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Latin Rectangles - Combinatorics - Exam, Exams of Mathematics

These are the notes of Exam and Its key important points are: Latin Rectangles, Addition Table, Orthogonal Mate, Recurrence Relation, Formula, Conjectured Formula, Examining, Extraordinary, Linear Recurrence Relation, Primes

Typology: Exams

2012/2013

Uploaded on 02/12/2013

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Combinatorics - MATH 0345
Exam 3
May 18, 2009
Name:
Honor Code Pledge
Signature
Directions: Please complete all but one problem.
1. Prove that the addition table of Z4is a Latin square without an orthogonal mate.
2. How many 2 by nLatin rectangles have first row equal to
0 1 2 · · · n1 ?
Estimate how many 3 by nrectangles have this as their first row.
3. Solve the recurrence relation hn=hn1+ 1,(n1); h0= 0,by examining the first
few values for a formula and then proving your conjectured formula by induction.
4. Call a subset Sof the integers {1,2, . . . , n}extraordinary provided its smallest integer
equals its size:
min{x:xS}=|S|.
For example, S={3,7,8}is extraordinary. Let gnbe the number of extraordinary
subsets of {1,2, . . . , n}. Determine a linear recurrence relation for gn.
5. Let p1p2. . . pkbe primes and n=p1×p2×. . . ×pk. Prove that N(n), the
largest number of MOLS of order n, is at least p11.
6. Solve the recurrence relation hn= 8hn116hn2,(n2) with initial value h0=1
and h1= 0.
1

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Combinatorics - MATH 0345

Exam 3

May 18, 2009

Name: Honor Code Pledge

Signature Directions: Please complete all but one problem.

  1. Prove that the addition table of Z 4 is a Latin square without an orthogonal mate.
  2. How many 2 − by − n Latin rectangles have first row equal to

0 1 2 · · · n − 1?

Estimate how many 3 − by − n rectangles have this as their first row.

  1. Solve the recurrence relation hn = −hn− 1 + 1, (n ≥ 1); h 0 = 0, by examining the first few values for a formula and then proving your conjectured formula by induction.
  2. Call a subset S of the integers { 1 , 2 ,... , n} extraordinary provided its smallest integer equals its size: min{x : x ∈ S} = |S|. For example, S = { 3 , 7 , 8 } is extraordinary. Let gn be the number of extraordinary subsets of { 1 , 2 ,... , n}. Determine a linear recurrence relation for gn.
  3. Let p 1 ≤ p 2 ≤... ≤ pk be primes and n = p 1 × p 2 ×... × pk. Prove that N (n), the largest number of MOLS of order n, is at least p 1 − 1.
  4. Solve the recurrence relation hn = 8hn− 1 − 16 hn− 2 , (n ≥ 2) with initial value h 0 = − 1 and h 1 = 0.