# 點算的奧秘：對稱群的循環式

[1 2 3], [1 3 2], [2 1 3], [2 3 1], [3 1 2], [3 2 1]

(2 5)(1 3 4); (2 5)(3 4 1); (2 5)(4 1 3); (5 2)(1 3 4); (5 2)(3 4 1); (5 2)(4 1 3)

5; 2 + 3; 1 + 4; 1 + 2 + 2; 1 + 1 + 3; 1 + 1 + 1 + 2; 1 + 1 + 1 + 1 + 1

(x x x x x); (x x)(x x x); (x)(x x x x); (x)(x x)(x x); (x)(x)(x x x); (x)(x)(x)(x x); (x)(x)(x)(x)(x) (註3)

(1 3 4 2 5); (2 5)(1 3 4); (2)(1 3 4 5); (2)(1 5)(3 4); (2)(5)(1 3 4); (2)(5)(1)(3 4); (2)(5)(1)(3)(4) □

 [(1 − 1)!]k1 [(2 − 1)!]k2 ... [(r − 1)!]kr = (0!)k1 (1!)k2 ... [(r − 1)!]kr    (1)

{{6,8},{2,9},{3,4,10},{7,5,1}}

(6 8)(2 9)(3 4 10)(1 7 5); (6 8)(2 9)(3 4 10)(1 5 7);
(6 8)(2 9)(3 10 4)(1 7 5); (6 8)(2 9)(3 10 4)(1 5 7)

r! / [(k1)! (1!)k1 (k2)! (2!)k2 ... (kr)! (r!)kr]    (2)

 r! (0!)k1 (1!)k2 ... [(r − 1)!]kr / [(k1)! (1!)k1 (k2)! (2!)k2 ... (kr)! (r!)kr] = r! / [(k1)! (1)k1 (k2)! (2)k2 ... (kr)! (r)kr]    (3)

「循環類型」「循環式」的數目
(x x x x x)5! / [1! (5)1] = 24
(x x)(x x x)5! / [1! (2)1 1! (3)1] = 20
(x)(x x x x)5! / [1! (1)1 1! (4)1] = 30
(x)(x x)(x x)5! / [1! (1)1 2! (2)2] = 15
(x)(x)(x x x)5! / [2! (1)2 1! (3)1] = 20
(x)(x)(x)(x x)5! / [3! (1)3 1! (2)1] = 10
(x)(x)(x)(x)(x)5! / [5! (1)5] = 1