By Dr. Wilfred Funk, Norman Lewis
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Example text
Then ∑A∈π |LA | = 2 (q + q + 1)(q + 1). In this sum each line was counted q + 1 times, once for each of its points. Therefore, |L | = q2 + q + 1. The axioms of the finite projective plane (projective space of dimension 2) can easily be extended to alow for higher dimensional projective spaces. Projective space of dimension d and order q is denoted by PG(d, q). So, projective plane of order q is PG(2, q) and in particular the Fano plane is just PG(2, 2). g. 15), qd+1 − 1 or |PG(d, q)| = . , it was shows only recently that a PG(2, 10) does not exist (a long computation by Lam, Swierz, Thiel in 1989).
Fn (m1 , m2 , . . , mn ). mn . Show that m1 · Fn−1 (m2 − 1, . . 3. 5. 4. Let Q = {1, 2, . . , mn} where m, n 2. Let {A1 , . . , An } be a partition of Q into n blocks of size m, and let {B1 , . . , Bn } be another partition of Q into n blocks of size m. Show that there is a permutation f of {1, . . , n} such that Ai ∩ B f (i) = ∅ for all i. 5. 10. 6. Find all doubly standard Latin squares of order 4. 7. Let λr×n denote the number of distinct Latin r × n rectangles on an n element set. Show that r−1 ∏(n − k)!
C) If ϕ : Q → Q is a bijection, then ϕ (R) is a Latin rectangle. 2. LATIN SQUARES 45 Let Q = {a1 , . . , an } be a set of integers and let a1 < . . < an . A Latin square over Q is said to be standard if the elements of the first row of the square are linearly ordered. It is said to be doubly standard if the elements of the first row and of the first column of the square are linearly ordered. Fig. 5 show a standard and a doubly standard Latin square of order 5. 10 (a) Every Latin square can be turned into a standard or a doubly standard Latin square by permuting rows and columns of the original square.