Partitions

A partition of a natural number n is a way of writing n as a sum of positive integers, where the order does not matter: two sums that differ only in the order of their summands are considered the same partition. This notion is closely related to that of a composition of n, but in a composition of n the order does matter. A summand of the partition is called a part.

Main functions

• p : Partition n is a structure, made of a multiset of integers which are all positive and add up to n.

Implementation details

The main motivation for this structure and its API is to show Euler's partition theorem, and related results.

The representation of a partition as a multiset is very handy as multisets are very flexible and already have a well-developed API.

Tags

Partition

References

https://en.wikipedia.org/wiki/Partition_(number_theory)

theorem Nat.Partition.ext {n : ℕ} (x : Nat.Partition n) (y : Nat.Partition n) (parts : x.parts = y.parts) : x = y

theorem Nat.Partition.ext_iff {n : ℕ} (x : Nat.Partition n) (y : Nat.Partition n) : x = y ↔ x.parts = y.parts

structure Nat.Partition (n : ℕ) : Type

• parts : Multiset ℕ

  positive integers summing to n

• parts_pos : ∀ {i : ℕ}, i ∈ s.parts → 0 < i

  proof that the parts are positive

• parts_sum : Multiset.sum s.parts = n

  proof that the parts sum to n

A partition of n is a multiset of positive integers summing to n.

instance Nat.Partition.decidableEqParition {n : ℕ} : DecidableEq (Nat.Partition n)

Equations

• Nat.Partition.decidableEqParition x x = match x, x with | p, q => Eq.mpr (_ : Decidable (p = q) = Decidable (p.parts = q.parts)) (Multiset.decidableEq p.parts q.parts)

def Nat.Partition.ofComposition (n : ℕ) (c : Composition n) : Nat.Partition n

A composition induces a partition (just convert the list to a multiset).

Equations

• Nat.Partition.ofComposition n c = { parts := ↑c.blocks, parts_pos := (_ : ∀ {i : ℕ}, i ∈ ↑c.blocks → 0 < i), parts_sum := (_ : Multiset.sum ↑c.blocks = n) }

theorem Nat.Partition.ofComposition_surj {n : ℕ} : Function.Surjective (Nat.Partition.ofComposition n)

def Nat.Partition.ofSums (n : ℕ) (l : Multiset ℕ) (hl : Multiset.sum l = n) : Nat.Partition n

Given a multiset which sums to n, construct a partition of n with the same multiset, but without the zeros.

Equations

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

def Nat.Partition.ofMultiset (l : Multiset ℕ) : Nat.Partition (Multiset.sum l)

A Multiset ℕ induces a partition on its sum.

Equations

• Nat.Partition.ofMultiset l = Nat.Partition.ofSums (Multiset.sum l) l (_ : Multiset.sum l = Multiset.sum l)

def Nat.Partition.indiscretePartition (n : ℕ) : Nat.Partition n

The partition of exactly one part.

Equations

• Nat.Partition.indiscretePartition n = Nat.Partition.ofSums n {n} (_ : Multiset.sum {n} = Multiset.sum {n})

instance Nat.Partition.instInhabitedPartition {n : ℕ} : Inhabited (Nat.Partition n)

Equations

• Nat.Partition.instInhabitedPartition = { default := Nat.Partition.indiscretePartition n }

theorem Nat.Partition.count_ofSums_of_ne_zero {n : ℕ} {l : Multiset ℕ} (hl : Multiset.sum l = n) {i : ℕ} (hi : i ≠ 0) : Multiset.count i (Nat.Partition.ofSums n l hl).parts = Multiset.count i l

The number of times a positive integer i appears in the partition ofSums n l hl is the same as the number of times it appears in the multiset l. (For i = 0, Partition.non_zero combined with Multiset.count_eq_zero_of_not_mem gives that this is 0 instead.)

theorem Nat.Partition.count_ofSums_zero {n : ℕ} {l : Multiset ℕ} (hl : Multiset.sum l = n) : Multiset.count 0 (Nat.Partition.ofSums n l hl).parts = 0

instance Nat.Partition.instFintypePartition (n : ℕ) : Fintype (Nat.Partition n)

Show there are finitely many partitions by considering the surjection from compositions to partitions.

Equations

• Nat.Partition.instFintypePartition n = Fintype.ofSurjective (Nat.Partition.ofComposition n) (_ : Function.Surjective (Nat.Partition.ofComposition n))

def Nat.Partition.odds (n : ℕ) : Finset (Nat.Partition n)

The finset of those partitions in which every part is odd.

Equations

• Nat.Partition.odds n = Finset.filter (fun c => ∀ (i : ℕ), i ∈ c.parts → ¬Even i) Finset.univ

def Nat.Partition.distincts (n : ℕ) : Finset (Nat.Partition n)

The finset of those partitions in which each part is used at most once.

Equations

• Nat.Partition.distincts n = Finset.filter (fun c => Multiset.Nodup c.parts) Finset.univ

def Nat.Partition.oddDistincts (n : ℕ) : Finset (Nat.Partition n)

The finset of those partitions in which every part is odd and used at most once.

Equations

• Nat.Partition.oddDistincts n = Nat.Partition.odds n ∩ Nat.Partition.distincts n