combinatorics - What is a combinatorial proof exactly?
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. $\endgroup$
What is combinatorics? - Mathematics Stack Exchange
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 ...
Good Book On Combinatorics - Mathematics Stack Exchange
In my opinion, Cohen's Basic Techniques of Combinatorial Theory is a good introduction for who first learning about the subject. If you want to see many outstanding ideas, I suggest Proofs that Really Count by Benjamin/Quinn. For complete studies, Stanley's famous 2 …
Where are good resources to study combinatorics?
2012年7月16日 · As far as book are concerned, my favorite basic combinatorics books are Basic Techniques of Combinatorial Theory by Daniel I.A. Cohen and Combinatorics and Graph Theory by Harris, Hirst and Mossinghof. Cohen, in particular, is a great resource for questions which will make you think deeply and expand your horizons.
Combinatorial interpretation of sum of squares, cubes
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 $0 \leq i,j < k \leq n$.
Intuitive (combinatorial) proof of $2^n=\\sum_{k=0}^n {n\\choose …
2018年1月30日 · Since my favorite answer (counting subsets) has already been given, i will try an inductive proof... The following is known as Pascal's Formula: $${n\choose k}={n-1 \choose k}+{n-1\choose k-1}$$ It's easy to see this by counting subsets...
combinatorics - general definition of combinatorial problem ...
I was looking for a general definition of combinatorial problems, and i found this definition : A combinatorial problem is defined by a couple (S,C) associated with a request R, where : S is a finite or infinite set of potential solutions. C is a finite set of properties to be satisfied. R is a request specifying the objective of the resolution.
combinatorics - A comprehensive list of binomial identities ...
2011年8月24日 · In general, Gould's work is a great resource for this sort of thing; he has spent much of his career collecting and proving combinatorial identities. Added: Another useful reference is John Riordan's Combinatorial Identities. It's hard to pick one of its 250 pages at random and not find at least one binomial coefficient identity there ...
combinatorics - How to calculate a combinatorial sum
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
Combinatorial Analysis: Fermat's Combinatorial Identity
I was looking through practice questions and need some guidance/assistance in Fermat's combinatorial identity. I read through this on the stack exchange, but the question was modified in the latest edition of my book. The book asks for a combinatorial argument (no computations needed) to establish the identity: