All Questions
Tagged with permanent co.combinatorics
10 questions
20
votes
2
answers
1k
views
Euler numbers and permanent of matrices
Motivated by Question 402249 of Zhi-Wei Sun, I consider the permanent of matrices
$$e(n)=\mathrm{per}\left[\operatorname{sgn} \left(\tan\pi\frac{j+k}n \right)\right]_{1\le j,k\le n-1},$$
where $n$ is ...
- 884
28
votes
3
answers
2k
views
Is every positive integer the permanent of some 0-1 matrix?
In the course of discussing another MO question we realized that we did not know the answer to a more basic question, namely:
Is it true that for every positive integer $k$ there exists a balanced ...
- 82.7k
6
votes
2
answers
309
views
Permanent of Nakayama algebras
See https://en.wikipedia.org/wiki/Nakayama_algebra for the definition of Nakayama algebras and define the permanent of such an algebra to be the permanent of its Cartan matrix.
(all algebras are ...
- 26.5k
3
votes
1
answer
308
views
Tangent numbers, secant numbers and permanent of matrices
Inspired by Question 402572, I consider the permanent of matrices
$$f(n)=\mathrm{per}(A)=\mathrm{per}\left[\operatorname{sgn} \left(\sin\pi\frac{j+2k}{n+1} \right)\right]_{1\le j,k\le n},$$
where $n$ ...
- 884
13
votes
1
answer
311
views
Permanent of the Coxeter matrix of a distributive lattice
Let $L$ be a finite distributive lattice with $n$ elements. Let $C=(c_{x,y})$ be the $n \times n$ matrix with entry 1 in case $x \leq y$ and 0 else.
The Coxeter matrix of $L$ is defined as the matrix $...
- 26.5k
12
votes
3
answers
892
views
Set partitions and permanents
Let $a(n)=$ Number of ordered set partitions of $[n]$ such that the smallest element of each block is odd.
...
- 884
7
votes
0
answers
349
views
Missing count in number of perfect matchings
Let $f(G)$ give number of perfect matchings of a graph $G$.
Denote $\mathcal N_{2n}=\{0,1,2,\dots,n!-1,n!\}$.
Denote collection of all $2n$ vertex balanced bipartite graph to be $\mathcal G_{2n}$.
...
- 13.9k
3
votes
0
answers
237
views
Multi-dimensional permanent of structured tensor
I am facing the multidimensional permanent
\begin{equation} \text{perm}(W) = \sum_{\sigma, \rho \in S_n} \prod_{j=1}^n W_{j, \sigma_j, \rho_j } \end{equation}
of a 3-tensor $W_{j,k,l}$ of ...
- 93
2
votes
2
answers
193
views
growth of the permanent of some band matrix
Consider such special band matrix of dimension $n$. It is a $0-1$ matrix, and only the first few diagonals are nonzero. Specifically,
$$ H_{ij} = 1 $$
if and only if $|i-j| \leq 2$.
How does the ...
- 265
0
votes
0
answers
245
views
Reduction from permanent to $(0,1)$-permanent and implication of $P \ne NP$
Valiant
shows reduction from counting the solutions of CNF formula $F$,$\#SAT(F)$
to computing permanent where $ Perm(A)= 4^{t(F)}\cdot \#SAT(F)$
for certain efficiently computable $t(F)$ and matrix $...
- 25.4k