feat(Combinatorics): Glaisher's theorem

Author: wwylele

Mathlib.lean

@@ -2948,6 +2948,7 @@ import Mathlib.Combinatorics.Enumerative.IncidenceAlgebra
 import Mathlib.Combinatorics.Enumerative.InclusionExclusion
 import Mathlib.Combinatorics.Enumerative.Partition
 import Mathlib.Combinatorics.Enumerative.Partition.GenFun
+import Mathlib.Combinatorics.Enumerative.Partition.Glaisher
 import Mathlib.Combinatorics.Enumerative.Stirling
 import Mathlib.Combinatorics.Extremal.RuzsaSzemeredi
 import Mathlib.Combinatorics.Graph.Basic

Mathlib/Combinatorics/Enumerative/Partition.lean

@@ -177,6 +177,14 @@ def distincts (n : ℕ) : Finset (Partition n) :=
 def oddDistincts (n : ℕ) : Finset (Partition n) :=
   odds n ∩ distincts n
 
+/-- The finset of those partitions in which every part satisfies a certain condition. -/
+def restricted (n : ℕ) (p : ℕ → Prop) [DecidablePred p] : Finset (n.Partition) :=
+  Finset.univ.filter fun x ↦ ∀ i ∈ x.parts, p i
+
+/-- The finset of those partitions in which every part is used less than `m` times. -/
+def countRestricted (n : ℕ) (m : ℕ) : Finset (n.Partition) :=
+  Finset.univ.filter fun x ↦ ∀ i ∈ x.parts, x.parts.count i < m
+
 end Partition
 
 end Nat

Mathlib/Combinatorics/Enumerative/Partition/Glaisher.lean

@@ -0,0 +1,168 @@
+/-
+Copyright (c) 2025 Weiyi Wang. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Weiyi Wang
+-/
+import Mathlib.Combinatorics.Enumerative.Partition.GenFun
+import Mathlib.RingTheory.PowerSeries.NoZeroDivisors
+
+/-!
+# Glaisher's theorem
+
+This file proves Glaisher's theorem: the number of partitions of an integer $n$ into parts not
+divisible by $d$ is equal to the number of partitions in which no part is repeated $d$ or more
+times.
+
+## Main declarations
+* `Nat.Partition.card_restricted_eq_card_countRestricted`: Glaisher's theorem.
+
+## References
+https://en.wikipedia.org/wiki/Glaisher%27s_theorem
+-/
+
+variable (M) [TopologicalSpace M]
+
+namespace PowerSeries.WithPiTopology
+
+section Semiring
+variable [Semiring M]
+
+theorem summable_X_pow_add_one_mul_add (i : ℕ) (k : ℕ) :
+    Summable fun j ↦ (X : M⟦X⟧) ^ ((i + 1) * (j + k)) := by
+  rcases subsingleton_or_nontrivial M with h | h
+  · simpa [Subsingleton.eq_zero] using summable_zero
+  apply summable_of_tendsto_order_atTop_nhds_top
+  refine ENat.tendsto_nhds_top_iff_natCast_lt.mpr (fun n ↦ Filter.eventually_atTop.mpr ⟨n + 1, ?_⟩)
+  intro m hm
+  rw [order_X_pow]
+  norm_cast
+  nlinarith
+
+end Semiring
+
+section CommRing
+variable [CommRing M]
+
+theorem multipliable_one_sub_X_pow : Multipliable fun n ↦ (1 : M⟦X⟧) - X ^ (n + 1) := by
+  rcases subsingleton_or_nontrivial M with h | h
+  · simpa [Subsingleton.eq_one] using multipliable_one
+  simp_rw [sub_eq_add_neg]
+  apply multipliable_one_add_of_tendsto_order_atTop_nhds_top
+  refine ENat.tendsto_nhds_top_iff_natCast_lt.mpr (fun n ↦ Filter.eventually_atTop.mpr ⟨n, ?_⟩)
+  intro m hm
+  rw [order_neg, order_X_pow]
+  norm_cast
+  nlinarith
+
+theorem tprod_one_sub_X_pow_ne_zero [T2Space M] [Nontrivial M] :
+    ∏' i, (1 - X ^ (i + 1)) ≠ (0 : M⟦X⟧) := by
+  by_contra! h
+  obtain h := PowerSeries.ext_iff.mp h 0
+  simp [coeff_zero_eq_constantCoeff, Multipliable.map_tprod (multipliable_one_sub_X_pow M) _
+    (WithPiTopology.continuous_constantCoeff M)] at h
+
+end CommRing
+
+end PowerSeries.WithPiTopology
+
+namespace Nat.Partition
+open PowerSeries Finset
+open scoped PowerSeries.WithPiTopology
+
+section Semiring
+variable [CommSemiring M]
+
+/-- The generating function of `Nat.Partition.restricted n p` is
+$$
+\prod_{i \mem p} \sum_{j = 0}^{\infty} X^{ij}
+$$ -/
+theorem hasProd_powerSeriesMk_card_restricted [T2Space M] [IsTopologicalSemiring M]
+    (p : ℕ → Prop) [DecidablePred p] :
+    HasProd (fun i ↦ if p (i + 1) then ∑' j : ℕ, X ^ ((i + 1) * j) else 1)
+    (PowerSeries.mk fun n ↦ (#(restricted n p) : M)) := by
+  convert hasProd_genFun (fun i c ↦ if p i then (1 : M) else 0) using 1
+  · ext1 i
+    split_ifs
+    · rw [tsum_eq_zero_add' ?_]
+      · simp
+      exact (WithPiTopology.summable_X_pow_add_one_mul_add M i 1)
+    · simp
+  · simp_rw [genFun, restricted, Finset.card_filter, Finset.prod_boole]
+    simp
+
+/-- The generating function of `Nat.Partition.countRestricted n m` is
+$$
+\prod_{i = 1}^{\infty} \sum_{j = 0}^{m - 1} X^{ij}
+$$ -/
+theorem hasProd_powerSeriesMk_card_countRestricted [T2Space M] {m : ℕ} (hm : 0 < m) :
+    HasProd (fun i ↦ ∑ j ∈ Finset.range m, X ^ ((i + 1) * j))
+    (PowerSeries.mk fun n ↦ (#(countRestricted n m) : M)) := by
+  rcases subsingleton_or_nontrivial M with h | h
+  · simp [Subsingleton.eq_one, HasProd]
+  convert hasProd_genFun (fun i c ↦ if c < m then (1 : M) else 0) using 1
+  · ext1 i
+    rw [Finset.sum_range_eq_add_Ico _ hm, Finset.sum_Ico_eq_sum_range]
+    congrm $(by simp) + ?_
+    trans ∑ k ∈ range (m - 1), (if k + 1 < m then (1 : M) else 0) • X ^ ((i + 1) * (k + 1))
+    · refine Finset.sum_congr rfl fun b hn ↦ ?_
+      rw [add_comm 1 b]
+      have : b + 1 < m := by grind
+      simp [this]
+    · exact (tsum_eq_sum (fun b hb ↦ smul_eq_zero_of_left (by simpa using hb) _)).symm
+  · simp_rw [genFun, countRestricted, Finset.card_filter, Finset.prod_boole]
+    simp
+
+end Semiring
+
+section Ring
+variable [CommRing M] [NoZeroDivisors M]
+
+private theorem aux_mul_one_sub_X_pow [T2Space M] [IsTopologicalRing M] {m : ℕ} (hm : 0 < m) :
+    (∏' i, if ¬m ∣ i + 1 then ∑' j, (X : M⟦X⟧) ^ ((i + 1) * j) else 1) *
+    ∏' i, (1 - X ^ (i + 1)) = ∏' i, (1 - X ^ ((i + 1) * m)) := by
+  rcases subsingleton_or_nontrivial M with h | h
+  · simp [Subsingleton.eq_one]
+  rw [← Multipliable.tprod_mul (hasProd_powerSeriesMk_card_restricted M (¬ m ∣ ·)).multipliable
+    (WithPiTopology.multipliable_one_sub_X_pow _)]
+  simp_rw [ite_not, ite_mul, pow_mul]
+  conv in fun b ↦ _ =>
+    ext b
+    rw [WithPiTopology.tsum_pow_mul_one_sub_of_constantCoeff_eq_zero (by simp)]
+  refine tprod_eq_tprod_of_ne_one_bij (fun i ↦ (i.val + 1) * m - 1) ?_ ?_ ?_
+  · intro a b h
+    rw [tsub_left_inj (by nlinarith) (by nlinarith), mul_left_inj' (hm.ne.symm), add_left_inj] at h
+    exact SetCoe.ext h
+  · suffices ∀ (i : ℕ), m ∣ i + 1 → ∃ a, (a + 1) * m - 1 = i by simpa
+    intro i hi
+    obtain ⟨j, hj⟩ := dvd_def.mp hi
+    refine ⟨j - 1, Nat.sub_eq_of_eq_add ?_⟩
+    rw [hj, mul_comm m j, Nat.sub_add_cancel (by grind)]
+  · intro i
+    have : (i + 1) * m - 1 + 1 = (i + 1) * m := Nat.sub_add_cancel (by nlinarith)
+    simp [this, pow_mul]
+
+omit [TopologicalSpace M] in
+theorem powerSeriesMk_card_restricted_eq_powerSeriesMk_card_countRestricted
+    {m : ℕ} (hm : 0 < m) :
+    (PowerSeries.mk fun n ↦ (#(restricted n (¬ m ∣ ·)) : M)) =
+    PowerSeries.mk fun n ↦ (#(countRestricted n m) : M) := by
+  let _ : TopologicalSpace M := ⊥
+  have _ : DiscreteTopology M := ⟨rfl⟩
+  refine ((hasProd_powerSeriesMk_card_restricted M (¬ m ∣ ·)).tprod_eq.symm.trans ?_).trans
+    (hasProd_powerSeriesMk_card_countRestricted M hm).tprod_eq
+  rcases subsingleton_or_nontrivial M with h | h
+  · simp [Subsingleton.eq_one]
+  apply mul_right_cancel₀ (WithPiTopology.tprod_one_sub_X_pow_ne_zero M)
+  apply (aux_mul_one_sub_X_pow M hm).trans
+  rw [← Multipliable.tprod_mul (hasProd_powerSeriesMk_card_countRestricted M hm).multipliable
+    (WithPiTopology.multipliable_one_sub_X_pow _)]
+  exact tprod_congr (fun i ↦ by simp_rw [pow_mul, geom_sum_mul_neg])
+
+end Ring
+
+theorem card_restricted_eq_card_countRestricted (n : ℕ) {m : ℕ} (hm : 0 < m) :
+    #(restricted n (¬ m ∣ ·)) = #(countRestricted n m) := by
+  simpa using PowerSeries.ext_iff.mp
+    (powerSeriesMk_card_restricted_eq_powerSeriesMk_card_countRestricted ℤ hm) n
+
+end Nat.Partition

docs/1000.yaml

@@ -3148,6 +3148,9 @@ Q5552370:
 
 Q5566491:
   title: Glaisher's theorem
+  decl: Nat.Partition.card_restricted_eq_card_countRestricted
+  authors: Weiyi Wang
+  date: 2025
 
 Q5567394:
   title: Gleason's theorem