Reduction to Fin

If we have a bijection X → Fin n, then the number of distinct colorings of X with colors in Y under the group action of G on X is equal to the number of distinct colorings of Fin n with colors in Y under the induced group action of G on Fin n. This allows us to use Fin n instead of more complex types when working with numbers of distinct colorings.

instance ReductionToFin.MulActionFin (X : Type u) (G : Type w) [enum : FinEnum X] [Monoid G] [MulAction G X] : MulAction G (Fin (FinEnum.card X))

Given a bijection enum.equiv : X → Fin enum.card and a group action of G on X, construct a group action of G on Fin enum.card with g • i ↦ enum.equiv (g • (enum.equiv⁻¹ i)).

Equations

• ReductionToFin.MulActionFin X G = MulAction.mk ⋯ ⋯

def ReductionToFin.equiv_of_quotient_of_quotient_Fin (X : Type u) (Y : Type v) (G : Type w) [enum : FinEnum X] [Group G] [MulAction G X] : Quotient (MulAction.orbitRel G (X → Y)) ≃ Quotient (MulAction.orbitRel G (Fin (FinEnum.card X) → Y))

A bijection between the distinct colorings of X with colors in Y under the group action of G on X and the distinct colorings of Fin enum.card with colors in Y under the induced group action of G on Fin enum.card.

Equations

• One or more equations did not get rendered due to their size.

instance ReductionToFin.instFintypeQuotientForallFinCardOrbitRelOfForall_polyaEnumerationTheorem (X : Type u) (Y : Type v) (G : Type w) [enum : FinEnum X] [Group G] [MulAction G X] [Fintype (Quotient (MulAction.orbitRel G (X → Y)))] : Fintype (Quotient (MulAction.orbitRel G (Fin (FinEnum.card X) → Y)))

An instance of Fintype for the distinct colorings of Fin enum.card with colors in Y under the induced group action of G on Fin enum.card. Required by numDistinctColorings_eq_numDistinctColorings_of_Fin.

Equations

• One or more equations did not get rendered due to their size.

theorem ReductionToFin.numDistinctColorings_eq_numDistinctColorings_of_Fin (X : Type u) (Y : Type v) (G : Type w) [enum : FinEnum X] [Group G] [MulAction G X] [Fintype (Quotient (MulAction.orbitRel G (X → Y)))] : DistinctColorings.numDistinctColorings X Y G = DistinctColorings.numDistinctColorings (Fin (FinEnum.card X)) Y G

The number of distinct colorings of X with colors in Y under the group action of G on X is equal to the number of distinct colorings of Fin enum.card with colors in Y under the induced group action of G on Fin enum.card.