Polya's Theory of Counting INEQUALITIES S8R t a new aspect of mathematical mechod G.POLYA G.Hardy,J.E.&G.Polyo Cambridge Mathematical Library George Polya (1887-1985)
Pólya’s Theory of Counting George Pólya (1887-1985)
Counting with Symmetry Rotation ǒh.6 0d Rotation Reflection:
Counting with Symmetry Rotation : Rotation & Reflection:
Symmetries
Symmetries
Symmetry rotation reflection configuration x [n]->[m] X (m]ln] positions colors permutation π：[n 1m on-to
Symmetry 0 1 2 3 4 5 rotation reflection configuration x : [n] ! [m] X = [m] [n] positions colors permutation ⇡ : [n] 1-1 ! on-to [n]
Permutation Groups group(G,）with binary operator·:G×G→G ·closure:π，o∈G→π·o∈G ·associativity::π·(o·T)=(π·o）·T 。identity:e∈G,Vπ∈G,e·T=π 。inverse:Vπ∈G,o∈G,π·o=o·π=e 0=π-1 commutative (abelian)group: π·0=0·π symmetric group S.:all permutations cyclic group Cn:rotations Dihedral group D:rotations reflections
Permutation Groups group (G, ·) with binary operator · : G ⇥ G ! G • closure: • associativity: • identity: • inverse: ⇡, 2 G ) ⇡ · 2 G ⇡ · ( · ⌧ )=(⇡ · ) · ⌧ = ⇡1 8⇡ 2 G, 9 2 G, ⇡ · = · ⇡ = e 9e 2 G, 8⇡ 2 G, e · ⇡ = ⇡ commutative (abelian) group: ⇡ · = · ⇡ symmetric group cyclic group Dihedral group Sn Cn Dn : all permutations : rotations : rotations & reflections