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Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.

3 votes
Accepted

Covering sequences of words

Try $w_0 = b$, $w_1 = bab$, $w_2 = baba^2bab$, $w_3 = baba^2baba^3baba^2bab$, and recursively $w_{i+1} = w_ia^{i+1}w_i$. Then $\displaystyle \limsup_{i \rightarrow \infty} Pr(w_i,a^n) = \lim_{i \rig …
D. Ror.'s user avatar
  • 399
3 votes

Sum of multinomial coefficients (even distribution)

If you take the averaged sum over all choices of signs $$\frac{1}{2^k} \sum_{\varepsilon_i = \pm 1} (\varepsilon_1x_1 + \cdots + \varepsilon_kx_k)^r$$ we see that only the terms with even exponents …
Community's user avatar
  • 1
1 vote

Bound of area of intersection

Since the geometry is irrelevant and you are just considering $n$ subsets of equal measure, $I(n,S,k)$ is also the maximum value such that, given any $n$ measure-$S$ subsets of the unit-circumference …
D. Ror.'s user avatar
  • 399
1 vote

Ratio between number of nodes and leaves in a rooted binary tree

We can prove a lower bound on the expression by considering only vertices immediately below a leaf: $$\frac{[\sum_{v \in T \text{ not a leaf}}2^{h(v)}L(T(v))] + L(T)}{N(T)} \geq \frac{[\sum_{v \text{ …
D. Ror.'s user avatar
  • 399