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Questions tagged [permanent]

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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 ...
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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 $... 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? 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 ... 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 $$\log\log(permanent(M))\leq\... 0answers 11 views On statistics of perfect matchings between planar 4 colorable and planar 3 colorable Does the mean for number of perfect matchings of random graphs that are planar and 3 colorable much higher than graphs that are planar are not 3 colorable? Does a planar graph if 3 colorable ... 1answer 53 views Maximum number of perfect matchings in a graph of genus g balanced k-partite graph What is the maximum number of perfect matchings a genus g balanced k-partite graph (number of vertices for each color in all possible k-colorings is within a difference of 1) can have? I am ... 0answers 35 views Statistics of perfect matching and incremental perfect matchings in bipartite planar graphs? Planar graph permanent can be reduced to determinants and so statistics should be amenable. Pick a uniformly random bipartite planar graph G with n vertices of each color and choose new ... 0answers 229 views Is the permanent of the matrix [(\frac{i+j}{2n+1})]_{0\le i,j\le n} always positive? Recall that the permanent of an n\times n matrix A=[a_{i,j}]_{1\le i,j\le n} is defined by$$\operatorname{per}A=\sum_{\sigma\in S_n}\prod_{i=1}^n a_{i,\sigma(i)}.$$In 2004, R. Chapman [Acta ... 0answers 43 views Volume interpretation of number of perfect matchings in bipartite planar graphs Permanent of biadjacency of bipartite graphs is the number of perfect matchings. In the case of planar graphs we can obtain an orientation with sign changes and get away with computing the determinant ... 0answers 243 views On the permanent \text{per}[i^{j-1}]_{1\le i,j\le p-1} modulo p^2 Let p be an odd prime. It is well-known that$$\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.$$I'm curious about the behavior of the permanent \text{per}[i^{j-... 2answers 318 views On the sum \sum_{\pi\in S_{n}}e^{2\pi i\sum_{k=1}^{n}k\pi(k)/n} Motivated by Question 316142 of mine, I consider the new sum$$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 ... 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 ... 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 ...
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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