# Tagged Questions

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### All $k$-regular subgraphs of $K_{n,n}$ have a perfect matching: a proof without Hall's Marriage Theorem?

There are several ways of describing this result: Theorem: For $k \in \{1,2,\ldots,n\}$, any $k$-regular subgraph of $K_{n,n}$ has a perfect matching (also known as a $1$-factor). I tend to ...
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### Tiling an $m\times n$ grid.

For natural numbers $m$ and $n$, an $m\times n$ grid of squares can be tiled with tiles of the form completely filling the grid, without overlapping, if and only if $m,n\geq2$ and $6\mid mn$. It ...
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### Intuition of Newton's identities, is it worth persuing these thoughts? Should be able to show it from special case.

I've "accidentally" stumbled on a special case of it (i=n) and it is very intuitive. Then there's some sort of "transpose" going on with the terms. So I'm thinking there might be something there. I ...
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### Combinatorics proof of “sum of (k choose m) with k from m up to n is equal to n+1 choose m+1”

I've already proved this statement algrebraically. I'm asked to prove it with combinatorics. So far I came up with, LHS= # ways to choose m apples from a total of m,m+1,...,n RHS= # ways to choose ...
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### An algebraic proof that a set has $2^n$ subsets. (I'm looking for an inductive argument.)

There will be duplicates of this, so let me explain why I am asking: I have become blind to what it may be, so I want hints. I am blind because I can do it "combinatorially". The question wishes me ...
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### Direct combinatorial proof of a sum identity on formal Lagrange polynomials

Let $k$ be a field and $K=k(x_0,x_1,\ldots, x_n)[x]$. Define $$\mathcal{L}_k(x)\triangleq \prod_{\substack{j=0\\ j\ne k}}^n\frac{x-x_j}{x_k-x_j}.$$ Is there a purely combinatorial way to show ...
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I trying to find the sign of this permutation: $\left(\begin{array}{cccccccccc} 1 & 2 & 3 & \cdots & \cdots & \cdots & \cdots & \cdots & n-1 & n\\ 2 & 4 & ... 3answers 571 views ### Combinatorial Proof of Multinomial Theorem - Without Induction or Binomial Theorem I've been trying to rout out an exclusively combinatorial proof of the Multinomial Theorem with bounteous details but only lighted upon this one - see P2. Any other helpful ones? ... 1answer 154 views ### Combinatorial proof of the fact$p$doesn't divide$ n \choose p^k$Let$p^k | n$and$p^{k+1} \nmid n$. Is there any combinatorial proof of the fact that$p \nmid {n \choose p^{k}} $? 3answers 269 views ### Further clarification needed on proof invovling generating functions and partitions (or alternative proof) Show with generating functions that every positive integer can be written as a unique sum of distinct powers of$2$. There are 2 parts to the proof that I don't understand. I will point them out ... 2answers 1k views ### How is Leibniz's rule for the derivative of a product related to the binomial formula? [duplicate] Possible Duplicate: “Binomial theorem”-like identities The binomial formula describes the expansion of the$n$th power of the sum$(a+b)$:$$(a+b)^n = \sum_{k = 0}^n {n\choose ... 3answers 320 views ### Proof of the duality of the dominance order on partitions Could anyone provide me with a nice proof that the dominance order$\leq$on partitions of an integer$n$satisfies the following: if$\lambda, \tau$are 2 partitions of$n$, then$\lambda \leq \tau ...
The general result in tropical geometry is $K$ algebraically closed valued field $I$ ideal of $K[x_1, \cdots, x_n]$ $V(I) = \lbrace \bar{a}\in K^n: f(\bar{a})=0 \text{ for all } f \in I \rbrace$. ...