Download Document

Pascal
Diagonal
Summation:
combinatorial proofs
Dr Gerard Tel
Informatics
Utrecht Univ.
Pascal triangle
Summation boxes
 Additive:
C(n,k) =
C(n-1,k) + C(n-1,k-1)
 Sum of Row n is
Twice Rowsum n-1:
2n

Combinatorial Coefficients
Combinatorial Coefficient
 Subsets k-of-n
 Multiplicative:
C(n,k) = n! / k! (n-k)!

Diagonal Sums

Fibonacci Number
Morse codes
Dash has duration of two dots
 Duration 1:
●
E
 Duration 2:
●● I
▬ T
 Duration 3:
●●● S
●▬ A
▬● N

Morse codes of duration 4
●●●●
H

●●▬
U

There are 5,
3 start with ●
●▬●
R

2 start with ▬
▬ ●●
▬▬
D
M
Morse codes of duration 5
Starting with ●
●●●●●
5
●●●▬
V
●●▬●
F
●▬●●
L
●▬ ▬
W
Starting with ▬
▬●●●
▬●▬
▬ ▬●
B
K
G
Together 8 codes.
Connection with
Fibonacci numbers?
Morse codes of duration 6
Starting with ●
●●●●●●
●●●●▬
4
●●●▬●
roger
●●▬●●
É
●●▬ ▬
Ü
●▬●●●
●▬●▬
●▬ ▬●
wait
Ä
P
Starting with ▬
▬●●●●
▬●●▬
▬●▬●
▬ ▬●●
▬▬▬
6
X
C
Z
O
Together 13 codes.
●▬ ▬ ▬
▬ ▬● ▬
▬ ●▬ ▬
J
Q
Y
Conclusions
• Morse numbers are
Fibonacci numbers
• Combinatorial arguments
look easy but are hard
• May increase intuition
• Summation: Poly(k) C(n,k)
with Absorption Rule
(Mathematica)