Counting with automorphisms Lectures for CO 430 / 630 March 24 - PowerPoint PPT Presentation

Counting with automorphisms Lectures for CO 430 / 630 March 24 – April 2, 2020

§1. Counting orbits §2. Type generating functions §3. Cycle index functions §4. Examples §5. Main theorem §6. Unlabelled trees §1. Counting orbits of group actions CO 430 / 630 Counting with automorphisms

§1. Counting orbits §2. Type generating functions §3. Cycle index functions §4. Examples §5. Main theorem §6. Unlabelled trees Group actions G a finite group. X a set. Definition. An action of G on X is a map G × X → X , written ( g , x ) �→ gx such that – 1 x = x for all x ∈ X , and – ( gh ) x = g ( hx ) for all g , h ∈ G , x ∈ X . For x ∈ X : ◮ the orbit of x is � x = { gx | g ∈ G } . ◮ the stabilizer of x is G x = { g ∈ G | gx = x } . Orbit–Stabilizer Lemma. # � x · # G x = # G . CO 430 / 630 Counting with automorphisms

§1. Counting orbits §2. Type generating functions §3. Cycle index functions §4. Examples §5. Main theorem §6. Unlabelled trees Example: Functions on G K a set. K G = set of functions G → K . We have an action of G on K G : for α ∈ K G , g ∈ G , g α : h �→ α ( g − 1 h ) . Special case: G = C n ← cyclic group of order n . K ← finite set of “colours”. K C n ↔ colouring sides of a regular n -gon. C n -action ↔ rotating the n -gon. Orbits ↔ equivalence classes of colourings up to rotation. CO 430 / 630 Counting with automorphisms

§1. Counting orbits §2. Type generating functions §3. Cycle index functions §4. Examples §5. Main theorem §6. Unlabelled trees Figure: Colourings of the square ( n = 4 ) with 2 colours CO 430 / 630 Counting with automorphisms

§1. Counting orbits §2. Type generating functions §3. Cycle index functions §4. Examples §5. Main theorem §6. Unlabelled trees Orbit counting � X = { � x | x ∈ X } ← set of orbits. For g ∈ G , the set of g-fixed points is X g = { x ∈ X | gx = x } . Orbit Counting Lemma (a.k.a. Burnside’s Lemma) Let X be a finite set with a G -action. � X = 1 # � # X g # G g ∈ G (Number of orbits = Average number of fixed points.) � � � 1 1 1 # X g = 1 = # G x Proof. # G # G # G g ∈ G ( g , x ) ∈ G × X x ∈ X gx = x � � � � � # G x 1 1 1 = # � = = x = x = X . # � # � # G x ∈ X x ∈ X x ∈ � x ∈ � x x ∈ � � � X X CO 430 / 630 Counting with automorphisms

§1. Counting orbits §2. Type generating functions §3. Cycle index functions §4. Examples §5. Main theorem §6. Unlabelled trees Example: Colourings of the n -gon up to rotation We compute the number of C n -orbits on X = K C n , | K | = k . For each c ∈ C n , X c = { f : C n → K | f ( i − c ) = f ( c ) ∀ i ∈ C n } = { f : C n → K | f ( i − d ) = f ( d ) ∀ i ∈ C n } , d = gcd ( c , n ) . Hence # X c = k gcd ( c , n ) . By the orbit counting lemma, � X = 1 k gcd ( c , n ) . # � n c ∈ C n Let φ be the totient function : φ ( m ) = # { ℓ ∈ [ m ] | gcd ( ℓ , m ) = 1 } . For each divisor d of n there are φ ( n / d ) elements c ∈ C n such that gcd ( c , n ) = d . Hence � X = 1 φ ( n / d ) k d . # � n d | n CO 430 / 630 Counting with automorphisms

§1. Counting orbits §2. Type generating functions §3. Cycle index functions §4. Examples §5. Main theorem §6. Unlabelled trees An example involving generating functions � Let S m be the set of functions f : C n → � such that c ∈ C n f ( c ) = m . How many C n -orbits? Let S = � C n ( all functions C n → � ), with weight function � wt : S → � , wt ( f ) = c ∈ C n f ( c ) . The OGF for S is S ( x ) = ( 1 − x ) − n . # S m = [ x m ]( 1 − x ) − n . For c ∈ C n , we have a bijection S c ↔ ( n d � ) C d , d = gcd ( c , n ) (see next slide). Hence the OGF for S c is ( 1 − x d ) − n / d . m = [ x m ]( 1 − x n / d ) − d . # S c Hence the number of C n orbits on S m is � [ x m ] 1 φ ( n / d )( 1 − x n / d ) − d . n d | n CO 430 / 630 Counting with automorphisms

§1. Counting orbits §2. Type generating functions §3. Cycle index functions §4. Examples §5. Main theorem §6. Unlabelled trees Figure: The bijection S c ↔ ( n d � ) C d 1 2 5 0 0 0 ↔ 5 2 15 6 1 1 3 2 0 5 Functions C 12 → � Functions ↔ fixed by 4 ∈ C 12 C 4 → 3 � CO 430 / 630 Counting with automorphisms

§1. Counting orbits §2. Type generating functions §3. Cycle index functions §4. Examples §5. Main theorem §6. Unlabelled trees Exercises 1. Let X be the set of colourings of the n -gon with two colours. The dihedral group D n of order 2 n acts on X by rotations and reflections. (Note: if n is even there are two different types of reflections — we can reflect across a line through two opposite vertices, or across a line through two opposite edges.) For n = 4, n = 6 and n = 9, use the orbit counting lemma to compute the number of D n -orbits. Answer: 6 , 13 and 46 2. Let R m be the set of colourings of the n -gon with m sides coloured red, and n − m sides coloured blue. The cyclic group C n acts on R m by rotation. Find a formula for the number of orbits, as the coefficient of x m in a generating function. � Answer: [ x m ] d | n φ ( n / d )( 1 + x n / d ) d CO 430 / 630 Counting with automorphisms

§1. Counting orbits §2. Type generating functions §3. Cycle index functions §4. Examples §5. Main theorem §6. Unlabelled trees §2. Type generating functions CO 430 / 630 Counting with automorphisms

§1. Counting orbits §2. Type generating functions §3. Cycle index functions §4. Examples §5. Main theorem §6. Unlabelled trees Isomorphism types of species S n the symmetric group of permutations of [ n ] . A a species. We have an action of S n on A [ n ] : for σ ∈ S n , α ∈ A [ n ] , σα = σ ∗ ( α ) . α = � Hence α , β ∈ A [ n ] are isomorphic iff � β . The set of isomorphism types of the species A is � � � � { � α | α ∈ A [ n ] } , A = A [ n ] = n ≥ 0 n ≥ 0 with weight function ord : � A → � , ord ( � α ) = n iff α ∈ A [ n ] . The type generating function of the species A is the OGF for � A : � α ) . � x ord ( � A ( x ) = α ∈ � � A CO 430 / 630 Counting with automorphisms

§1. Counting orbits §2. Type generating functions §3. Cycle index functions §4. Examples §5. Main theorem §6. Unlabelled trees Examples of type generating functions ◮ E – species of sets. One isomorphism type of each order. � 1 x n = ( 1 − x ) − 1 � E ( x ) = n ≥ 0 ◮ L • – species of rooted linear orders. L also has one isomorphism type of each order n , but there are n non-isomorphic ways to root. � nx n = x ( 1 − x ) − 2 � L • ( x ) = n ≥ 0 ◮ S – species of permutations. Two permutations are isomorphic iff they have the same cycle type (same number of cycles of each size). Isomorphism types of order n ↔ partitions of n . � � ( 1 − x i ) − 1 S ( x ) = i ≥ 1 CO 430 / 630 Counting with automorphisms

§1. Counting orbits §2. Type generating functions §3. Cycle index functions §4. Examples §5. Main theorem §6. Unlabelled trees Properties and non-properties Proposition. Let A and B be species. 1. The type generating function of A + B is � A ( x ) + � B ( x ) . 2. The type generating function of A ∗ B is � A ( x ) � B ( x ) . Unfortunately, for other species operations we can’t determine the TGF of the output species just from the TGF of the input species. This is why TGFs are usually harder to compute than EGFs. , but E • and L • have Example. E and L have the same TGF different TGFs. CO 430 / 630 Counting with automorphisms

§1. Counting orbits §2. Type generating functions §3. Cycle index functions §4. Examples §5. Main theorem §6. Unlabelled trees Type generating functions via orbit counting lemma For σ ∈ S n , write A σ = { α ∈ A [ n ] | σ ∗ ( α ) = α } . By the orbit counting lemma, the number of S n -orbits on A [ n ] is � 1 # A σ . n ! σ ∈ S n Hence � � 1 # A σ x n . � A ( x ) = n ! n ≥ 0 σ ∈ S n This is usually difficult to use directly. Goal: Develop a generalization of this formula which will allow us to compute TGFs in a manner similar to EGFs. We will need to work with objects that are more general than formal power series. CO 430 / 630 Counting with automorphisms

§1. Counting orbits §2. Type generating functions §3. Cycle index functions §4. Examples §5. Main theorem §6. Unlabelled trees Exercises 1. Find the first few terms of the type generating function for the species T of trees. T ( x ) = x + x 2 + x 3 + 2 x 4 + 3 x 5 + ... Answer: � 2. Compute the type generating functions for E [ E + ] and L [ E + ] . Notice that TGFs of E and L are the same, but the TGFs of E [ E + ] and L [ E + ] are different! � i ≥ 1 ( 1 − x i ) − 1 and ( 1 − x )( 1 − 2 x ) − 1 Answer: CO 430 / 630 Counting with automorphisms

§1. Counting orbits §2. Type generating functions §3. Cycle index functions §4. Examples §5. Main theorem §6. Unlabelled trees §3. Cycle index functions CO 430 / 630 Counting with automorphisms