Polya's Theory of Counting INEQUALITIES a new aspect of mathematical method G.POLYA G.Hordy.J.E.Limlewood G.Polya Cambridge Mathematical Library George Polya (1887-1985)
Pólya’s Theory of Counting George Pólya (1887-1985)
Counting with Symmetry Rotation Rotation Reflection: 9
Counting with Symmetry Rotation : Rotation & Reflection:
Symmetries
Symmetries
Symmetry rotation reflection 3 8 configuration x [n]>[m] X (m]inl positions colors permutation π：。回
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·:GxG→G 。closure:T,o∈G→T·o∈G 。associativity:T·(o·T)=(r·o）·T 。identity:]e∈G,Vπ∈G,e·π=π ●inverse: Vπ∈G,0∈G,π·g=0·π=e -1 0三π commutative (abelian)group: π·0=0·m 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