Equations of Cobinatorics

[lottoarchitect]

Several current WR wheels come from existing ones by combining smaller covering designs. There are equations that define the way this can be achieved ("by-hand" construction). This contrasts computer programs that approach the construction of a wheel using linear / greedy / hill-climbing / simulated annealing or other techniques.

A covering is declared as C(v,k,t,m,L,=b) where
v=total numbers in wheel
k=block size (size of each ticket)
t=matches/guarantee
m=correct numbers drawn within the v set
L=1 for minimal wheels (omitted)
b=total tickets needed

Also, a t-design, t-(v,k,L,=b) or (v,k,t,L)=b is a covering with the special property of t=m and each combination covered only once (L=1) and it is minimal of course.

I have found the following equations regarding how to mix wheels together.

Main theorem:
C(v1,k,t,m1,L,=b1) + C(v2,k,t,m2,L,=b2) = C(v1+v2,k,t,m1+m2-1,L,=b1+b2)

Also, for t-designs, the following apply:

C(v+2,k,t)<=C(v,k,t)+C(v,k-2,t-1)
C(v+1,k+1,t)<=C(v,k,t)
C(v+1,k,t)<=C(v,k,t)+C(v,k-1,t-1)
C(v-1,k,t)<=C(v,k,t)

If you know of other such equations, feel free to post them.

cheers

 

[thornc]

?? forgive my ignorance... what is exactly your request??

 

[lottoarchitect]

These equations are mathematically proven and can be applied to design new coverings. For example

C(75,5,2,4)<=3*C(25,5,2,2)<=3*30=90

which uses the main theorem, thus we can construct the above covering using a smaller one. I ask if there are more such equations which I was not able to find.

 

[Grandmaster]

Lottoarchitect,

In my archives I have some pdf files that could help you a lot searching for equations of Combinatorics..


I have the full pdf article "Lotto Design Tables", with contains many theorems.


Abstract:
An LD(n, k, p, t; b) lotto design is a set of b k-sets (blocks) of an
n-set such that any p-set intersects at least one k-set in t or more
elements. Let L(n, k, p, t) denote the minimum number of blocks in
any LD(n, k, p, t; b) lotto design. We will list the known lower and
upper bound theorems for lotto designs. Since many of these bounds
are recursive, we will incorporate this information in a set of tables for
lower and upper bounds for lotto designs with small parameters. We
will also use back-track algorithms, greedy algorithms and simulated
annealing to improve the tables.

Also I have the full "Handbook of Combinatorial designs", this pdf file is a bible of covering designs.

 

[lottoarchitect]

Lottoarchitect,

In my archives I have some pdf files that could help you a lot searching for equations of Combinatorics..


I have the full pdf article "Lotto Design Tables", with contains many theorems.


Abstract:
An LD(n, k, p, t; b) lotto design is a set of b k-sets (blocks) of an
n-set such that any p-set intersects at least one k-set in t or more
elements. Let L(n, k, p, t) denote the minimum number of blocks in
any LD(n, k, p, t; b) lotto design. We will list the known lower and
upper bound theorems for lotto designs. Since many of these bounds
are recursive, we will incorporate this information in a set of tables for
lower and upper bounds for lotto designs with small parameters. We
will also use back-track algorithms, greedy algorithms and simulated
annealing to improve the tables.

Also I have the full "Handbook of Combinatorial designs", this pdf file is a bible of covering designs.

That's wonderful Grandmaster! Where can I find that handbook?

 

[Grandmaster]

LA,

The Handbook of Combinatorial designs has a cost of $129.95, I have one in my archives.

 

[Grandmaster]

<deleted>

 

[lottoarchitect]

Ok thanks for the information. I'll check this out