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Well, and for what it worth an elementary proof. Let $\mathcal{F}_k$ denote the set of forests with $k$ trees, vertex set $\{1,2,\ldots,n+1\}$, and one labelled root in every component. Define the wei …

Another way to find the value of the sum $$ S:=\sum_{T} \prod_{i=1}^{n+1} d_i(T)! $$ using Cayley formula $$ \sum_{T} x_1^{d_1(T)}x_2^{d_2(T)}\cdots x_{n+1}^{d_{n+1}(T)} = x_1x_2\cdots x_{n+1} (x_1+x …

All these conditions are equivalent to six lines $AB'C''=a$, $CB'A''=b$, $CA'B''=c$, $BA'C''=d$, $BC'A''=e$, $AC'B''=f$ being tangent to the same conic (by Brianchon theorem and its inverse). For thes …

I think, it does. By change of coordinates, you may suppose that $Z$ contains the origin and the tangent vector space is the hyperplane $\{x_n=0\}$. Then, by implicit function theorem, for small enoug …

Let $N$ be a positive integer and $c_1,\ldots,c_N$ non-negative real numbers. Denote $f(x)=((x+c_1)\ldots (x+c_N))^{-1}$. Lemma 1. For all integer $d\geqslant 0$ and all $x>0$ we have $(-1)^df^{(d)}(x …

Denote these subsets which sum up to $s$ by $\alpha, \beta, \gamma$. Partition each subset $\alpha, \beta, \gamma$ onto two disjoint parts: $\alpha=\alpha_1\sqcup \alpha_2$,$\beta=\beta_1\sqcup \beta_ …

Let $A_0\subset A$ be the set where $\ell$ attains a minimum on $A$. It is a face of some dimension $k<d$. If $k=d-1$, we are done. Assume that $k<d-1$. Without loss of generality, $0\in A_0$ and more …

For what it worth, here is a combinatorial proof. We start with a known Lemma 1. Let $m$ be an even positive integer. Then the number of permutations of $[m]$ with only odd cycles equals to the number …

Consider all partitions $\lambda$ with distinct parts not exceeding $n$ and sum up $(-1)^{r(\lambda)+1}q^{|\lambda|}$, where $r$ goes for the number of parts. You may count the sum by fixing the small …

Note that $u(n)=|\Theta(n+2)|$ where $\Theta(k)$ is the set of partitions of $k$ without 1's with one part being labelled (i.e., multisets of integers greater than 1 summing up to $k$ with one labelle …

You can simply work in the unique factorization domain $R:=\mathbb{Z}[\omega]$, where $\omega=(-1+\sqrt{-3})/2$ is a cubic root of 1. $R$ has 6 units $\pm 1$, $\pm \omega$, $\pm \omega^2$. If 3 divide …

Well, Binet–Cauchy plus Lindström–Gessel–Viennot work indeed. Let $i,j$ vary from 0 to $m-1$, and $k$ vary from $0$ to $m+n-1$. Denote also $k^*=m+n-1-k$, so $k^*$ also varies from 0 to $m+n-1$. Consi …

Here is a proof of this fact. We start with a standard Lemma 1. Any prime divisor $q$ of $1+x+x^2$ for an integer $x$ is either equal to 3 or is congruent to 1 modulo $3$. Proof. If, on the contrary, …

Here is a proof that the variance of $d_T(v)$ does not exceed $\frac14(\deg v-1)$. For every edge $e\in E$ take a variable $x_e$ and consider the polynomial $$P:=\sum_T \prod_{e\in T} x_e,$$ where the …

This is exactly Rado theorem on independent transversal (for the corresponding linear matroid). Strictly speaking, to use it you should deal with finite sets, but a dimension of a vector subspace alwa …