Intereting Posts

If $f^2$ is Riemann Integrable is $f$ always Riemann Integrable?
Family of definite integrals involving Dedekind eta function of a complex argument, $\int_0^{\infty} \eta^k(ix)dx$
Limit $\lim_{x\to\infty}x\tan^{-1}(f(x)/(x+g(x)))$
What are some meaningful connections between the minimal polynomial and other concepts in linear algebra?
Is the intersection of two f.g. projective submodules f.g.?
When are two norms equivalent on a Banach space?
How to determine pointwise limit/uniform convergence.
Taking the automorphism group of a group is not functorial.
Independence of $H(f)=\int_M \alpha \wedge f^* \beta$ on choice of $d\alpha=f^*\beta$?
Natural derivation of the complex exponential function?
Categorical description of algebraic structures
Eigenvalues of a $2 \times 2$ block matrix where every block is an identity matrix
Why does “separable” imply the “countable chain condition”?
“Support function of a set” and supremum question.
Proving Cauchy condensation test

I encountered this particular combinatorial identity :$$\begin{align}&1\dbinom{n-1}{r-1}+2\dbinom{n-2}{r-1}+\cdots+(n+1-r)\dbinom{r-1}{r-1}\\=&\dbinom{n}{r}+\dbinom{n-1}{r}+\cdots+\dbinom{r}{r}\\=& \dbinom{n+1}{r+1}\end{align}$$How does one figure it out?

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- Combinatorial proof for $\binom{n}{a}\binom{a}{k}\binom{n-a}{b-k} = \binom{n}{b}\binom{b}{k}\binom{n-b}{a-k}$?
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- Combinatorial proof involving partitions and generating functions
- Generating functions for combinatorics
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- Limit of $\sum_{i=1}^n \left(\frac{{n \choose i}}{2^{in}}\sum_{j=0}^i {i \choose j}^{n+1}\right)$
- Find smallest number which is divisible to $N$ and its digits sums to $N$
- Product of simplicial complexes?

$\binom{n+1}{r+1}$ is of course the number of ways to choose $r+1$ numbers from the set $\{0,1,\ldots,n\}$. Now break those sets into categories according to the smallest number chosen. If $k$ is the smallest number chosen, the remaining $r$ numbers must come from the $(n-k)$-element set $\{k+1,\ldots,n\}$, so there are $\binom{n-k}r$ ways to choose them. The smallest number chosen can be anywhere from $0$ through $n-r$, so

$$\binom{n+1}{r+1}=\sum_{k=0}^{n-r}\binom{n-k}r\;.$$

Finally, categorize the $(r+1)$-element subsets according to their **second-smallest** elements. Suppose that the second-smallest element of a set is $k$. Then $r-1$ elements must be chosen from the $n-k$ elements in $\{k+1,\ldots,n\}$, and the smallest element must be chosen from the $k$ elements in $\{0,1,\ldots,k-1\}$; these choices can be made in a total of $k\binom{n-k}{r-1}$ ways. The second-smallest element can be anything from $1$ through $n-r+1$, so

$$\binom{n+1}{r+1}=\sum_{k=1}^{n-r+1}k\binom{n-k}{r-1}\;.$$

This is what they call the Hockey-Stick Identity or the Chu-Shih-Chieh’s Identity as I have encountered it in the book Principle and Techniques in Combinatorics by Chen and Koh. You can read about it from here. ðŸ™‚

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