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co.combinatorics – Two magnificient weighted sums over binary phrases retort

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co.combinatorics – Two magnificient weighted sums over binary phrases

This query builds off of the earlier MO query Number of collinear methods to fullfil a grid.

Let $A(m,n)$ denote the clique of binary phrases $alpha=(alpha_1,alpha_2,ldots,alpha_{m+n-2})$ consisting of $m-1$ $0’s$ and $n-1$ $1’s$. Evidently $#A(m,n) = binom{m+n-2}{m-1}$.

For $alpha in A(m,n)$ and $1leq i leq m+n-2$, clique
$$ b^alpha_i := #{1leq j < icolon alpha_ineqalpha_j} +1.$$
$$ c^alpha_i := #{1leq j leq icolon alpha_i=alpha_j}=(i+1)-b^alpha_i.$$

The decision of the above-linked query implies that
$$ sum_{alpha in A(m,n)} frac{b^alpha_1b^alpha_2 cdots b^alpha_{m+n-2}}{b^alpha_{m+n-2} (b^alpha_{m+n-2}+b^alpha_{m+n-3})cdots (b^alpha_{m+n-2}+b^alpha_{m+n-3}+cdots+b^alpha_1) } = frac{mn}{(m+n-1)!}$$
Meanwhile, on this retort, it’s defined that
$$ sum_{alpha in A(m,n)} frac{1}{c^alpha_{m+n-2} (c^alpha_{m+n-2}+c^alpha_{m+n-3})cdots (c^alpha_{m+n-2}+c^alpha_{m+n-3}+cdots+c^alpha_1) } = frac{2^{m-1}2^{n-1}}{(2m-2)! (2n-2)!}$$

Considering the similarities of those two magnificient weighted sums over binary phrases, we query:

Question: Is there a extra widespread formulation which specializes to the above two formulation?

EDIT:

Since for any $alpha in A(m,n)$ we maintain
$$ {c_1^{alpha},c_2^{alpha},ldots,c_{m+n-2}^{alpha}} = {1,2,ldots,m-1,1,2,ldots,n-1},$$
we are able to rewrite that second sum to breathe
$$ sum_{alpha in A(m,n)} frac{c^alpha_1 c^alpha_2 cdots c^alpha_{m+n-2}}{c^alpha_{m+n-2} (c^alpha_{m+n-2}+c^alpha_{m+n-3})cdots (c^alpha_{m+n-2}+c^alpha_{m+n-3}+cdots+c^alpha_1) } = frac{2^{m-1}(m-1)!2^{n-1}(n-1)!}{(2m-2)! (2n-2)!},$$
to breathe aircraft extra just like the primary sum.

If we clique
$$ d^{alpha}_i = xb^{alpha}_i+yc^{alpha}_i,$$
then the above commentary explains that
$$ sum_{alpha in A(m,n)} frac{d^alpha_1 d^alpha_2 cdots d^alpha_{m+n-2}}{d^alpha_{m+n-2} (d^alpha_{m+n-2}+d^alpha_{m+n-3})cdots (d^alpha_{m+n-2}+d^alpha_{m+n-3}+cdots+d^alpha_1)}$$
has a product formulation for $x,yin {0,1}$. Maybe it has a product formulation for widespread $x,y$.

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