Title: Theory / Unimportant / Math / Combinatorics / Aigner / 3 Catalan+5=
Tags: récupéré hasSeenConjDef

Definition: \(a_{0,0}=1\)
Page: 289
Chapter: 7
Variables: \(\mathbb F\) a field of characteristic 0
\(\sigma=(s_0,\dots)\in\mathbb F^{\mathbb N}\)
\(\tau=(t_1,\dots)\mathbb F^{\mathbb N}\) with \(t_i\ne 0\) for all \(i\)
Notation: \(A^{\sigma,\tau}\)
Definition2: \(a_{0,k}=0\) for \(k>0\)
Conjdef: The \(\mathbb{N\times N}\) matrix with
Definition3: \(a_{n,k}=a_{n-1,k-1}+s_ka_{n-1,k}+t_{k+1}a_{n-1,k+1}\)
Typ: Lower triangular

Name: A Catalan matrix
Definition: A matrix of the form \(A^{\sigma,\tau}\).
Page: 291, 292
Chapter: 7
Section: 1
Variables: \(\mathbb F\) a field of characteristic 0
Index: , 2
Kind: , Prop
Definition2: Lower triangular matrix such that \(ATA^T=H\) for:
-\(T\) diagonal, with non-0 on the diagonal and \(T_0=1\),
-\(H\) an Hankel matrix

Name: The Catalan sequence associated with \(A^{\sigma,\tau}\)
Definition: \((a_{n,0})_{n\in\mathbb N}\)
Page: 291
Chapter: 7
Section: 1
Variables: \(\mathbb F\) a field of characteristic 0
\(A^{\sigma,\tau}\) a Catalan matrix
Denoted by: \(B\)

Name: A Catalan sequence
Definition: A Catalan sequence associated with some \(A^{\sigma,\tau}\)
Page: 291
Chapter: 7
Section: 1
Variables: \(\mathbb F\) a field of characteristic 0

Smaller0: \(a_{n+1,n}\)
Greater0: \(\sum_{i=0}^ns_i\)
Page: 291
Chapter: 7
Section: 1
Variables: \(\mathbb F\) a field of characteristic 0
\(\sigma=(s_0,\dots)\in\mathbb F^{\mathbb N}\)
\(\tau=(t_1,\dots)\in\mathbb F^{\mathbb N}\) with \(t_i\ne 0\) for all \(i\)
\(A=A^{\sigma,\tau}\)
Smaller to greater: equals
Context: Catalan sequence

Definition: \(a_{n,0}\)
Page: 291
Chapter: 7
Section: 1
Variables: \(\mathbb F\) a field of characteristic 0
\(\sigma=(s_0,\dots)\in\mathbb F^{\mathbb N}\)
\(\tau=(t_1,\dots)\mathbb F^{\mathbb N}\) with \(t_i\ne 0\) for all \(i\)
\(A=A^{\sigma,\tau}\)
\(n\in\mathbb N\)
Notation: \(B_n\)
Denoted by: \(B_n^{\sigma,\tau}\)

Definition: \(\prod_{i=1}^n t_i\)
Page: 291
Chapter: 7
Section: 1
Variables: \(\mathbb F\) a field of characteristic 0
\(\sigma=(s_0,\dots)\in\mathbb F^{\mathbb N}\)
\(\tau=(t_1,\dots)\mathbb F^{\mathbb N}\) with \(t_i\ne 0\) for all \(i\)
\(A=A^{\sigma,\tau}\)
\(n\in\mathbb N\)
Notation: \(T_n\)

Smaller0: \(\sum_{k\ge0}a_{m,k}a_{n,k}T_k\)
Greater0: \(B_{m+n}\)
Page: 291
Chapter: 7
Section: 1
Variables: \(\mathbb F\) a field of characteristic 0
\(\sigma=(s_0,\dots)\in\mathbb F^{\mathbb N}\)
\(\tau=(t_1,\dots)\mathbb F^{\mathbb N}\) with \(t_i\ne 0\) for all \(i\)
\(A=A^{\sigma,\tau}\)
\(m,n\in\mathbb N\)
Smaller to greater: =
Name: Fundamental lemma for Catalan matrices

Definition: The diagonal matrix whose entries are \((T_i)_i\)
Page: 292
Chapter: 7
Section: 1
Variables: \(\mathbb F\) a field of characteristic 0
\(\sigma=(s_0,\dots)\in\mathbb F^{\mathbb N}\)
\(\tau=(t_1,\dots)\mathbb F^{\mathbb N}\) with \(t_i\ne 0\) for all \(i\)
\(A=A^{\sigma,\tau}\)
\(m,n\in\mathbb N\)
Context: Catalan Matrices
Notation: \(T\)

Name: The Hankel matrix of the Catalan numbers \(B_n\)
Definition: The matrix whose \((i,j)\)-th entry is \(B_{i+j}\)
Page: 292
Chapter: 7
Section: 1
Variables: \(\mathbb F\) a field of characteristic 0
\(\sigma=(s_0,\dots)\in\mathbb F^{\mathbb N}\)
\(\tau=(t_1,\dots)\mathbb F^{\mathbb N}\) with \(t_i\ne 0\) for all \(i\)
\(A=A^{\sigma,\tau}\)
\(m,n\in\mathbb N\)
Context: Catalan Matrices
Notation: \(H\)
Definition2: \(ATA^T\) with \(A^T\) the transposef

Smaller0: \(ATA^T=H\) with \(H\) Henkel, \(T\) diagonal
Greater0: \(\frac{T_k}{T_k-1}\)
Page: 292
Chapter: 7
Section: 1
Variables: \(\mathbb F\) a field of characteristic 0
\(\sigma=(s_0,\dots)\in\mathbb F^{\mathbb N}\)
\(\tau=(t_1,\dots)\mathbb F^{\mathbb N}\) with \(t_i\ne 0\) for all \(i\)
\(A=A^{\sigma,\tau}\)
\(n\in\mathbb N\)
Index: 2
Kind: Prop
Prefix Greater: \(t_k=\)
Greater1: \(a_{k+1,k}-a_{k,k-1}\)
Connect Greater0: and \(s_k=\)