The Finsupp counterpart of Multiset.antidiagonal.

The antidiagonal of s : α →₀ ℕ consists of all pairs (t₁, t₂) : (α →₀ ℕ) × (α →₀ ℕ) such that t₁ + t₂ = s.

def Finsupp.antidiagonal' {α : Type u} [DecidableEq α] (f : α →₀ ℕ) : (α →₀ ℕ) × (α →₀ ℕ) →₀ ℕ

The Finsupp counterpart of Multiset.antidiagonal: the antidiagonal of s : α →₀ ℕ consists of all pairs (t₁, t₂) : (α →₀ ℕ) × (α →₀ ℕ) such that t₁ + t₂ = s. The finitely supported function antidiagonal s is equal to the multiplicities of these pairs.

Equations

• Finsupp.antidiagonal' f = Multiset.toFinsupp (Multiset.map (Prod.map ⇑Multiset.toFinsupp ⇑Multiset.toFinsupp) (Multiset.antidiagonal (Finsupp.toMultiset f)))

instance Finsupp.instHasAntidiagonal {α : Type u} [DecidableEq α] : Finset.HasAntidiagonal (α →₀ ℕ)

The antidiagonal of s : α →₀ ℕ is the finset of all pairs (t₁, t₂) : (α →₀ ℕ) × (α →₀ ℕ) such that t₁ + t₂ = s.

Equations

• Finsupp.instHasAntidiagonal = { antidiagonal := fun (f : α →₀ ℕ) => (Finsupp.antidiagonal' f).support, mem_antidiagonal := ⋯ }

@[simp]

theorem Finsupp.antidiagonal_zero {α : Type u} [DecidableEq α] : Finset.antidiagonal 0 = {(0, 0)}

theorem Finsupp.sum_antidiagonal_swap {α : Type u} [DecidableEq α] {M : Type u_1} [AddCommMonoid M] (n : α →₀ ℕ) (f : (α →₀ ℕ) → (α →₀ ℕ) → M) : (Finset.sum (Finset.antidiagonal n) fun (p : (α →₀ ℕ) × (α →₀ ℕ)) => f p.1 p.2) = Finset.sum (Finset.antidiagonal n) fun (p : (α →₀ ℕ) × (α →₀ ℕ)) => f p.2 p.1

theorem Finsupp.prod_antidiagonal_swap {α : Type u} [DecidableEq α] {M : Type u_1} [CommMonoid M] (n : α →₀ ℕ) (f : (α →₀ ℕ) → (α →₀ ℕ) → M) : (Finset.prod (Finset.antidiagonal n) fun (p : (α →₀ ℕ) × (α →₀ ℕ)) => f p.1 p.2) = Finset.prod (Finset.antidiagonal n) fun (p : (α →₀ ℕ) × (α →₀ ℕ)) => f p.2 p.1

@[simp]

theorem Finsupp.antidiagonal_single {α : Type u} [DecidableEq α] (a : α) (n : ℕ) : Finset.antidiagonal (Finsupp.single a n) = Finset.map (Function.Embedding.prodMap { toFun := Finsupp.single a, inj' := ⋯ } { toFun := Finsupp.single a, inj' := ⋯ }) (Finset.antidiagonal n)