By René Schoof

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**Extra resources for Algebra 2. The symmetric groups Sn**

**Sample text**

1 · 2 · · · · · n. Here as a convention, 0! := 1. 4. COMBINATORICS 25 2) Arrangements An arrangement of n elements taken k at a time is an ordered arrangement of k elements from n given ones. The number of arrangements of n taken k is Akn = n! = n(n − 1) · · · (n − k + 1), 0 ≤ k ≤ n. (n − k)! 3) Combinations A combination of n elements taken k at a time is an arrangement of k elements from n given ones. The numbers of combinations of n taken k is n k Cnk := = n! , 0 ≤ k ≤ n. (n − k)! The following hold: • n k = n n−k .

9 (1975) Find all real x which satisfy 3 x3 + m3 x3 + n3 x3 + p3 3 (x − m)(x − n)(x − p) = . 10 (1976) Find all integer solutions of the system xx+y = y 12 , y y+x = x3 . 11 (1976) Let k and n be positive integers and x1 , . . , xk positive real numbers satisfying x1 + · · · + xk = 1. Prove that −n n+1 . x−n 1 + · · · + xk ≥ k CHAPTER 3. 13 1 − x 1− x−1 1 > . x x (1977) Consider real numbers a0 , a1 , . . , an+1 that satisfy a0 = an+1 = 0, |ak−1 − 2ak + ak+1 | ≤ 1 (k = 1, . . , n). 14 k(n − k + 1) , ∀k = 0, 1, .

CHAPTER 2. 4 Graph 1) Deﬁnitions A graph is a set of a ﬁnite number of points called vertices and links connecting some pairs of vertices called edges. Vertices of a graph is usually denoted by A1 , . . , An , while its edges denoted by u1 , . . , um ; each edge u connecting two vertices Ai and Aj is denoted by u = Ai Aj . A edge u = Ai Aj is called a circuit if Ai ≡ Aj . Two or more edges connecting the same pair of vertices are called multiple edges. A single graph is a graph having neither circuits nor multiple edges.

### Algebra 2. The symmetric groups Sn by René Schoof

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