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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
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1 vote
1 answer
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$A$ 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
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2 votes
0 answers
631 views

Note: I have edited the post below in order to include sharper (conjectured) inequalities, using $|G_1 \cap G_2|$. Let $[n] = \{1, \dots, n\}$ and let $\sim$ be an equivalence relation on $[n]$. Then $...
Malkoun's user avatar
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1 vote
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164 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
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1 vote
1 answer
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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
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3 votes
2 answers
423 views

Is there an analogoue to Jacobi's formula for the matrix permanent?
Sela Fried's user avatar
  • 135
3 votes
0 answers
251 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
  • 5,377
2 votes
1 answer
404 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
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1 answer
366 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
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1 vote
0 answers
206 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
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10 votes
1 answer
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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
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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
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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
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0 answers
67 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
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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
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2 votes
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414 views

Let $A$ be a $0/1$ square matrix which can be permuted to a non singular or a singular lower triangular matrix. Determinant is either $0$ or $1$. Can we provide tighter upper bounds on its spectral ...
Turbo's user avatar
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2 votes
0 answers
573 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 ...
Jacob.Z.Lee's user avatar
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1 answer
417 views

Recently, several of my conjectures in Question 402572 and Question 403336 were proved by Fu, Lin and Sun available from Proofs of five conjectures relating permanents to combinatorial sequences. ...
Deyi Chen's user avatar
  • 1,248
12 votes
3 answers
989 views

Let $a(n)=$ Number of ordered set partitions of $[n]$ such that the smallest element of each block is odd. ...
Deyi Chen's user avatar
  • 1,248
2 votes
0 answers
80 views

Suppose we have two bipartite graphs $G_1$ and $G_2$ with perfect matching count $P_1$ and $P_2$ respectively then their disjoint union gives a bipartite graph with perfect matching $P_1P_2$. Is ...
Turbo's user avatar
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6 votes
1 answer
598 views

For a matrix $[a_{j,k}]_{1\le j,k\le n}$ over a field, its permanent is defined by $$\mathrm{per}[a_{j,k}]_{1\le j,k\le n}:=\sum_{\pi\in S_n}\prod_{j=1}^n a_{j,\pi(j)}.$$ In a recent preprint of mine, ...
Zhi-Wei Sun's user avatar
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3 votes
1 answer
374 views

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 $n$ ...
Deyi Chen's user avatar
  • 1,248
10 votes
1 answer
761 views

The permanent $\mathrm{per}(A)$ of a matrix $A$ of size $n\times n$ is defined to be: $$\mathrm{per}(A)=\sum_{\tau\in S_n}\prod_{j=1}^na_{j,\tau(j)}.$$ Let $$A=\left[\tan\pi\frac{j+k}n\right]_{1\le j,...
Deyi Chen's user avatar
  • 1,248
1 vote
0 answers
166 views

For a matrix $[a_{j,k}]_{1\le j,k\le n}$, its permanent is given by $$\mathrm{per}[a_{j,k}]_{1\le j,k\le n}=\sum_{\tau\in S_n}\prod_{j=1}^na_{j,\tau(j)}.$$ Let $p$ be an odd prime. I have proved the ...
Zhi-Wei Sun's user avatar
  • 18.1k
20 votes
2 answers
1k views

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 ...
Deyi Chen's user avatar
  • 1,248
3 votes
1 answer
523 views

Recall that the permanent of a matrix $A=[a_{j,k}]_{1\le j,k\le n}$ is given by $$\mathrm{per}(A)=\sum_{\tau\in S_n}\prod_{j=1}^na_{j,\tau(j)}.$$ Let $n$ be an odd integer greater than one. In 2019 I ...
Zhi-Wei Sun's user avatar
  • 18.1k
4 votes
0 answers
203 views

Fix reals $a,b\in(1,2)$ satisfying $1<b<a<ab<2$. What fraction of $0/1$ matrices of dimensions $n\times n$ have permanents in $[b2^m,a2^m]$ at some $m\in\{0,1,2,\dots,\lfloor\log_2n!\...
Turbo's user avatar
  • 1
4 votes
1 answer
214 views

Suppose that $p\ge 5$ is a prime, $n$ a positive integer divisible by $p-1$, and $L<\mathbb F_p^n$ a subspace of dimension $d=n/(p-1)$. Do there exist vectors $l_1,\dotsc,l_n\in L$ such that the ...
Seva's user avatar
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2 votes
2 answers
298 views

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 ...
S. Kohn's user avatar
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1 vote
0 answers
136 views

Is there a function which describes the number of $\{0,1\}^{n\times n}\cap\mathbb Z^{n\times n}$ matrices having permanent $1$? I think it might be $\mathsf{poly}(n!)$ bounded. Is there a function ...
Turbo's user avatar
  • 1
3 votes
0 answers
93 views

$\DeclareMathOperator{\Per}{Per}$ $\newcommand{\oI}{{\overline I}}$ $\newcommand{\oJ}{{\overline J}}$ Is it possible to classify pairs $(A,B)$ of square, nonsingular matrices over a field of prime ...
Seva's user avatar
  • 23.5k
2 votes
0 answers
159 views

Let $\mathcal T_n=\{M\in\{0,1\}^{n\times n}:\mathsf{Per}(M)=\mathsf{Det}(M)\wedge\mathsf{Det}(M)\in\{0,1\}\}$ (restricted set unimodular or singular having permanent and determinant identical). $\...
Turbo's user avatar
  • 1
2 votes
0 answers
131 views

Given a $0/1$ matrix in $\mathbb Z^{n\times n}$ the standard interpretation of permanent of the matrix is the number of perfect matchings in the underlying $2n$ vertex balanced bipartite graph with ...
Turbo's user avatar
  • 1
7 votes
1 answer
329 views

$\DeclareMathOperator\perm{perm}\DeclareMathOperator\diag{diag}$Using MacMahon's master theorem, the properties of complex gaussian integrals, and Cauchy's integral theorem one can show that the ...
motherboard's user avatar
  • 281
5 votes
0 answers
100 views

Assume $M$ is a $n \times n$-matrix with entries in $\mathbb{Z}$ such that $M^k$ is the identity matrix for some $k \geq 1$. Question 1: Is the permanent of $M$ non-zero? This is tested for many ...
Mare's user avatar
  • 28.2k
7 votes
1 answer
404 views

It is well known that $\det(A \otimes B) = \det(A)^m \det(B)^n$ when $A$ and $B$ are square matrices of size $n$ and $m$ where $\otimes$ denotes the Kronecker product. Question: Is there a similar ...
Mare's user avatar
  • 28.2k
22 votes
3 answers
2k views

$\DeclareMathOperator\perm{perm}$Let $G$ be a finite group. Define the determinant $\det(G)$ of $G$ as the determinant of the character table of $G$ over $\mathbb{C}$ and define the permanent $\perm(G)...
Mare's user avatar
  • 28.2k
1 vote
1 answer
851 views

I'm trying to find an efficient algorithm/technique to calculate, or approximate, the permanent of a matrix. After reading some literature, it seems nothing exists faster than Ryser's algorithm in the ...
chasmani's user avatar
  • 113
7 votes
1 answer
348 views

Let $\operatorname{PSD}_n$ be the cone of $n\times n$ semidefinite positive matrices. For any $X\in \operatorname{PSD}_n$, define $$f(X)=\log(\det(X)).$$ Then $f$ is a concave function on $\...
Bill Bradley's user avatar
  • 4,589
-2 votes
1 answer
450 views

The Lieb's permanent dominance conjecture states that the expression $$\frac{d_{\chi}^HA}{\chi(e)}\le per(A)$$ holds for all positive semidefinite matrices $A$, where $d_{\chi}^HA=\sum_\limits{\sigma\...
vidyarthi's user avatar
  • 2,145
20 votes
1 answer
2k views

Van der Waerden's conjecture (now a theorem of Egorychev and Falikman) states that the permanent of a doubly stochastic matrix is at least $n!/n^n$. The Wikipedia article, as well as many other ...
Timothy Chow's user avatar
  • 88.5k
13 votes
1 answer
376 views

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 $...
Mare's user avatar
  • 28.2k
2 votes
0 answers
158 views

Permanent mod $2$ of biadjacency gives polynomial time algorithm of $\#PM(G)\mod 2$ of perfect matchings of bipartite graph. Is there a similar efficient strategy for general graphs?
Turbo's user avatar
  • 1
1 vote
0 answers
98 views

We can find Pfaffian orientation and take determinant to compute permanent in $O(n^\omega)$ time where $\omega$ is exponent of matrix multiplication. We know that permanent of $O(n)$ vertex planar ...
Turbo's user avatar
  • 1
1 vote
1 answer
183 views

What is the maximum number of perfect matchings a genus $g$ balanced $k$-partite graph (number of vertices for each color in all possible $k$-colorings is within a difference of $1$) can have? I am ...
Turbo's user avatar
  • 1
2 votes
0 answers
103 views

Planar graph permanent can be reduced to determinants and so statistics should be amenable. Pick a uniformly random bipartite planar graph $G$ with $n$ vertices of each color and choose new additional ...
Turbo's user avatar
  • 1
8 votes
1 answer
425 views

Recall that the permanent of an $n\times n$ matrix $A=[a_{i,j}]_{1\le i,j\le n}$ is defined by $$\operatorname{per}A=\sum_{\sigma\in S_n}\prod_{i=1}^n a_{i,\sigma(i)}.$$ In 2004, R. Chapman [Acta ...
Zhi-Wei Sun's user avatar
  • 18.1k
4 votes
0 answers
115 views

Permanent of biadjacency of bipartite graphs is the number of perfect matchings. In the case of planar graphs we can obtain an orientation with sign changes and get away with computing the determinant ...
Turbo's user avatar
  • 1
10 votes
1 answer
404 views

Let $p$ be an odd prime. It is well-known that $$\det[i^{j-1}]_{1\le i,j\le p-1}=\prod_{1\le i<j\le p-1}(j-i)\not\equiv0\pmod p.$$ I'm curious about the behavior of the permanent $\text{per}[i^{j-...
Zhi-Wei Sun's user avatar
  • 18.1k
3 votes
2 answers
672 views

Motivated by Question 316142 of mine, I consider the new sum $$S(n):=\sum_{\pi\in S_{n}}e^{2\pi i\sum_{k=1}^{n}k\pi(k)/n}$$ for any positive integer $n$, where $S_n$ is the symmetric group of all the ...
Zhi-Wei Sun's user avatar
  • 18.1k

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