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.

noncomputable 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.

@[implicit_reducible]

noncomputable 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.

@[simp]

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

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) : ∏ p ∈ Finset.antidiagonal n, f p.1 p.2 = ∏ p ∈ Finset.antidiagonal n, f p.2 p.1

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) : ∑ p ∈ Finset.antidiagonal n, f p.1 p.2 = ∑ p ∈ Finset.antidiagonal n, f p.2 p.1

@[simp]

theorem Finsupp.antidiagonal_single {α : Type u} [DecidableEq α] (a : α) (n : ℕ) : (Finset.antidiagonal fun₀ | a => n) = Finset.map ({ toFun := single a, inj' := ⋯ }.prodMap { toFun := single a, inj' := ⋯ }) (Finset.antidiagonal n)

theorem Finsupp.image_prodMap_embDomain_antidiagonal {α : Type u} [DecidableEq α] {β : Type u_1} [DecidableEq β] (f : α ↪ β) (y : α →₀ ℕ) : Finset.image (Prod.map (embDomain f) (embDomain f)) (Finset.antidiagonal y) = Finset.antidiagonal (embDomain f y)