Questions tagged [permanent]
The permanent tag has no usage guidance.
83 questions
3
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Rank relation to maximum subpermanent and subdeterminant?
Given a $\pm1$ matrix $M$ of rank $r$ let the largest subdeterminant be $d$ and let the largest subpermanent be $p$.
Are there relations/bounds that connect $r$, $d$ and $p$?
Are there geometric and ...
- 13.9k
3
votes
1
answer
335
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Multi-dimensional permanent
Is there a particularly natural / "correct" way of generalizing permanents to tensors? (I mean of course, 'square' tensors.) There seem to be very few resources on the matter. There needs to be a ...
- 1,203
3
votes
1
answer
220
views
On particular sumset properties of permanent?
Denote $\mathcal R_2[n]=\mathcal R[n] + \mathcal R[n]$ to be sumset of integers in $\mathcal R[n]$ where $\mathcal R[n]$ to be set of permanents possible with permanents of $n\times n$ matrices with $...
- 13.9k
2
votes
1
answer
416
views
How hard is it to compute these prime factor related problems?
We know that computing number of prime factors implies efficient factoring algorithm (How hard is it to compute the number of prime factors of a given integer?).
Let $\omega(n)$ be number of distinct ...
- 13.9k
2
votes
0
answers
364
views
Permanent of a matrix
Let $n \geq 2$ $a,b$ complex numbers (or in some other ring if you wish).
What is the permanent of the matrix
$$M(a,b,n)=
\begin{bmatrix}
a & a & a & ... & a & a \\
a &...
- 26.5k
28
votes
3
answers
2k
views
Is every positive integer the permanent of some 0-1 matrix?
In the course of discussing another MO question we realized that we did not know the answer to a more basic question, namely:
Is it true that for every positive integer $k$ there exists a balanced ...
- 82.7k
6
votes
2
answers
309
views
Permanent of Nakayama algebras
See https://en.wikipedia.org/wiki/Nakayama_algebra for the definition of Nakayama algebras and define the permanent of such an algebra to be the permanent of its Cartan matrix.
(all algebras are ...
- 26.5k
25
votes
3
answers
2k
views
Interpretations and models of permanent
The standard interpretation of permanent of a $0/1$ matrix if considered as a biadjacency matrix of a bipartite graph is number of perfect matchings of the graph or if considered as a adjacency matrix ...
- 13.9k
2
votes
1
answer
139
views
An inequality between permanents of non-negative matrices
Let $A$ be a non-negative (all entries $\geq 0$) square matrix. Is it always true that
$$
(a_{11}+a_{12}+a_{21}+a_{22})^2\geq 4a_0a_2
$$
where
$a_{ij}$ is the permanent of a matrix obtained by ...
- 3,041
2
votes
0
answers
64
views
Minimum size of genus $g$ bipartite graphs with $2^n$ perfect matchings
Given $n\in\Bbb Z_{\geq0}$ let $2T_{n,g}$ be size of smallest number of vertices of genus $g$ bipartite graph with $T_{n,g}$ vertices of each color such that number of perfect matchings is $2^n$.
Eg: ...
- 13.9k
2
votes
1
answer
118
views
What $n$-linear sums can be extracted from a product of linear polynomials in $m$ variables?
Let $\boldsymbol{c}_1, ..., \boldsymbol{c}_n$ be $n$ orthonormal, $m$-dimensional complex vectors, with $\boldsymbol{c}_i = (c_{i,1}, ..., c_{i,m})$.
Consider the following polynomial in $x_1,..., x_m$...
- 342
6
votes
0
answers
175
views
Permanent bound for Laplacian matrix of signed graph
In 1986, Prof. RB Bapat shown that (see here) if $G$ is a simple connected graph on $n$ vertices, then, the permanent per$\big(L(G)\big)\ge 2(n-1)\kappa(G)$, where $L(G)$ is the Laplacian matrix of $G$...
- 640
7
votes
3
answers
346
views
Concentration Bound of $0/1$ permanent
If I pick a random $0/1$ $n\times n$ matrix with $0$ occuring with probability $p$ then what does the distribution of the permanent look like?
- 13.9k
3
votes
1
answer
170
views
Permanent of distorted matrix
Let $J$ be all $1$ matrix. Suppose permanent of $M$ is $p$ and $a\in\Bbb Z$. Is there a closed formula or at least a faster than Ryser's technique to find $Permanent(M+aJ)$?
- 13.9k
6
votes
2
answers
310
views
On additive/multiplicative property of permanent
With $\Bbb K$ a commutative rings there a way to characterize $A,B\in\Bbb K^{n\times n}$ with $$Per(AB)=Per(A)Per(B)?$$
John provides a reasonable concept coverage multiplicative property.
How about ...
user94040
3
votes
0
answers
372
views
On the precise concentration of permanent of $\pm1$ matrices
Obtain $M\in\{-1,+1\}^{n\times n}$ by unbiased coin flipping.
What is known about the distribution of permanent $\mathsf{Perm}(M)$? It seems to be bimodal.
Given a function $g(n)$ what is the ...
user94040
6
votes
2
answers
180
views
Permanent of a complete graph with negative cliques
Let $K_n$ be the complete graph on $n$ vertices. Inside $K_n$ there are $k$ negative(edge weight is equal to -1) complete subgraphs $K_{n_1}, K_{n_2},...,K_{n_k}$, which are vertex disjoint. Let $A(G)$...
- 640
7
votes
0
answers
349
views
Missing count in number of perfect matchings
Let $f(G)$ give number of perfect matchings of a graph $G$.
Denote $\mathcal N_{2n}=\{0,1,2,\dots,n!-1,n!\}$.
Denote collection of all $2n$ vertex balanced bipartite graph to be $\mathcal G_{2n}$.
...
- 13.9k
2
votes
1
answer
561
views
Vanishing of permanent of a Vandermonde matrix [Edited]
Does there exist an explicit criterion (or a good sufficient condition) for proving that a Vandemonde matrix:
$$A(x_1,\dots,x_n):=\left[ \begin{array}{llll}1 & x_1 &\dots& x_1^{n-1}\\ 1 &...
- 4,484
1
vote
0
answers
138
views
FPTAS for approximating the permanent of a matrix
My question concerns approximating permanent of an $n$-by-$n$ matrix.
Several approximation algorithms have been proposed in the literature for this purpose, whose time complexity depend on $n$ and ...
- 11
-1
votes
1
answer
446
views
What is wrong with the argument that zero permanent is polynomial?
This Lecture summarizes some well known facts about $\#P$ completeness of permanent.
Given a CNF formula $\phi$ on $n$ variables, they construct
matrix $A$ such that:
$$perm(A)=4^{3m} \#SAT(\phi)$$
...
- 25.4k
1
vote
0
answers
68
views
Zero as a repeated permanental root for a matrix over a finite field
All,
Suppose $A \in Mat(n, \mathbb{F}_{q})$ for $q$ prime, $q \geq 5$, and $n \geq 2^{q-2}$ . Let $\pi_A(x)$ be the permanental polynomial for $A$. That is,
\begin{equation*}
\pi_{A}(x)=per(xI-A).
\...
- 11
6
votes
1
answer
335
views
Distribution of the permanent modulo $p$
We know that the order of $SL_n({\mathbb F}_p)$ is
$$p^{n(n-1)/2}(p^n-1)(p^{n-1}-1)\cdots(p^2-1).$$
Dividing by $p^{n^2}$, we deduce the probability that $\det$ takes the value $1$ over $M_n({\mathbb ...
- 52.4k
2
votes
0
answers
179
views
Does this permanent have a closed form?
What is the closed form of this permanent? (similar to the Cauchy determinant)
\begin{aligned}
f(z_1,z_2,\cdots,z_N,w_1,w_2,\cdots,w_N)=\left[
\small{\begin{matrix}
\frac{1}{(z_1-w_1)^2} && \...
- 181
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
7
votes
1
answer
505
views
About an identity which gives immediate proof of the permanent lemma
Let $A$ be a $n \times n$ matrix over field $F$. Let $a_1, \cdots, a_n$ be the column vectors of $A$. For any subset $S \subseteq [n] = \{1, 2, \cdots, n\}$, let $a_S = \sum_{i \in S} a_i$. Alon's ...
- 143
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
28
votes
1
answer
794
views
Are there any nontrivial near-isometries of the $n$-dimensional cube?
Consider the $n$-dimensional Hamming cube, $C = \{-1,1\}^n$. Given an $n \times n$ orthogonal matrix $O$, I'll measure "how close $O$ is to being an isometry of $C$" by the following scoring function:...
- 9,833
18
votes
3
answers
2k
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Silly me & Van der Waerden conjecture
So I walked into this very innocent-looking combinatorics problem,
and quite soon I ended up with the problem to prove that any doubly stochastic $n \times n$ matrix has a non-zero permanent.
Now ...
- 15.8k
7
votes
1
answer
383
views
A generalization of van der Waerden's conjecture
I am wondering if the following generalization of van der Waerden's conjecture is true.
Suppose A is an n x n non-negative matrix with all column sums equal to 1, and the sum of row i equal to $T_i$. ...
- 173
7
votes
4
answers
644
views
Permanent identities for special classes of matrices
The permanent $P(M)$ of a matrix $M$ of size $n$ is defined to be:
$$
P(M) := \sum_{\sigma \in S_n}\prod_{i=1}^n M_{i\sigma(i)}
$$
If you have a matrix of the form
$$
M_{ij} := A_i + B_j
$$
where ...
- 333
13
votes
2
answers
946
views
Computing a large permanent
Is there a practical way to compute the permanent of a large ($91 \times 91$) $(0,1)$ matrix?
I have tried to use the matlab function written by Luke Winslow which works great for smaller matrices ...
- 6,962
4
votes
2
answers
620
views
Is Ryser's conjecture on permanent minimizers still open?
Let $A(k,n)$ be the set of $\{0,1\}$ matrices of order $n$ with all their line sums equal to $k$.
Conjecture number 5 on the list from Minc's book, attributed to Ryser, says that if $A(k,n)$ ...
- 6,962