Questions tagged [permanent]

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5
votes
1answer
85 views

When is the log-permanent concave?

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 $\...
0
votes
1answer
180 views

On the permanent dominance conjecture for symmetric group

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\...
18
votes
1answer
1k views

Is Van der Waerden's conjecture really due to Van der Waerden?

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 ...
12
votes
1answer
210 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 $...
2
votes
0answers
106 views

Mod $2$ of $\#PM(G)$ for arbitrary G?

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?
1
vote
0answers
50 views

Planar graphs with perfect matching count in linear time?

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 ...
1
vote
1answer
63 views

Maximum number of perfect matchings in a graph of genus $g$ balanced $k$-partite graph

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 ...
2
votes
0answers
49 views

Statistics of perfect matching and incremental perfect matchings in bipartite planar graphs?

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 ...
6
votes
0answers
232 views

Is the permanent of the matrix $[(\frac{i+j}{2n+1})]_{0\le i,j\le n}$ always positive?

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 ...
2
votes
0answers
43 views

Volume interpretation of number of perfect matchings in bipartite planar graphs

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 ...
9
votes
0answers
246 views

On the permanent $\text{per}[i^{j-1}]_{1\le i,j\le p-1}$ modulo $p^2$

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-...
3
votes
2answers
332 views

On the sum $\sum_{\pi\in S_{n}}e^{2\pi i\sum_{k=1}^{n}k\pi(k)/n}$

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 ...
7
votes
3answers
391 views

Distribution of sum of two permutation matrices

Determinant and permanent of sum of two $n\times n$ permutation matrices can be arbitrarily different. What is the distribution of determinant of sum and difference of two $n\times n$ permutation ...
3
votes
0answers
59 views

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 ...
3
votes
1answer
150 views

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 ...
3
votes
1answer
202 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 $...
2
votes
1answer
249 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 ...
2
votes
0answers
224 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 &...
23
votes
3answers
1k 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 ...
6
votes
2answers
241 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 ...
23
votes
2answers
1k views

Interpretations 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 ...
2
votes
1answer
102 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 ...
2
votes
0answers
57 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: ...
2
votes
1answer
103 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 $x_1,..., x_m$...
5
votes
0answers
134 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 $G$...
7
votes
3answers
263 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?
3
votes
1answer
134 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 $Permanent(M+aJ)$?
6
votes
2answers
237 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 ...
3
votes
0answers
303 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 ...
6
votes
2answers
163 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 $A(G)$...
7
votes
0answers
329 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 $\mathcal G_{2n}$. ...
2
votes
1answer
343 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 &...
1
vote
0answers
119 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 ...
0
votes
0answers
197 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: $$perm(A)=4^{3m} \#SAT(\phi)$$ ...
1
vote
0answers
64 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). \...
6
votes
1answer
277 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 ...
2
votes
0answers
158 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} && \...
3
votes
0answers
200 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 ...
7
votes
1answer
389 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 ...
0
votes
0answers
187 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 $...
28
votes
1answer
707 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:...
16
votes
3answers
2k views

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 ...
7
votes
1answer
346 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 $T_i$. ...
7
votes
4answers
489 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 ...
13
votes
2answers
817 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 ...
4
votes
2answers
552 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 $A(k,n)$ ...