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Results tagged with co.combinatorics
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user 27513
Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.
3
votes
A circulant coin weighing problem
Here's a rather indirect suggestion, using ideas from compressed sensing. Let $\Phi$ be your weighing matrix. Denote by $v$ your vector of coin weights and by $u$ the all ones vector. Observe that $\P …
- 1,300
3
votes
Large Intersecting Subsets of a Set
As Gerhard has pointed out, if there exists a Hadamard matrix of order $4k$, then there exists a symmetric $(4k-1, 2k-1, k-1)$ design. There are $4k-1$ blocks in this design, each contains $2k-1$ poin …
- 1,300
2
votes
A combinatorial optimization problem
The "Hungarian method" is a well known technique for solving such problems. Google throws up many hits. An optimal solution can be found in time O(n^3). References etc. can be found on wikipedia.
htt …
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1
vote
Has this kind of design been studied before?
Under your hypotheses, the Frankl-Wilson theorem says that $|\mathcal{B}| \leq |X|$. More generally, if there are $s$ possible intersection sizes, $|\mathcal{B}| \leq \binom{|X|}{s}$.
I'm not aware …
- 1,300
8
votes
1
answer
249
views
Pairs of roots of unity whose real part satisfies a polynomial identity
Some motivation: The matrix $M$ is Butson Hadamard if the entries of $M$ are $k^{\textrm{th}}$ roots of unity (for some $k$), and distinct pairs of rows are orthogonal under the usual Hermitian inner …
- 1,300
2
votes
Permutations of $(Z/pZ)^*$
A similar concept is an orthomorphism of a group $G$. This is an automorphism $\theta: G \rightarrow G$ with the property that $g^{-1}\theta(g)$ is a bijection (equivalently an automorphism). Two orth …
- 1,300
5
votes
Accepted
Creating a Latin rectangle from a projective plane
By a theorem of Singer, the automorphism group of $PG(2,q)$ contains a cyclic subgroup $\langle\sigma\rangle$ which acts regularly on points and regularly on lines. Fix a base block, $B$, and list its …
- 1,300
9
votes
Cardinality of the maximum points of the determinant on matrices with entries in [-1, 1]
This response was too long for a comment, but is far from a complete answer.
Much of the research on the maximal determinant problem has focused on the Gram matrix, and used the theory of positive def …
- 1,300
6
votes
Isomorphism testing in STS(13)
There is a construction given in Theorem II.2.2.10 of the Handbook of
Combinatorial designs for a STS of order a prime of form $6t+1$. It is clear
that the resulting STS has a cyclic automorphism ac …
- 1,300
4
votes
Accepted
For which finite projective planes can the incidence structure be written as a circulant mat...
To answer your questions:
1) A projective plane admits a circulant incidence matrix if and only if the automorphism group contains a cyclic group acting regularly on points and regularly on blocks. E …
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