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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 $x,y$.

we’ll proffer you the answer to co.combinatorics – Two magnificient weighted sums over binary phrases query by way of our community which brings all of the solutions from a number of reliable sources.

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