109074.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

19 declarations, 1 contain sorry, 0 declared as axioms

Numerator of $\binom{6n-2}{2n} / \left(2 \binom{4n-1}{2n}\right)$

The sequence $a(n)$ is the numerator of $\binom{6n-2}{2n} / \left(2 \binom{4n-1}{2n}\right)$. The OEIS entry records the conjecture that this ratio equals $\text{A005156}(n) / \text{A005156}(n-1)$, where $\text{A005156}(n)$ is the number of $(2n+1) \times (2n+1)$ vertically symmetric alternating sign matrices (VSASMs). This follows from Robbins's VSASM product formula, proved by Kuperberg (2002).

References:

A109074
A005156
G. Kuperberg, "Symmetry classes of alternating-sign matrices under one roof," arXiv:math/0008184, Ann. of Math. 156 (2002), 835–866.

Imports (1)

FormalConjecturesUtil

def OeisA109074.frac The rational number defined by $\binom{6n-2}{2n} / \left(2 \binom{4n-1}{2n}\right)$, whose numerator is A109074 line 47 source
Source documentation
The rational number defined by $\binom{6n-2}{2n} / \left(2 \binom{4n-1}{2n}\right)$, whose numerator is A109074.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Rat
Proof uses: Nat.choose, HSub.hSub, HMul.hMul, OfNat.ofNat, HDiv.hDiv
9 instances, coercions and auxiliary constants left out.
```
def frac (n : ℕ) : ℚ :=
…
```
def OeisA109074.a The primary defining sequence a line 56 source
Source documentation
The primary defining sequence a. $a(n)$ is the numerator of $\binom{6n-2}{2n} / \left(2 \binom{4n-1}{2n}\right)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Int.natAbs, Rat.num, OeisA109074.frac
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA109074.a_0 line 60 source
extra axioms: _private.0.OeisA109074.a_0._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA109074.a, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA109074.a_1 line 63 source
extra axioms: _private.0.OeisA109074.a_1._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA109074.a, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA109074.a_2 line 66 source
extra axioms: _private.0.OeisA109074.a_2._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA109074.a, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA109074.a_3 line 69 source
extra axioms: _private.0.OeisA109074.a_3._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA109074.a, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA109074.a_4 line 72 source
extra axioms: _private.0.OeisA109074.a_4._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA109074.a, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.

def OeisA109074.trinaryLists Lists of length k with entries in {-1, 0, 1} line 77 source

Source documentation

Lists of length k with entries in {-1, 0, 1}.

no axioms what this means

Statement uses: Nat, List, Int

2 instances, coercions and auxiliary constants left out.

def trinaryLists : ℕ → List (List ℤ)
  | 0 => [[]]
  | k + 1 =>
    (trinaryLists k).flatMap fun xs => [(-1 : ℤ) :: xs, 0 :: xs, 1 :: xs]

/--
Checks that all prefix sums of `xs` starting from `acc` lie in `{0, 1}` and the final sum is `1`.
-/

def OeisA109074.isAsmLineAux Checks that all prefix sums of xs starting from acc lie in {0, 1} and the final sum is 1 line 85 source
Source documentation
Checks that all prefix sums of xs starting from acc lie in {0, 1} and the final sum is 1.

no axioms what this means
Statement uses: Int, List, Bool
2 instances, coercions and auxiliary constants left out.
```
def isAsmLineAux : ℤ → List ℤ → Bool
  | acc, [] => acc == 1
  | acc, x :: xs =>
    let s := acc + x
…
```
def OeisA109074.isAsmLine A list of integers in {-1, 0, 1} is a valid alternating sign matrix row/column if all of its prefix sums lie in {0, 1} and its total sum is 1 line 95 source
Source documentation
A list of integers in {-1, 0, 1} is a valid alternating sign matrix row/column if all of its prefix sums lie in {0, 1} and its total sum is 1.

no axioms what this means
Statement uses: List, Int, Bool
Proof uses: OeisA109074.isAsmLineAux, OfNat.ofNat
1 instances, coercions and auxiliary constants left out.
```
def isAsmLine (xs : List ℤ) : Bool :=
…
```
def OeisA109074.symAsmRows All vertically symmetric alternating sign matrix rows of length 2 * n + 1 line 101 source
Source documentation
All vertically symmetric alternating sign matrix rows of length 2 * n + 1.

no axioms what this means
Statement uses: Nat, List, Int
Proof uses: List.flatMap, List.filterMap, HAppend.hAppend, List.cons, List.nil, List.reverse, ite, Option, Eq, Bool, OeisA109074.isAsmLine, Bool.true, Option.some, Option.none, Neg.neg, OfNat.ofNat, OeisA109074.trinaryLists
5 instances, coercions and auxiliary constants left out.
```
def symAsmRows (n : ℕ) : List (List ℤ) :=
…
```
def OeisA109074.countVsasmRows Number of ways to complete k remaining vertically symmetric ASM rows given the current column partial sums colSums line 111 source
Source documentation
Number of ways to complete k remaining vertically symmetric ASM rows given the current column partial sums colSums.

no axioms what this means
Statement uses: List, Int, Nat
2 instances, coercions and auxiliary constants left out.
```
def countVsasmRows (rows : List (List ℤ)) : ℕ → List ℤ → ℕ
  | 0, colSums => if colSums.all (· == 1) then 1 else 0
  | k + 1, colSums =>
    let step := countVsasmRows rows k
…
```
def OeisA109074.b A005156 (offset 0): the number of $(2n+1) \times (2n+1)$ alternating sign matrices symmetric about the vertical axis (VSASMs) line 125 source
Source documentation
A005156 (offset 0): the number of $(2n+1) \times (2n+1)$ alternating sign matrices symmetric about the vertical axis (VSASMs).

no axioms what this means
Statement uses: Nat
Proof uses: OeisA109074.countVsasmRows, OeisA109074.symAsmRows, HAdd.hAdd, HMul.hMul, OfNat.ofNat, List.replicate, Int
6 instances, coercions and auxiliary constants left out.
```
def b (n : ℕ) : ℕ :=
…
```
theorem OeisA109074.b_0 line 129 source
no axioms what this means
Statement uses: Eq, Nat, OeisA109074.b, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA109074.b_1 line 132 source
no axioms what this means
Statement uses: Eq, Nat, OeisA109074.b, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA109074.b_2 line 135 source
no axioms what this means
Statement uses: Eq, Nat, OeisA109074.b, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA109074.b_3 line 138 source
no axioms what this means
Statement uses: Eq, Nat, OeisA109074.b, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA109074.b_4 line 141 source
extra axioms: _private.0.OeisA109074.b_4._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA109074.b, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA109074.conjecture Documented declaration: conjecture sorry line 153 source
Source documentation
"It is conjectured that $\binom{6n-2}{2n} / \left(2 \binom{4n-1}{2n}\right) = \text{A005156}(n+1)/\text{A005156}(n)$."
The OEIS comment indexes A005156 from $1$; for the $0$-indexed sequence b this states frac (n + 1) = b (n + 1) / b n. This holds by the product formula $$\text{A005156}(n) = \frac{1}{2^n} \prod_{k=1}^{n} \frac{(6k-2)!\,(2k-1)!}{(4k-1)!\,(4k-2)!}$$ conjectured by Robbins and proved by Kuperberg (2002).

depends on sorry what this means
Statement uses: Nat, Eq, Rat, OeisA109074.frac, HAdd.hAdd, OfNat.ofNat, HDiv.hDiv, OeisA109074.b
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture (n : ℕ) :
    frac (n + 1) = (b (n + 1) : ℚ) / (b n : ℚ) := by
…
```
Mentions: OeisA109074.frac, OeisA109074.b