TITLE: rational function identity
QUESTION [17 upvotes]: I just had to make use of an elementary rational function identity (below).  The proof is a straightforward exercise, but that isn't the point.  First, "my" identity is almost surely
not original, but I don't have a reference for it.  Perhaps someone knows it (like a lost cat without a collar) or, more likely, could spot this as a special case of a more general identity.  Second, the obvious proof is not much of an explanation: a combinatorial identity often arises for a conceptual reason, and I'd be happy to hear if anyone sees mathematics behind this one.  
Let $f(x_1,\ldots,x_n)=\prod_{p=1}^n\big(\sum_{i=p}^n x_i\big)^{-1}$.  Then
$$
f(x_1,\ldots,x_n)+f(x_2,x_1,x_3,\ldots,x_n)+\cdots+f(x_2,\ldots,x_n,x_1)=\big(\sum_{i=1}^n x_i\big)/x_1\cdot f(x_1,\ldots,x_n),
$$
where $x_1$ appears as the $i$th argument to $f$ in the $i$th summand on the left side, for $1\leq i\leq n$.  But why?

REPLY [10 votes]: A simple proof of the Sh(a,b) cat, using iterated integrals, is as follows.
Note that
$$
f(x_1,\ldots,x_n)=\int_{1>t_1>\cdots>t_n>0} dt_1\cdots dt_n
\ t_1^{x_1-1}\cdots t_n^{x_n-1}\ .
$$
Littlewood's identity follows from changing variables using the permutation so as to keep the
integrand fixed. Then one has a sum of simplices (corresponding to all possible relative orderings of the variables) which recombines into a cube of integration
$[0,1]^n$.
The proof of the Sh(a,b) identity follows the same idea. Here the total volume of integration is a product of simplices which is broken into a union of simplices.
This is probably well known to people working with moulds, operads, etc.
An additional remark:
Littlewood's identity follows from Lemma II.2 in my article
"Trees forests and jungles: a botanical garden for cluster expansions" 
with V. Rivasseau.
To see this, extract the coefficient of the highest degree monomial in the v variables (notations of that article), then specialize the u variables to the case where
$u_{i, i+1}=x_i$ and all other pair variables are zero (killing all edges of the complete graph which are not in a `spanning chain').
The Lemma in our article is related to many other topics in mathematical physics such as the Wilson-Polchinski renormalization group equation,
see e.g. these slides.