The reduced $Gr_2$-simplex recurrence conjecture

The four-neighbor recurrence holds for every reduced Gr₂-simplex entry, with no boundary exceptions. In fact, its entries admit the following explicit positive integral formula:

ξ_{ℓ,r,j} =(ℓ−1)!ℓ!(ℓ−r+1)/ [(r−j)!(j−1)!(ℓ−j+1)!(ℓ−r+j)!],

for 1≤j≤r≤ℓ, and ξ_{ℓ,r,j}=0 outside this range. Here ℓ=1 denotes the reduced singleton layer, as dictated by the actual reduced generating function.

For nonnegative A,B,C set d=A+B+C and

q(A,B,C)=d!(d+1)!(B+1)/ [A!C!(A+B+1)!(B+C+1)!].

Let q vanish if any argument is negative and define

Q_d(a,b,c)=∑_{A+B+C=d}q(A,B,C)a^A b^B c^C.

First, the source's explicit stable-envelope formula in Example 3.2 gives, with A=i−1,B=j−i−1,C=n−j,

F_n(i,j)=q(A−1,B,C)+2q(A,B−1,C)+q(A,B,C−1).

Indeed its displayed factorial bracket expands exactly as

(B+1)A(A+B+1) +2B(A+B+1)(B+C+1) +(B+1)C(B+C+1),

which proves the coefficient identity including all boundary cases. Consequently the original layer polynomial factors as

H_n(a,b,c)=(a+2b+c)Q_{n−3}(a,b,c).

Thus Q_d is precisely the reduced layer.

For a direct recurrence proof, interpret Q_d as the weight enumerator of nonnegative length-d walks starting at height zero with steps

U: height +1, weight b; D: height −1, weight ac/b; L_a: height 0, weight a; L_c: height 0, weight c.

The exponent B of b is the terminal height. For fixed A,B,C and s down-steps, the ballot principle gives

d!(B+1)/[(A−s)!(C−s)!s!(B+s+1)!]

walks. Summing over 0≤s≤min(A,C) and applying Chu–Vandermonde yields q(A,B,C), proving that these are exactly the reduced coefficients and are positive integers.

Appending the last step now gives the polynomial recurrence

Q_{d+1} =(a+b+c)Q_d+(ac/b)[Q_d−Q_d(a,0,c)].

The subtraction excludes precisely the height-zero walks for which a down-step is impossible. Identify the reduced position (ℓ,r,j) with the monomial

a^{r−j}b^{ℓ−r}c^{j−1}

in Q_{ℓ−1}. Extracting its coefficient from the four terms of the polynomial recurrence proves

ξ_{ℓ,r,j} =ξ_{ℓ−1,r,j} +ξ_{ℓ−1,r−1,j} +ξ_{ℓ−1,r−1,j−1} +ξ_{ℓ−1,r−2,j−1}

for every ℓ,r,j, using zero outside the triangular array. This is exactly the conjectured exception-free recurrence and simultaneously establishes the closed coefficient formula, integrality, positivity, and the companion divisibility conjecture.