How to prove combinatorial identity $\sum_{k=0}^s{s\choose k}{m\choose k}{k\choose m-s}={2s\choose s}{s\choose m-s}$?

Question

The following combinatorial identity have been verified via maple, but I can not prove it. Who can prove it without WZ mehtod? $$\sum_{k=0}^s{s\choose k}{m\choose k}{k\choose m-s}={2s\choose s}{s\choose m-s}$$.

Answer 1

Use subset-of-a-subset**, isolate non-index term, and then use Vandermonde*** to arrive at the result. $$\begin{align} \sum_{k=0}^s \color{blue}{\binom sk}\binom mk \color{blue}{\binom k{m-s}} &=\sum_{k=0}^{s}\color{blue}{\binom sk{\binom k{m-s}}} \binom mk \\ &=\sum_{k=0}^{s}\color{blue}{\binom s{m-s}\binom{2s-m}{k-m+s}}\binom mk\\ &=\binom s{m-s}\sum_{k=0}^{s}\binom{2s-m}{\color{red}k-m+s}\binom m{m\color{red}{-k}}\\ &=\binom s{m-s}\binom{2s}s \\&=\binom{2s}s\binom s{m-s}\qquad \blacksquare \end{align}$$

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** Subset-of-a-subset: $$\color{blue}{\binom ab\binom bc=\binom ac \binom {a-c}{b-c}}$$

*** Vandermonde: $$\sum_{r=0}^{a-b} \binom a{\color{red}r+b} \binom c{d\color{red}{-r}}=\binom {a+c}{b+d}$$

Answer 2

Let $n=m-s$; we may assume that $0\le n\le s$, as otherwise both sides are zero. The righthand side is the number of ordered pairs $\langle A,B\rangle$ such that $A$ is an $s$-subset of $[2s]$ and $B$ is an $m$-subset of $[s]$.

So is the lefthand side, though this is less obvious. Note that the $k$ term is zero unless $n\le k\le s$. For those $k$ we first pick $k$ elements of $[s]$ and then pick $n$ of those elements to be $B$. This leaves $k-n$ of the chosen elements unused; they will be $A\cap([s]\setminus B)$. The remaining

$$s-(k-n)=s+n-k=m-k$$

elements of $A$ must come from $B\cup([2s]\setminus[s])$, a set that has $s+n=m$ elements. Altogether there are

$$\binom{s}k\binom{k}n\binom{m}{m-k}=\binom{s}k\binom{k}n\binom{m}{k}$$

to complete the choice of $A$ and $B$ for this value of $k$, and summing over $k$ now yields the result.

Answer 3

By way of enrichment here is another algebraic proof using basic complex variables.

Suppose we are trying to show that $$\sum_{k=0}^n {n\choose k} {m\choose k} {k\choose m-n} = {2n\choose n}{n\choose m-n}.$$

Introduce the integral representations $${m\choose k} = \frac{1}{2\pi i} \int_{|z|=\epsilon} \frac{(1+z)^m}{z^{k+1}} \; dz \quad\text{and}\quad {k\choose m-n} = \frac{1}{2\pi i} \int_{|w|=\epsilon} \frac{(1+w)^k}{w^{m-n+1}} \; dw.$$ where $m\ge n.$

This gives for the sum the representation $$\frac{1}{2\pi i} \int_{|z|=\epsilon} \frac{(1+z)^m}{z} \frac{1}{2\pi i} \int_{|w|=\epsilon} \frac{1}{w^{m-n+1}} \sum_{k=0}^n {n\choose k} \left(\frac{1+w}{z}\right)^k \; dw \; dz \\ = \frac{1}{2\pi i} \int_{|z|=\epsilon} \frac{(1+z)^m}{z} \frac{1}{2\pi i} \int_{|w|=\epsilon} \frac{1}{w^{m-n+1}} \frac{(1+w+z)^n}{z^n} \; dw \; dz.$$ Extracting the inner residue we obtain $$\frac{1}{2\pi i} \int_{|z|=\epsilon} \frac{(1+z)^m}{z^{n+1}} {n\choose m-n} (1+z)^{2n-m} \; dz \\ = {n\choose m-n} \frac{1}{2\pi i} \int_{|z|=\epsilon} \frac{(1+z)^{2n}}{z^{n+1}} \; dz \\ = {2n\choose n} {n\choose m-n}.$$

We have not made use of the properties of complex integrals here so this computation can also be presented using just algebra of generating functions.

There is a similar but somewhat more advanced calculation at this MSE link.

Apparently this method is due to Egorychev although some of it is probably folklore.