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        Questions tagged [permanent]

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        18
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        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
        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
        0answers
        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
        0answers
        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
        0answers
        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
        0answers
        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
        1answer
        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
        0answers
        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
        0answers
        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
        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
        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
        2answers
        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
        3answers
        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
        0answers
        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
        1answer
        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 ...

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