 0, Cd ,11,`,n = Cd ,1
`,1,n = Cd`,n.
Observation
In general, C 2,21,3,n 6= C 2,2
3,1,n.
Problem
Study Cd ,ek,` if at most one of d , e, k , ` is 1 and others are at least 2.
Find other interpretations of the number Cd ,ek,`,n, using noncrossing
partitions, polygon triangulations, etc.
Find (q, t)-analogues of the number Cd ,ek,`,n.
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 24 / 25
Questions
Conjecture
For all k , ` ≥ 1 and n ≥ 0, C 1,1k,`,n = C 1,1
k+`−1,1,n.
For all d , ` ≥ 1 and n ≥ 0, Cd ,11,`,n = Cd ,1
`,1,n = Cd`,n.
Observation
In general, C 2,21,3,n 6= C 2,2
3,1,n.
Problem
Study Cd ,ek,` if at most one of d , e, k , ` is 1 and others are at least 2.
Find other interpretations of the number Cd ,ek,`,n, using noncrossing
partitions, polygon triangulations, etc.
Find (q, t)-analogues of the number Cd ,ek,`,n.
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 24 / 25
Questions
Conjecture
For all k , ` ≥ 1 and n ≥ 0, C 1,1k,`,n = C 1,1
k+`−1,1,n.
For all d , ` ≥ 1 and n ≥ 0, Cd ,11,`,n = Cd ,1
`,1,n = Cd`,n.
Observation
In general, C 2,21,3,n 6= C 2,2
3,1,n.
Problem
Study Cd ,ek,` if at most one of d , e, k , ` is 1 and others are at least 2.
Find other interpretations of the number Cd ,ek,`,n, using noncrossing
partitions, polygon triangulations, etc.
Find (q, t)-analogues of the number Cd ,ek,`,n.
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 24 / 25
Questions
Conjecture
For all k , ` ≥ 1 and n ≥ 0, C 1,1k,`,n = C 1,1
k+`−1,1,n.
For all d , ` ≥ 1 and n ≥ 0, Cd ,11,`,n = Cd ,1
`,1,n = Cd`,n.
Observation
In general, C 2,21,3,n 6= C 2,2
3,1,n.
Problem
Study Cd ,ek,` if at most one of d , e, k , ` is 1 and others are at least 2.
Find other interpretations of the number Cd ,ek,`,n, using noncrossing
partitions, polygon triangulations, etc.
Find (q, t)-analogues of the number Cd ,ek,`,n.
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 24 / 25
Questions
Conjecture
For all k , ` ≥ 1 and n ≥ 0, C 1,1k,`,n = C 1,1
k+`−1,1,n.
For all d , ` ≥ 1 and n ≥ 0, Cd ,11,`,n = Cd ,1
`,1,n = Cd`,n.
Observation
In general, C 2,21,3,n 6= C 2,2
3,1,n.
Problem
Study Cd ,ek,` if at most one of d , e, k , ` is 1 and others are at least 2.
Find other interpretations of the number Cd ,ek,`,n, using noncrossing
partitions, polygon triangulations, etc.
Find (q, t)-analogues of the number Cd ,ek,`,n.
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 24 / 25
Questions
Conjecture
For all k , ` ≥ 1 and n ≥ 0, C 1,1k,`,n = C 1,1
k+`−1,1,n.
For all d , ` ≥ 1 and n ≥ 0, Cd ,11,`,n = Cd ,1
`,1,n = Cd`,n.
Observation
In general, C 2,21,3,n 6= C 2,2
3,1,n.
Problem
Study Cd ,ek,` if at most one of d , e, k , ` is 1 and others are at least 2.
Find other interpretations of the number Cd ,ek,`,n, using noncrossing
partitions, polygon triangulations, etc.
Find (q, t)-analogues of the number Cd ,ek,`,n.
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 24 / 25
Thank you!
Jia Huang (UNK) Nilpotent ideals and Catalan numbers October 19, 2017 25 / 25
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