# Questions tagged [permanent]

The permanent tag has no usage guidance.

46
questions

**18**

votes

**1**answer

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

**1**answer

187 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

**0**answers

96 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

**0**answers

48 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 ...

**0**

votes

**0**answers

35 views

### Biadjacency permanent upper bound in terms of genus of graph?

Take $M$ to be biadjacency of a planar balanced bipartite on $2n$ vertices with genus $g$.
Is it true for every $\epsilon\in(0,1)$ there is a $c_\epsilon>0$ such that $$\log\log(permanent(M))\leq\...

**0**

votes

**0**answers

11 views

### On statistics of perfect matchings between planar $4$ colorable and planar $3$ colorable

Does the mean for number of perfect matchings of random graphs that are planar and $3$ colorable much higher than graphs that are planar are not $3$ colorable?
Does a planar graph if $3$ colorable ...

**1**

vote

**1**answer

53 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

**0**answers

35 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

**0**answers

229 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

**0**answers

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

**0**answers

243 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

**2**answers

318 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

**3**answers

339 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

**0**answers

48 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

**1**answer

123 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 ...