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1 -Enumerative Combinatorics - July 31, 2006
2 CONTENTS
3 CONTENTS e
4 1. A. B. C. D.
5 2. a, b, c (ab)c a(bc)
6 (ab)c a(bc)
7 a, b, c, d ((ab)c)d (a(bc))d a((bc)d) a(b(cd)) (ab)(cd)
8 3. (Catalan number) C n = n C 1 = 1, C 2 = 1, C 3 = 2, C 4 = 5, C 5 = 14, C 6 = 42, C 7 = 132, C 8 =
9 Eugène Charles Catalan ( )
10 4. * (ab)c = ab c a(bc) = abc (ab)(cd) = ab cd
11 (ab)c = ab c a b c a(bc) = abc a b c (ab)(cd) = ab cd a b c d
12 5. ab c d abc d ab cd abc d abcd ((ab)c)d (a(bc))d (ab)(cd) a((bc)d) a(b(cd)) b, c, d, e,...
13 C n n C 4 = 5
14 (((ab)c)d)e ((a(bc))d)e ((ab)(cd))e (a((bc)d))e (a(b(cd)))e ((ab)c)(de) (a(bc))(de) ab c d e abc d e ab cd e abc d abcd e ab c de abc de ab cd e abc d e abcd e ab cde abc de abcd e abcde (ab)((cd)e) a(((bc)d)e) a((b(cd))e) (ab)(c(de)) a((bc)(de)) a(b((cd)e)) a(b(c(de))) C 5 = 14
15 6. ab cd ab c d abcd abc d abc d (ab)(cd) (a(bc))d (a(b(cd)) a((bc)d) (a(bc))d
16 6. ab cd ab c d abcd abc d abc d (ab)(cd) (a(bc))d (a(b(cd)) a((bc)d) (a(bc))d
17 C n (n + 1) C 4 = 5
18 7. C 1 = 1, C 2 = 1, C 3 = 2, C 4 = 5, C 5 = 14, C 6 = 42, C 7 = 132, C 8 = C 1 = 1, C 2 = 1, C 2 = 1, C 1 = 1, = 2 = C 3
19 C 1 = 1, C 2 = 1, C 3 = 2, C 3 = 2, C 2 = 1, C 1 = 1, = 5 = C 4 C 1 = 1, C 2 = 1, C 3 = 2, C 4 = 5, C 4 = 5, C 3 = 2, C 2 = 1, C 1 = 1, = 14 = C 5 C 1 = 1, C 2 = 1, C 3 = 2, C 4 = 5, C 5 = 14, C 5 = 14, C 4 = 5, C 3 = 2, C 2 = 1, C 1 = 1, = 42 = C 6
20
21 C 4 C 1 C 3 C 2 C 2 C 3 C 4 C 1 C 5 = C 4 C 1 + C 3 C 2 + C 2 C 3 + C 1 C 4
22 8.
23 x 2 + 5x + 3 x 1 + 2x + 3x 2 + 4x
24 y = C 1 x + C 2 x 2 + C 3 x = x + x 2 + 2x 3 + 5x x x x n x n C n
25 y 2 = (x + x 2 + 2x 3 + 5x x x ) (x + x 2 + 2x 3 + 5x x x ) = x 2 + ( )x 3 + ( )x 4 +( )x = x 2 + 2x 3 + 5x x x = y x y 2 y + x = 0 y
26 1 ± 1 4x 2 y = 1 1 4x 2 C 1 = 1, C 2 = 1, C 3 = 2, C 4 = 5, C 5 = 14, C 6 = 42, C 7 = 132, C 8 =
27 a n a 0 + a 1 x + a 2 x C n a + bn ar n 1 1 4x 2 a 1 x + a 1 rx bx (1 x) 2 1 n! n! 1 n ex mc n (1 + x) m
28 9. (1 + x) 2 = 1 + 2x + x 2 (1 + x) 3 = 1 + 3x + 3x 2 + x 3 (1 + x) 4 = 1 + 4x + 6x 2 + 4x 3 + x 4 (1 + x) 5 = 1 + 5x + 10x x 3 + 5x 4 + x 5 (1 + x) 6 = 1 + 6x + 15x x x 4 + 6x 5 + x 6
29 (1 + x) m = m C 0 + m C 1 x + m C 2 x 2 + m C 3 x 3 + m C 4 x m C n m n mc n = m! (n!)(m n)! 5C 2 = 5! (2!)(5 2)! = (2 1)(3 2 1) = = 10
30 1 1 4x 2 m 1 4x = (1 4x) 1/2 (1 4x) 1/2 = 1 2x 2x 2 4x 3 10x 4 28x 5... = 1 ( 2 C 1 ) x 1 3 ( 4C 2 ) x ( 6C 3 ) x ( 8C 4 ) x 4...
31 C n = 1 n (2n 2C n 1 ) C 4 = 1 4 ( 6C 3 ) = (3 2 1)(3 2 1) = 20 4 = 5
32 = π =
33 1 rx a 0. 9 = = = = 1 (0. 9 = 1) a = 1, r = 1, x = = 1 + ( 1) + ( 1) 2 + ( 1) 3 + ( 1) = 1 2
34 = 1 12
35 11. n! 1 ex e x = 1 + x 1! + x2 2! + x3 3! + x4 4! +... x = 1 e 1 = ! + 1 2! + 1 3! + 1 4! +... π
36 e = e =
37 12. e πi + 1 = 0 i = 1, i 2 = 1 π e i 0
38 e πi = 1 e x 1 n! 1 = 1 + πi 1! + (πi)2 + (πi)3 + (πi) ( 2! ) 3! ( 4! = 1 π2 2! + π4 π 4!... + i 1! π3 3!... )
39
40 13.
41 1 2 (AbBa) 2 6 = (AbBcCa, AcBaCb) 24 9 = ( AbBcCdDa, AbBdCaDc, AcBaCdDb, AcBdCbDa, AdBaCbDc, AdBcCaDb, AbBaCdDc, AcBdCaDb, AdBcCbDa ) =
42 m ! + 1 4!... + ( 1)m 1 m! e 1 = 1 e 1 e m = 7 1 e m 1 e ( m 1 m ) m ( = 1 1 ) m 1 m e
43 e A. B. C. D.
44
45 ( ) 246 (2002/11)
46
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