feat(Combinatorics/Enumerative/Bell): sum over partition shapes (leanprover-community#39693)

Kill TODO in `Mathlib/Combinatorics/Enumerative/Bell.lean` which proves `Nat.bell` as a sum of `Multiset.bell` over partition shapes

Author: yuanyi-350

Mathlib/Combinatorics/Enumerative/Bell.lean

@@ -1,10 +1,12 @@
 /-
 Copyright (c) 2024 Antoine Chambert-Loir & María-Inés de Frutos—Fernández. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Antoine Chambert-Loir, María-Inés de Frutos—Fernández, Yu Shao, Beibei Xiong, Weijie Jiang
+Authors: Antoine Chambert-Loir, María-Inés de Frutos—Fernández, Yu Shao, Beibei Xiong, Weijie Jiang,
+Yi Yuan
 -/
 module
 
+public import Mathlib.Combinatorics.Enumerative.Partition.Basic
 public import Mathlib.Data.Nat.Choose.Multinomial
 
 /-! # Bell numbers for multisets
@@ -74,15 +76,12 @@ private theorem bell_mul_eq_lemma {x : ℕ} (hx : x ≠ 0) :
         ring_nf
 
 theorem bell_mul_eq (m : Multiset ℕ) :
-    m.bell * (m.map (fun j ↦ j !)).prod * ∏ j ∈ (m.toFinset.erase 0), (m.count j)!
-      = m.sum ! := by
+    m.bell * (m.map (fun j ↦ j !)).prod * ∏ j ∈ (m.toFinset.erase 0), (m.count j)! = m.sum ! := by
   unfold bell
   rw [← Nat.mul_right_inj (a := ∏ i ∈ m.toFinset, (i * count i m)!) (by positivity)]
-  simp only [← mul_assoc]
-  rw [Nat.multinomial_spec]
-  simp only [mul_assoc]
-  rw [mul_comm]
-  apply congr_arg₂
+  simp only [← mul_assoc, Nat.multinomial_spec]
+  rw [mul_assoc, mul_assoc, mul_comm]
+  congr
   · rw [mul_comm, mul_assoc, ← Finset.prod_mul_distrib, Finset.prod_multiset_map_count]
     suffices this : _ by
       by_cases hm : 0 ∈ m.toFinset
@@ -94,30 +93,50 @@ theorem bell_mul_eq (m : Multiset ℕ) :
         nth_rewrite 3 [← Finset.erase_eq_of_notMem hm]
         exact this
     rw [← Finset.prod_mul_distrib]
-    apply Finset.prod_congr rfl
-    intro x hx
+    congr! 1 with x hx
     rw [← mul_assoc, bell_mul_eq_lemma]
     simp only [Finset.mem_erase, ne_eq, mem_toFinset] at hx
     simp only [ne_eq, hx.1, not_false_eq_true]
-  · apply congr_arg
-    rw [Finset.sum_multiset_count]
+  · rw [Finset.sum_multiset_count]
     simp only [smul_eq_mul, mul_comm]
 
 theorem bell_eq (m : Multiset ℕ) :
-    m.bell = m.sum ! / ((m.map (fun j ↦ j !)).prod *
-      ∏ j ∈ (m.toFinset.erase 0), (m.count j)!) := by
+    m.bell = m.sum ! / ((m.map fun j ↦ j !).prod * ∏ j ∈ m.toFinset.erase 0, (m.count j)!) := by
   rw [← Nat.mul_left_inj, Nat.div_mul_cancel _]
   · rw [← mul_assoc]
     exact bell_mul_eq m
   · rw [← bell_mul_eq, mul_assoc]
     apply Nat.dvd_mul_left
   · rw [← Nat.pos_iff_ne_zero]
-    apply Nat.mul_pos
-    · simp only [CanonicallyOrderedAdd.multiset_prod_pos, mem_map, forall_exists_index, and_imp,
-        forall_apply_eq_imp_iff₂]
-      exact fun _ _ ↦ Nat.factorial_pos _
-    · apply Finset.prod_pos
-      exact fun _ _ ↦ Nat.factorial_pos _
+    exact Nat.mul_pos (by simp [Nat.factorial_pos]) (by positivity)
+
+private theorem bell_cons_mul_count (m : Multiset ℕ) {a : ℕ} (ha : a ≠ 0) :
+    (a ::ₘ m).bell * (a ::ₘ m).count a = (m.sum + a).choose a * m.bell := by
+  let rest := ∏ j ∈ (m.toFinset.erase 0).erase a, (m.count j)!
+  have hrest : rest = ∏ j ∈ ((a ::ₘ m).toFinset.erase 0).erase a, ((a ::ₘ m).count j)! := by
+    unfold rest
+    congr! 1 with j hj
+    · grind [Multiset.toFinset_cons]
+    · simp [Finset.mem_erase.mp hj]
+  let c := (m.map (· !)).prod * (m.count a)! * rest
+  have hm0 : m.bell * c = m.sum ! := by
+    have hsplit : (m.count a)! * rest = ∏ j ∈ m.toFinset.erase 0, (m.count j)! := by
+      by_cases hmem : a ∈ m.toFinset.erase 0
+      · rw [← Finset.mul_prod_erase _ _ hmem]
+      · have hcount : m.count a = 0 := by grind [Multiset.count_eq_zero_of_notMem]
+        simp [rest, Finset.erase_eq_of_notMem hmem, hcount]
+    simpa [c, hsplit, mul_assoc] using Multiset.bell_mul_eq m
+  have hm : m.sum ! * a ! = m.bell * a ! * c := by grind
+  have hc : 0 < a ! * c := Nat.mul_pos (by positivity) <|
+    Nat.mul_pos (by simp [Nat.factorial_pos]) (by positivity)
+  apply Nat.eq_of_mul_eq_mul_right hc
+  calc
+    _ = (m.sum + a)! := by
+      have hq := Multiset.bell_mul_eq (a ::ₘ m)
+      rw [← Finset.mul_prod_erase _ _ (a := a) (by simp [*]), ← hrest] at hq
+      simpa [c, Nat.factorial_succ, add_comm, mul_assoc, mul_left_comm] using hq
+    _ = ((m.sum + a).choose a * m.bell) * (a ! * c) := by
+      simp [← Nat.add_choose_mul_factorial_mul_factorial, mul_assoc, hm]
 
 end Multiset
 
@@ -128,7 +147,7 @@ of `n`-element subsets. -/
 def uniformBell (m n : ℕ) : ℕ := bell (replicate m n)
 
 theorem uniformBell_eq (m n : ℕ) : m.uniformBell n =
-    ∏ p ∈ (Finset.range m), Nat.choose (p * n + n - 1) (n - 1) := by
+    ∏ p ∈ (Finset.range m), choose (p * n + n - 1) (n - 1) := by
   unfold uniformBell bell
   rw [toFinset_replicate]
   split_ifs with hm
@@ -172,43 +191,96 @@ theorem uniformBell_eq_div (m : ℕ) {n : ℕ} (hn : n ≠ 0) :
     uniformBell m n = (m * n)! / (n ! ^ m * m !) := by
   rw [eq_comm]
   apply Nat.div_eq_of_eq_mul_left
-  · exact Nat.mul_pos (Nat.pow_pos (Nat.factorial_pos n)) m.factorial_pos
+  · exact Nat.mul_pos (Nat.pow_pos n.factorial_pos) m.factorial_pos
   · rw [← mul_assoc, ← uniformBell_mul_eq _ hn]
 
 /--
 The `n`th standard Bell number,
 which counts the number of partitions of a set of cardinality `n`.
-
-## TODO
-
-Prove that `Nat.bell n` is equal to the sum of `Multiset.bell m`
-over all multisets `m : Multiset ℕ` such that `m.sum = n`.
 -/
 protected def bell : ℕ → ℕ
   | 0 => 1
-  | n + 1 => ∑ i : Fin n.succ, choose n i * Nat.bell (n - i)
+  | n + 1 => ∑ i ≤ n, choose n i * (n - i).bell
 
 theorem bell_succ (n : ℕ) :
-    Nat.bell (n + 1) = ∑ i : Fin n.succ, Nat.choose n i * Nat.bell (n - i) := by
+    (n + 1).bell = ∑ i ≤ n, choose n i * (n - i).bell := by
   rw [Nat.bell]
 
 theorem bell_succ' (n : ℕ) :
-    Nat.bell (n + 1) = ∑ ij ∈ Finset.antidiagonal n, Nat.choose n ij.1 * Nat.bell ij.2 := by
+    (n + 1).bell = ∑ ij ∈ Finset.antidiagonal n, choose n ij.1 * ij.2.bell := by
   rw [Nat.bell_succ,
-    Finset.Nat.sum_antidiagonal_eq_sum_range_succ (fun x y => Nat.choose n x * Nat.bell y) n,
-    Finset.sum_range]
-
+    ← Nat.range_succ_eq_Iic,
+    ← Finset.Nat.sum_antidiagonal_eq_sum_range_succ (fun x y ↦ choose n x * y.bell) n]
 
 @[simp]
 theorem bell_zero : Nat.bell 0 = 1 := by
   simp [Nat.bell]
 
 @[simp]
 theorem bell_one : Nat.bell 1 = 1 := by
-  simp [Nat.bell]
+  have : Finset.Iic 0 = {0} := Eq.symm (Finset.eq_of_veq rfl)
+  simp [Nat.bell, this]
 
 @[simp]
 theorem bell_two : Nat.bell 2 = 2 := by
-  simp [Nat.bell]
+  have : Finset.Iic 1 = {0, 1} := Finset.eq_of_veq rfl
+  simp [Nat.bell, this]
+
+theorem bell_eq_sum_erase {n : ℕ} (p : (n + 1).Partition) :
+    p.parts.bell = ∑ a ∈ p.parts.toFinset, n.choose (a - 1) * (p.parts.erase a).bell := by
+  apply Nat.eq_of_mul_eq_mul_left n.succ_pos
+  calc
+  _ = (∑ a ∈ p.parts.toFinset, p.parts.count a * a) * p.parts.bell := by
+    rw [succ_eq_add_one, mul_eq_mul_right_iff]
+    left
+    simpa [smul_eq_mul, p.parts_sum] using Finset.sum_multiset_count p.parts
+  _ = ∑ a ∈ p.parts.toFinset, a * (p.parts.count a * p.parts.bell) := by grind [Finset.sum_mul]
+  _ = ∑ a ∈ p.parts.toFinset, (n + 1) * (n.choose (a - 1) * (p.parts.erase a).bell) := by
+    congr! 1 with a ha
+    have ha0 : a ≠ 0 := by grind
+    have hsum : (p.parts.erase a).sum + a = n + 1 := by
+      simpa [p.parts_sum, add_comm] using congrArg Multiset.sum (cons_erase (mem_dedup.mp ha))
+    grind [Nat.add_one_mul_choose_eq, cons_erase, bell_cons_mul_count]
+  _ = _ := by rw [Finset.mul_sum]
+
+private def sigmaPartitionWithPartEquiv (n : ℕ) :
+    (Σ i : Fin n.succ, {p : (n + 1).Partition // (i + 1 : ℕ) ∈ p.parts}) ≃
+    Σ p : (n + 1).Partition, {a : ℕ // a ∈ p.parts.toFinset} where
+  toFun x := ⟨x.2.1, ⟨(x.1 + 1 : ℕ), by simpa using x.2.2⟩⟩
+  invFun x := ⟨⟨x.2.1 - 1, by grind⟩, ⟨x.1, by grind⟩⟩
+  left_inv x := by simp
+  right_inv x := by grind
+
+/-- `Nat.bell n` is equal to the sum of `Multiset.bell m` over all multisets `m : Multiset ℕ` such
+that `m.sum = n`. -/
+theorem bell_eq_sum_partition (n : ℕ) : n.bell = ∑ p : n.Partition, p.parts.bell := by
+  refine Nat.strong_induction_on n ?_
+  rintro (_ | n) ih
+  · simp
+  rw [Nat.bell_succ]
+  calc
+  _ = ∑ i ≤ n, ∑ q : (n - i).Partition, n.choose i * q.parts.bell := by
+    congr! with i
+    simp [ih (n - i) _, Finset.mul_sum]
+  _ = ∑ i ≤ n, ∑ p : {p : (n + 1).Partition // (i + 1 : ℕ) ∈ p.parts},
+      choose n i * (p.1.parts.erase (i + 1)).bell := by
+    congr! with i hi
+    have : i ≤ n := Finset.mem_Iic.mp hi
+    have h1 : 1 ≤ (i + 1 : ℕ) := by lia
+    have h2 : (i + 1 : ℕ) ≤ n + 1 := by lia
+    have hsub : n + 1 - (i + 1 : ℕ) = n - i := by lia
+    exact hsub ▸ (Fintype.sum_equiv (Partition.partitionWithPartEquiv h1 h2) _ _ (fun _ ↦ rfl)).symm
+  _ = ∑ x : Σ p : (n + 1).Partition, p.parts.toFinset,
+      choose n (x.2.1 - 1) * (x.1.parts.erase x.2.1).bell := by
+    rw [← Nat.range_succ_eq_Iic, Finset.sum_range, ← Fintype.sum_sigma']
+    refine Fintype.sum_equiv (sigmaPartitionWithPartEquiv n) _ _ ?_
+    simp [sigmaPartitionWithPartEquiv]
+  _ = ∑ p : (n + 1).Partition, ∑ a : p.parts.toFinset, choose n (a - 1) * (p.parts.erase a).bell :=
+    Fintype.sum_sigma' fun (p : (n + 1).Partition) (a : p.parts.toFinset) ↦
+      choose n (a - 1) * (p.parts.erase a.1).bell
+  _ = _ := by
+    congr! with p
+    rw [bell_eq_sum_erase p]
+    exact p.parts.toFinset.sum_coe_sort (fun a ↦ choose n (a - 1) * (p.parts.erase a).bell)
 
 end Nat

Mathlib/Combinatorics/Enumerative/Partition/Basic.lean

@@ -229,6 +229,33 @@ theorem countRestricted_two (n : ℕ) : countRestricted n 2 = distincts n := by
 def oddDistincts (n : ℕ) : Finset n.Partition :=
   odds n ∩ distincts n
 
+/-- If `1 ≤ a` and `a ≤ n`, partitions of `n` containing `a` as a part are equivalent to
+partitions of `n - a`. The forward map removes one occurrence of `a`, and the inverse adds `a` as
+a part. -/
+def partitionWithPartEquiv {n a : ℕ} (ha1 : 1 ≤ a) (ha : a ≤ n) :
+    {p : n.Partition // a ∈ p.parts} ≃ (n - a).Partition where
+  toFun p := by
+    refine ⟨p.1.parts.erase a, ?_, ?_⟩
+    · intro _ hi
+      exact p.1.parts_pos (p.1.parts.erase_subset a hi)
+    · have hs : a + (p.1.parts.erase a).sum = n := by
+        simpa [p.1.parts_sum] using congrArg Multiset.sum (Multiset.cons_erase p.2)
+      lia
+  invFun q := ⟨⟨a ::ₘ q.parts, by grind, by simp [q.parts_sum, ha]⟩, by simp⟩
+  left_inv p := Subtype.ext <| Partition.ext <| cons_erase p.property
+  right_inv q := Partition.ext <| erase_cons_head a q.parts
+
+@[simp]
+theorem partitionWithPartEquiv_apply_parts {n a : ℕ} (ha1 : 1 ≤ a) (ha : a ≤ n)
+    (p : {p : n.Partition // a ∈ p.parts}) :
+    (partitionWithPartEquiv ha1 ha p).parts = p.1.parts.erase a := by
+  dsimp [partitionWithPartEquiv]
+
+@[simp]
+theorem partitionWithPartEquiv_symm_apply_parts {n a : ℕ} (ha1 : 1 ≤ a) (ha : a ≤ n)
+    (p : (n - a).Partition) : ((partitionWithPartEquiv ha1 ha).symm p).1.parts = a ::ₘ p.parts := by
+  dsimp [partitionWithPartEquiv]
+
 end Partition
 
 end Nat