Consider two d-uplets $u = (u\_1,...,u\_d)$ and $v = (v\_1, ..., v\_d)$ both living in $\mathbb{N}^d$ with $d$ a positive integer. They both verify $$(\*) \sum\_{i=1}^d u\_i = \sum\_{i=1}^d v\_i = k$$ with $k$ a positive integer, $k\geq d$ supposed to tend to infinity later on.
Given a constant $c \in \mathbb{N}$ such that $c<k$ ($c$ could also depend on $k$, e.g. $ c= \lceil k/d \rceil$), I would like to upper bound the cardinal of the following sets:
$$\{(u,v) | u \neq v, (\*), \sum\_{i=1}^d |u\_i -v\_i| \leq c \}. $$


It could also be written as bounding the cardinal of the set $$\{ u | u \neq v, (\*), \sum\_{i=1}^d |u\_i -v\_i| \leq c \} $$ for any $v$.


My knowledge of combinatorics is very low, I'd be glad to receive any reference or idea that could lead to the solution of this problem or similar ones. I do not even have intuition for the magnitude order of this quantity.



Counting the number of pair of d-uplets with upper bounded distance
This is not the best upper bound, but for the second set, ($\ast$) tells us about the norm of $u$ and $v$ and $\sum\_{i=1}^{d}|u\_i-v\_i|$ is the $L\_1$ distance between $u$ and $v$ (Manhattan metric). Ignoring $\ast$ we get an upper bound by counting the lattice points in the ball of radius $c$. Now considering $\ast$, we intersect this ball with the surface of a radius $k$ sphere (in this metric), this sharpens our bound. Hope this help you having a linear programming perspective.


