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MAST90030: Advanced Discrete Mathematics
Assignment 4
Question 1. (50 marks)
For n, t ∈ N, let M be an n × n matrix with entries 
given by
for i, j ∈ [n].
(a) Describe a set of non-intersecting 3-tuples of binomial paths which are enumerated by det M in the t = 3 case. Sketch all of these 3-tuples. (20 marks)
(b) You are told that det 
for some values a, b ∈ N. Conjecture how a and b depend on n and t. (10 marks)
(c) Describe a bijection from which you then prove the identity (you don’t have to prove that your map is actually a bijection). (20 marks)
Question 2. (50 marks)
For n ∈ N0, define
(a) By evaluating f (n) explicitly for a few small values of n, conjecture a simple function of n which equals f (n). (10 marks)
(b) Define a signed set Ω for which ||Ω|| = f (n).
Hint: 
is the number of Fibonacci pavings of an n-board using exactly k dimers, and each factor of 2 can be interpreted as arising from colouring certain pavers in two different ways (such as red and blue). (10 marks)
(c) Define a sign-reversing involution on your signed set Ω (you don’t have to prove it is sign-reversing). (15 marks)
(d) Describe the fixed point set of your sign-reversing involution, and determine its cardinality. Indicate how this proves the identity you conjectured in part (a). (15 marks)