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Results tagged with co.combinatorics
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user 135311
Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.
4
votes
0
answers
532
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How should the proof of the XYZ theorem be understood?
The XYZ Theorem of Shepp [1] states the following for a given poset $P$. Consider the probability space of all the linear extensions of $P$, where each possible extension is equally likely. Then for a …
- 4,219
3
votes
Does $\sum_{|S| \text{ even}} f(S) \le \sum_{|T| \text{ even}} f(T)$ hold for all nondecreas...
Assuming the inequality you would like to prove has the sum over odd subsets on the RHS, then you could consider the following counter-example for $n=3$: $f(\emptyset)=0$, $f(\{i\})=1$, $f(\{i, j\})=2 …
- 571
11
votes
3
answers
589
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Non-singular matrix with restricted entries
Given a set $S$ of integers with $1 \not\in S$, let us consider the set $\mathcal{M}$ of all the symmetric matrices $M$, such that:
All the diagonal entries of $M$ are equal to $1$.
All the off-diag …
- 37.7k
7
votes
1
answer
516
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Packing equal-size disks in a unit disk
Inspired by the delicious buns and Siu Mai in bamboo steamers I saw tonight in a food show about Cantonese Dim Sum, here is a natural question. It probably has been well studied in the literature, but …
- 5,429
26
votes
1
answer
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Generalization of Cauchy's eigenvalue interlacing theorem?
Cauchy's Interlacing Theorem says that given an $n \times n$ symmetric matrix $A$, let $B$ be an $(n-1) \times (n-1)$ principal submatrix of it, then the eigenvalues of $A$ and those of $B$ interlace. …
- 1,719