1. Find the cyclic index of automorphism group of the finite projective plane with 7 vertex.
2. Let ? be a permutation on n objects. Define ?k ? i (?? ??id ).
a. Let an be the number of permutation on n objects with ?3 ? i or ?4 ? i . Find the function generator ?anxn .
b. Let bn be the number of permutation on n objects with ?k ? i when k is odd. Find ?bnxn .
c. By b, find bn .
d. Find bn directly.
3. Prove: for all k?r?0 there is n0(k,r) which for all n?n0(k,r) if F ??? ?[nk] ???
is family of sets so for all S,T ?F : S?T ?r , then F ?? ??kn??r r??? (hint: first
prove that this is saved under shifting).
4. Find integer n so that for all counting on the cube 2[n] with seven colors:
a. There are 3 different sets AB, , A?B with same color.
b. There are 3 different sets A,B, A?B with same color (
A?B ? (A B)?(B A)).