By Joyner W.D.

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In this chapter we present some basic notation and properties of permutations. Notation: We may denote a permutation f : T → T by a 2 × n array: f↔ 1 2 ... n f (1) f (2) ... f (n) 33 34 CHAPTER 3. PERMUTATIONS Example 36 The identity permutation, denoted by I, is the permutation which doesn’t do anything: 1 2 ... n 1 2 ... n I= Definition 37 Let ef (i) = #{j > i | f (i) > f (j)}, 1 ≤ i ≤ n − 1. Let swap(f ) = ef (1) + ... + ef (n − 1). We call this the swapping number (or length of the permutation f since it counts the number of times f swaps the inequality in i < j to f (i) > f (j).

Ar . Such a permutation is called cyclic. The number r is called the length of the cycle. We call two such cycles (a1 a2 ... ar ) and (b1 b2 ... , bt } are disjoint. 42 CHAPTER 3. PERMUTATIONS Lemma 51 If f and g are disjoint cyclic permutations of T then f g = gf . proof: This is clear since the permutations f and g of T affect disjoint collections of integers, so the permutations may be performed in either order. ✷ Lemma 52 The cyclic permutation (a1 a2 ... ar ) has order r. , f r¡1 (a1 ) = ar , f r (a1 ) = a1 , by definition of f .

2. ) Let X denote rotation clockwise by 90 degrees of the face labeled x, where x ∈ {r, l, f, b, u, d} (so, for example, if x = f then X = F ). Use the cycle notation to determine the permutations of the facets given by (a) R, (b) L, (c) F , (d) B, (e) U , (f ) D. Lemma 55 A cyclic permutation is even if and only if the length of its cycle is odd. A general permutation f : T → T is odd if and only if the number of cycles of even length in its cycle decomposition is odd. 46 CHAPTER 3. PERMUTATIONS We shall not prove this here.