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c8720e4 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 | import Mathlib
import OAI.RepresentationTheory.PartialPermutation.DiagramCounts
namespace OAI
section
namespace PartialPermutation
namespace DiagramCounting
noncomputable section
open Finset
lemma shift_injective : Function.Injective (fun x : ℕ × ℕ => (x.1+1,x.2)) := by
rintro ⟨i,j⟩ ⟨k,l⟩ h
simp only [Prod.mk.injEq] at h
exact Prod.ext (Nat.add_right_cancel h.1) h.2
def dropTop (μ : YoungDiagram) : YoungDiagram where
cells := μ.cells.preimage (fun x : ℕ × ℕ => (x.1+1,x.2)) shift_injective.injOn
isLowerSet := by
intro y x hxy hy
simp only [Finset.mem_coe,Finset.mem_preimage] at hy ⊢
have hh : (x.1+1,x.2) ≤ (y.1+1,y.2) := ⟨Nat.add_le_add_right hxy.1 1,hxy.2⟩
exact μ.isLowerSet hh hy
lemma mem_dropTop (μ : YoungDiagram) (i j : ℕ) :
(i,j) ∈ dropTop μ ↔ (i+1,j) ∈ μ := by
change (i,j) ∈ μ.cells.preimage (fun x : ℕ × ℕ => (x.1+1,x.2)) shift_injective.injOn ↔ _
exact Finset.mem_preimage
def lowerEquiv (μ : YoungDiagram) : (dropTop μ).cells ≃ {x : μ.cells // x.1.1≠0} where
toFun x := ⟨⟨(x.1.1+1,x.1.2), (mem_dropTop μ _ _).mp x.2⟩,Nat.succ_ne_zero _⟩
invFun x := ⟨(x.1.1.1-1,x.1.1.2), by
apply (mem_dropTop μ _ _).mpr
have he : x.1.1.1-1+1=x.1.1.1 := by have h:=x.2; omega
rw [he]; exact x.1.2⟩
left_inv x := by apply Subtype.ext; simp
right_inv x := by
apply Subtype.ext; apply Subtype.ext; apply Prod.ext
· have h:=x.2; dsimp; omega
· rfl
def rowEquiv (μ : YoungDiagram) : {x : μ.cells // x.1.1=0} ≃ Fin (μ.rowLen 0) where
toFun x := ⟨x.1.1.2, by have h:=μ.mem_iff_lt_rowLen.mp x.1.2; simpa [x.2] using h⟩
invFun i := ⟨⟨(0,i), μ.mem_iff_lt_rowLen.mpr i.2⟩,rfl⟩
left_inv x := by apply Subtype.ext; apply Subtype.ext; exact Prod.ext x.2.symm rfl
right_inv _ := rfl
lemma dropTop_card (μ : YoungDiagram) :
(dropTop μ).cells.card = μ.cells.card-μ.rowLen 0 := by
have h := Fintype.card_congr (lowerEquiv μ)
rw [Fintype.card_subtype_compl] at h
have he := Fintype.card_congr (rowEquiv μ)
simpa only [Fintype.card_coe,Fintype.card_fin,he] using h
lemma eq_of_dropTop_row (μ ν : YoungDiagram) (hd : dropTop μ=dropTop ν)
(hr : μ.rowLen 0=ν.rowLen 0) : μ=ν := by
apply YoungDiagram.ext
apply Finset.ext
rintro ⟨i,j⟩
cases i with
| zero => exact μ.mem_iff_lt_rowLen.trans (hr ▸ ν.mem_iff_lt_rowLen.symm)
| succ i =>
change (i+1,j) ∈ μ ↔ (i+1,j) ∈ ν
rw [← mem_dropTop μ, ← mem_dropTop ν,hd]
lemma transpose_card (μ : YoungDiagram) : μ.transpose.cells.card=μ.cells.card := by
have e : μ.cells ≃ μ.transpose.cells := {
toFun := fun x => ⟨x.1.swap, by simpa only [YoungDiagram.mem_cells,
YoungDiagram.mem_transpose,Prod.swap_swap] using x.2⟩
invFun := fun x => ⟨x.1.swap,YoungDiagram.mem_transpose.mp x.2⟩
left_inv := fun _ => rfl
right_inv := fun _ => rfl }
simpa only [Fintype.card_coe] using (Fintype.card_congr e).symm
def ShapeDeficit (n k : ℕ) :=
{μ : Shape n // μ.1.cells.card-max (μ.1.rowLen 0) (μ.1.colLen 0)=k}
instance (n k : ℕ) : Fintype (ShapeDeficit n k) := by
classical
unfold ShapeDeficit
infer_instance
def deficitCode (n k : ℕ) (μ : ShapeDeficit n k) : Bool × Shape k :=
if h : μ.1.1.colLen 0 ≤ μ.1.1.rowLen 0 then
(true,⟨dropTop μ.1.1, by rw [dropTop_card]; simpa only [max_eq_left h] using μ.2⟩)
else
(false,⟨dropTop μ.1.1.transpose, by
rw [dropTop_card,transpose_card,YoungDiagram.rowLen_transpose]
simpa only [max_eq_right (le_of_not_ge h)] using μ.2⟩)
lemma deficitCode_injective (n k : ℕ) : Function.Injective (deficitCode n k) := by
intro μ ν he
unfold deficitCode at he
split_ifs at he with hμ hν hν
· have hd : dropTop μ.1.1=dropTop ν.1.1 := congrArg (fun t : Bool × Shape k => t.2.1) he
have hcμ := μ.2
have hcν := ν.2
rw [μ.1.2,max_eq_left hμ] at hcμ
rw [ν.1.2,max_eq_left hν] at hcν
have hleμ := rowLen_le_card μ.1.1 0
have hleν := rowLen_le_card ν.1.1 0
rw [μ.1.2] at hleμ
rw [ν.1.2] at hleν
apply Subtype.ext; apply Subtype.ext
exact eq_of_dropTop_row _ _ hd (by omega)
· have hh := congrArg Prod.fst he; cases hh
· have hh := congrArg Prod.fst he; cases hh
· have hd : dropTop μ.1.1.transpose=dropTop ν.1.1.transpose :=
congrArg (fun t : Bool × Shape k => t.2.1) he
have hcμ := μ.2
have hcν := ν.2
rw [μ.1.2,max_eq_right (le_of_not_ge hμ)] at hcμ
rw [ν.1.2,max_eq_right (le_of_not_ge hν)] at hcν
have hleμ := colLen_le_card μ.1.1 0
have hleν := colLen_le_card ν.1.1 0
rw [μ.1.2] at hleμ
rw [ν.1.2] at hleν
have htr := eq_of_dropTop_row _ _ hd (by
simp only [YoungDiagram.rowLen_transpose]; omega)
apply Subtype.ext; apply Subtype.ext
have hh := congrArg YoungDiagram.transpose htr
simpa only [YoungDiagram.transpose_transpose] using hh
lemma deficit_count_le (n k : ℕ) : Fintype.card (ShapeDeficit n k) ≤ 2*partitionCount k := by
simpa only [Fintype.card_prod,Fintype.card_bool,partitionCount] using
Fintype.card_le_of_injective (deficitCode n k) (deficitCode_injective n k)
end
end DiagramCounting
end PartialPermutation
end
end OAI
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