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85 questions
Newest Active Bountied Unanswered
4 votes
1 answer
249 views

Let $A$ be a $0/1$ matrix in $\mathbb Z^{n\times n}$ such that $I+A$ is invertible $\bmod 3$. Consider $Q=(I-A)(I+A)^{-1}$. Let $Det(M)$ and $Per(M)$ be determinant and permanent respectively of ...
xoxo's user avatar
  • 199
1 vote
1 answer
201 views

$A$ is a complex matrix of order $2n\times 2n$. $S=\sigma_x\otimes I_{n}$, where $\sigma_x$ is Pauli matrix. Suppose we have $S^T A S=\bar{A}$, show that the permanent $per(A)$ is non-negative. When $...
Aztec's user avatar
  • 61
6 votes
0 answers
863 views

Let $[n] = \{1, \dots, n\}$ and let $\sim$ be an equivalence relation on $[n]$. Then $\sim$ induces a partition of $[n]$ into equivalence classes. Let $G$ denote the product of the groups of ...
Malkoun's user avatar
  • 5,497
1 vote
0 answers
170 views

In the description of the book "Combinatorics and Complexity of Partition Functions", by Alexander Barvinok, it is written "...The main focus of the book is on efficient ways to compute ...
Malkoun's user avatar
  • 5,497
1 vote
1 answer
311 views

The Van der Waerden conjecture is a lower estimate of the permanent of a doubly stochastic matrix. In this article in Wikipedia it is stated that Egorychev's proof uses the Alexandrov-Fenchel ...
asv's user avatar
  • 23.3k
3 votes
2 answers
430 views

Is there an analogoue to Jacobi's formula for the matrix permanent?
Sela Fried's user avatar
  • 135
3 votes
0 answers
255 views

Let $H$ denote an $n$ by $n$ hermitian positive semidefinite matrix. Let $G$ and $K$ be two subgroups of the symmetric group $\Sigma_n$. Define $$ f_{G, K}(H) = \sum_{(\sigma, \tau) \in G \times K} \...
Malkoun's user avatar
  • 5,497
2 votes
1 answer
418 views

The permanent of an $n$-by- $n$ matrix $A=\left(a_{i j}\right)$ is defined as $$ \operatorname{perm}(A)=\sum_{\sigma \in S_{n}} \prod_{i=1}^{n} a_{i, \sigma(i)} $$ The sum here extends over all ...
Alexandr Dorofeev's user avatar
  • 123
5 votes
1 answer
373 views

Let $n>1$ be an integer, and let $\zeta$ be a primitive $n$th root of unity. By $(3.4)$ of arXiv:2206.02589, $1$ and those $n+1-2s\ (s=1,\ldots,n-1)$ are all the eigenvalues of the matrix $M=[m_{jk}...
Zhi-Wei Sun's user avatar
  • 19.2k
1 vote
0 answers
207 views

Motivated by my study of determinants and permanents, here I present several conjectures on $p$-adic congruences involving permutations. As usual, we let $S_n$ be the symmetric group consisting of all ...
Zhi-Wei Sun's user avatar
  • 19.2k
10 votes
1 answer
338 views

Suppose $A=(a_{ij})$ is a symmetric (0,1)-matrix with 1's along the diagonal, and let $A_{ij}$ be the matrix obtained by removing the $i$-th row and $j$-th column. Based on substantial numerical ...
ngm's user avatar
  • 115
7 votes
2 answers
439 views

Let $A = (a_{i,j})$ be a double stochastic matrix with positive entries. That is, all entries are positive real numbers, and each row and column sums to one. A permanent of a matrix $A = (a_{i,j})$ is ...
user avatar
25 votes
2 answers
1k views

A somewhat strange elementary polynomial inequality came up recently in my work, and I wonder if anyone has seen other things that are reminiscent of what follows. Given $n$ non-negative reals $a_1, ...
BPN's user avatar
  • 563
2 votes
0 answers
68 views

Permanents can be interpreted as counting directed cycle covers of an asymmetric graph with unit cost edge weights. That interpretation leads to a branch and bound algorithm for calculating the ...
Manfred Weis's user avatar
  • 14.3k
1 vote
0 answers
93 views

The slides here provide a way to get a pfaffian orientation from Minimum Spanning Tree. MST can be found in linear time if graph is planar and weights are $1$ and the slides give a linear time ...
Turbo's user avatar
  • 14k

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