2016年1月11日 · The other answers also talk about other ways to prove using Combinatorial Arguments. Combinatorics is a wide branch in Math, and a proof based on Combinatorial arguments can use many various tools, such as Bijection, Double Counting, Block Walking, et cetera, so a combinatorial proof may involve any (or a combination) of these. $\endgroup$

Aspects of combinatorics include counting the structures of a given kind and size (enumerative combinatorics), deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory), finding "largest", "smallest", or "optimal" objects (extremal combinatorics and ...

Here's a combinatorial proof for $$\sum_{k=1}^n k^2 = \binom{n+1}{2} + 2 \binom{n+1}{3},$$ which is just another way of expressing the sum. Both sides count the number of ordered triples $(i,j,k)$ with $0 \leq i,j < k \leq n$.

2024年4月12日 · $\begingroup$ Might not work for your tastes, but Lovasz has a lovely book called "Combinatorial Problems and Exercises", that develops a lot of theory through problems (which have both extensive hints and solutions provided). I found it quite useful in developing intuitions about combinatorial objects.

2016年7月22日 · Combinatorial Proof regarding Falling Factorials and sums of Falling Factorials Hot Network Questions How, anatomically, can an alien species have pointed digits (without claws or nails)?

2014年10月9日 · The book Combinatorial Identities from John Riordan ($1968$) is a wonderful classic with thousands of binomial identities which are systematically organised. But it does not typically provide combinatorial proofs. It's a great reference to search for different classes of combinatorial identities.

Combinatorial proof of $\sum_{k=0}^{n} \binom{3n-k}{2n} = \binom{3n+1}{n}$ Hot Network Questions How (in)efficient would a rocket be that flew to orbital heights, hovered for a while, and then fell back down instead of going into orbit and back?

Give a combinatorial proof of the following identity: $$3^n=\sum_{i=0}^{n}\binom{n}{i}2^{n-i}$$ I can't see any counting argument that would yield $3^n$, and the right hand side is also pretty opaque. For some reason I really really suck at doing these proofs - I just started my first combinatorics course.

Combinatorial (and Algebraic Proof) of an Identity Involving Lah Numbers 2 Find a Closed form for the Combinatorial Sum $\sum_{k=0}^m\binom{n-k}{m-k} $ and Provide a Combinatorial Proof of the Result

2021年1月14日 · $\begingroup$ Following the clue given by @QiaochuYuan, I found the answer presented as Identity 17 in Combinatorial sums and finite differences by Michael Z. Spivey (2007). $\endgroup$ – Matcha Commented Jan 14, 2021 at 2:23