# 點算的奧秘：著色問題的推廣應用

[],[1],[1],[1,1],[1,1,1]   (1) (註1)

[1,1,1]-[]-[1]-[1,1]-[1];     [1]-[1]-[1,1,1]-[1,1]-[]   (2)

|F / S5| = Σg ∈ S5 |Fix(g)| / |S5|     (3)

|Fix[(x)(x)(x)(x)(x)]| = C(5 + 7 − 1, 7) = C(11, 7) = 330

 |Fix[(x)(x)(x)(x x)]| = C(3 + 7 − 1, 7) + C(3 + 5 − 1, 5) + C(3 + 3 − 1, 3) + C(3 + 1 − 1, 1) = 36 + 21 + 10 + 3 = 70

 |Fix[(x)(x)(x x x)]| = C(2 + 7 − 1, 7) + C(2 + 4 − 1, 4) + C(2 + 1 − 1, 1) = 8 + 5 + 2 = 15

|Fix[(x)(x x)(x x)]| = 4 + 3 + 2 + 1 = 10

|Fix[(x)(x x x x)]| = 2

|Fix[(x x)(x x x)]| = 1

|Fix[(x x x x x)]| = 0

「循環類型」g「循環式」的數目|Fix(g)|
(x)(x)(x)(x)(x)1330
(x)(x)(x)(x x)1070
(x)(x)(x x x)2015
(x)(x x)(x x)1510
(x)(x x x x)302
(x x)(x x x)201
(x x x x x)240

 |F / S5| = (330 × 1 + 70 × 10 + 15 × 20 + 10 × 15 + 2 × 30 + 1 × 20 + 0 × 24) / 120 = 13 □

|Fix[(x)(x)(x)(x)(x)]| = 57 = 78125

|Fix[(x)(x)(x)(x x)]| = 37 = 2187

|Fix[(x)(x)(x x x)]| = 27 = 128
|Fix[(x)(x x)(x x)]| = 17 = 1
|Fix[(x)(x x x x)]| = 17 = 1

|Fix[(x x)(x x x)]| = 0

|Fix[(x x x x x)]| = 0

「循環類型」g「循環式」的數目|Fix(g)|
(x)(x)(x)(x)(x)178125
(x)(x)(x)(x x)102187
(x)(x)(x x x)20128
(x)(x x)(x x)151
(x)(x x x x)301
(x x)(x x x)200
(x x x x x)240

 |F / S5| = (78125 × 1 + 2187 × 10 + 128 × 20 + 1 × 15 + 1 × 30 + 0 × 20 + 0 × 24) / 120 = 855

「循環類型」g「循環式」的數目Z[g*]
(x)(x)(x)(x)(x)1z15
(x)(x)(x)(x x)10z13z2
(x)(x)(x x x)20z12z3
(x)(x x)(x x)15z1z22
(x)(x x x x)30z1z4
(x x)(x x x)20z2z3
(x x x x x)24z5

Z[S5](z1, z2, z3, z4, z5) = (z15 + 10z13z2 + 20z12z3 + 15z1z22 + 30z1z4 + 20z2z3 + 24z5) / 120

 Z[S5](3, 3, 3, 3, 3) = (35 + 10 × 33 × 3 + 20 × 32 × 3 + 15 × 3 × 32 + 30 × 3 × 3 + 20 × 3 × 3 + 24 × 3) / 120 = 21

 Z[S5](o + g + w, o2 + g2 + w2, o3 + g3 + w3, o4 + g4 + w4, o5 + g5 + w5) = [(o + g + w)5 + 10(o + g + w)3(o2 + g2 + w2) + 20(o + g + w)2(o3 + g3 + w3) + 15(o + g + w)(o2 + g2 + w2)2 + 30(o + g + w)(o4 + g4 + w4) + 20(o2 + g2 + w2)(o3 + g3 + w3) + 24(o5 + g5 + w5)] / 120

|Fix[(x)(x)(x)(x)(x)]| = 5!

|Fix[(x x x x x)]| = 0

「循環類型」g「循環式」的數目|Fix(g)|
(x)(x)(x)(x)(x)15!
(x x x x x)40

 |F / C5| = (5! + 0 × 4) / 5 = 4! = 24