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Consider a binary tree constructed as the following. Given a node with a some value $x$, I construct two children nodes each having value $l(x)$ and $r(x)$ respectively. I repeat the same on the ...

Let $\mathbb N_0^\ast$ be the set of all finite words/sequences over $\mathbb N_0$ and $\varepsilon$ the empty word. For a word $a=(a_1,\ldots,a_n)$ we define $\operatorname{len}a:=n$, $\Sigma a:=\...

This is a cross-post from math.stackexchange.com$^{[1]}$, since the bounty there didn't lead to any new insights. For reference, The Frog game is the generalization of the Frog Jumping (see it on ...

I have come across the following stochastic process which seems very elementary, although I do not know any appropriate terminology for it; I greatly appreciate any suggestions! Suppose I am given ...

While working on some computations on Hilbert schemes, I came across the following combinatorial problem. Let $D(k,n)$ be the weighted number of binary trees (children are left/right) with $n$ ...

Let $ c_{n,k} $ be the Simsun permutations$^1$ defined by the following relations: $\displaystyle c_{n,0} = 1, \hspace{0.1cm} (n \ge 1);$ $$ c_{n,k} = (k+1) c_{n-1,k} +(n-2k+1) c_{n-1,k-1}, \hspace{0....