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Let $A$ and $B$ be nonempty finite sets of cardinalities $n$ and $m$, respectively. The distance between two functions $f, g : A \to B$ is defined as the number of disagreements between them, that is, …

The identity can be rewritten as $(a+b+c)^3=a^3+b^3+c^3+3(a+b)(a+c)(b+c)$ by means of a linear change of variables $a:=(−x+y+z)/2$, etc. Let $T$ be a circle of length $a+b+c$, and let's chop it into …

Let $n$, $k$, $r_1, \dots, r_k$ be positive integers. For each $i \in [k]:=\{1,\dots,k\}$, suppose we are given $n$ permutations of the the set $[r_i]$, that is $f_1^{(i)}, \dots, f_n^{(i)}$ in $\math …

I need the following "color interpolation lemma". Actually I know a way to prove it, but I'm not very satisfied with that proof. Lemma. Let $G=(V,E)$ be a (properly) colored graph with colors $1, …

Take an $n \times n$ random matrix whose entries are i.i.d. with uniform distribution in $[0,1]$. Look at the sums of the elements of each row and then permute the rows so that these sums form an incr …