Rational Catalan Combinatorics 2

104 Pages·2012·1.17 MB·English

Rational Catalan Combinatorics 2
Drew Armstrong et al.
UniversityofMiami
www.math.miami.edu/∼armstrong
June 19, 2012
This talk will further advertise a definition.
Here is it.
Definition
Let 0<a<b be coprime and consider x =a/(b−a)∈Q.
Then we define the Catalan number
(cid:18) (cid:19)
1 a+b (a+b−1)!
Cat(x):= = .
a+b a,b a!b!
!
Please note the a,b-symmetry.
This talk will further advertise a definition.
Here is it.
Definition
Let 0<a<b be coprime and consider x =a/(b−a)∈Q.
Then we define the Catalan number
(cid:18) (cid:19)
1 a+b (a+b−1)!
Cat(x):= = .
a+b a,b a!b!
!
Please note the a,b-symmetry.
This talk will further advertise a definition.
Here is it.
Definition
Let 0<a<b be coprime and consider x =a/(b−a)∈Q.
Then we define the Catalan number
(cid:18) (cid:19)
1 a+b (a+b−1)!
Cat(x):= = .
a+b a,b a!b!
!
Please note the a,b-symmetry.
Special cases.
When b = 1 mod a...
(cid:73) Eug`ene Charles Catalan (1814-1894)
(a<b)=(n<n+1) gives the good old Catalan number
(cid:16)n(cid:17) 1 (cid:18)2n+1(cid:19)
Cat(n)=Cat = .
1 2n+1 n
(cid:73) Nicolaus Fuss (1755-1826)
(a<b)=(n<kn+1) gives the Fuss-Catalan number
(cid:18) (cid:19) (cid:18) (cid:19)
n 1 (k +1)n+1
Cat = .
(kn+1)−n (k +1)n+1 n
Special cases.
When b = 1 mod a...
(cid:73) Eug`ene Charles Catalan (1814-1894)
(a<b)=(n<n+1) gives the good old Catalan number
(cid:16)n(cid:17) 1 (cid:18)2n+1(cid:19)
Cat(n)=Cat = .
1 2n+1 n
(cid:73) Nicolaus Fuss (1755-1826)
(a<b)=(n<kn+1) gives the Fuss-Catalan number
(cid:18) (cid:19) (cid:18) (cid:19)
n 1 (k +1)n+1
Cat = .
(kn+1)−n (k +1)n+1 n
Special cases.
When b = 1 mod a...
(cid:73) Eug`ene Charles Catalan (1814-1894)
(a<b)=(n<n+1) gives the good old Catalan number
(cid:16)n(cid:17) 1 (cid:18)2n+1(cid:19)
Cat(n)=Cat = .
1 2n+1 n
(cid:73) Nicolaus Fuss (1755-1826)
(a<b)=(n<kn+1) gives the Fuss-Catalan number
(cid:18) (cid:19) (cid:18) (cid:19)
n 1 (k +1)n+1
Cat = .
(kn+1)−n (k +1)n+1 n
Euclidean Algorithm & Symmetry.
Definition
Again let x =a/(b−a) for 0<a<b coprime.
Then we define the derived Catalan number
(cid:18) (cid:19) (cid:40)
1 b Cat(1/(x −1)) if x >1
Cat(cid:48)(x):= =
b a Cat(x/(1−x)) if x <1
This is a “categorification” of the Euclidean algorithm.
Remark
If we define Cat:Q\[−1,0]→N by Cat(−x −1):=Cat(x) then the
formula is simpler:
Cat(cid:48)(x)=Cat(1/(x −1))=Cat(x/(1−x)).
Euclidean Algorithm & Symmetry.
Definition
Again let x =a/(b−a) for 0<a<b coprime.
Then we define the derived Catalan number
(cid:18) (cid:19) (cid:40)
1 b Cat(1/(x −1)) if x >1
Cat(cid:48)(x):= =
b a Cat(x/(1−x)) if x <1
This is a “categorification” of the Euclidean algorithm.
Remark
If we define Cat:Q\[−1,0]→N by Cat(−x −1):=Cat(x) then the
formula is simpler:
Cat(cid:48)(x)=Cat(1/(x −1))=Cat(x/(1−x)).
Euclidean Algorithm & Symmetry.
Definition
Again let x =a/(b−a) for 0<a<b coprime.
Then we define the derived Catalan number
(cid:18) (cid:19) (cid:40)
1 b Cat(1/(x −1)) if x >1
Cat(cid:48)(x):= =
b a Cat(x/(1−x)) if x <1
This is a “categorification” of the Euclidean algorithm.
Remark
If we define Cat:Q\[−1,0]→N by Cat(−x −1):=Cat(x) then the
formula is simpler:
Cat(cid:48)(x)=Cat(1/(x −1))=Cat(x/(1−x)).