All Questions
Tagged with permanent co.combinatorics
35 questions
1
vote
1
answer
209
views
Deciding if given number is a permanent of matrix
The permanent of an $n$-by- $n$ matrix $A=\left(a_{i j}\right)$ is defined as
$$
\operatorname{perm}(A)=\sum_{\sigma \in S_{n}} \prod_{i=1}^{n} a_{i, \sigma(i)}
$$
The sum here extends over all ...
10
votes
1
answer
268
views
A bound for the permanent of a nonnegative matrix
Suppose $A=(a_{ij})$ is a symmetric (0,1)-matrix with 1's along the diagonal, and let $A_{ij}$ be the matrix obtained by removing the $i$-th row and $j$-th column.
Based on substantial numerical ...
- 105
7
votes
2
answers
366
views
On permanent of a square of a doubly stochastic matrix
Let $A = (a_{i,j})$ be a double stochastic matrix with positive entries. That is, all entries are positive real numbers, and each row and column sums to one. A permanent of a matrix $A = (a_{i,j})$ is ...
user125543
23
votes
1
answer
1k
views
Symmetric polynomial inequality arising from the fixed-point measure of a random permutation
A somewhat strange elementary polynomial inequality came up recently in my work, and I wonder if anyone has seen other things that are reminiscent of what follows.
Given $n$ non-negative reals $a_1, ...
- 543
2
votes
0
answers
354
views
Spectral norm bound for lower triangular matrix
Let $A$ be a $0/1$ square matrix which can be permuted to a non singular or a singular lower triangular matrix. Determinant is either $0$ or $1$. Can we provide tighter upper bounds on its spectral ...
- 13.9k
2
votes
0
answers
489
views
The calculation of permanent of a matrix
The permanent of an $n$-by- $n$ matrix $A=\left(a_{i j}\right)$ is defined as
$$
\operatorname{perm}(A)=\sum_{\sigma \in S_{n}} \prod_{i=1}^{n} a_{i, \sigma(i)}
$$
The sum here extends over all ...
- 767
5
votes
1
answer
346
views
Permanents and Kummer-like congruence
Recently, several of my conjectures in Question 402572 and Question 403336 were proved by Fu, Lin and Sun available from Proofs of five conjectures relating permanents to combinatorial sequences.
...
- 884
12
votes
3
answers
892
views
Set partitions and permanents
Let $a(n)=$ Number of ordered set partitions of $[n]$ such that the smallest element of each block is odd.
...
- 884
2
votes
0
answers
60
views
Sum of number of perfect matchings and a constant constuction
Suppose we have two bipartite graphs $G_1$ and $G_2$ with perfect matching count $P_1$ and $P_2$ respectively then their disjoint union gives a bipartite graph with perfect matching $P_1P_2$.
Is ...
- 13.9k
3
votes
1
answer
308
views
Tangent numbers, secant numbers and permanent of matrices
Inspired by Question 402572, I consider the permanent of matrices
$$f(n)=\mathrm{per}(A)=\mathrm{per}\left[\operatorname{sgn} \left(\sin\pi\frac{j+2k}{n+1} \right)\right]_{1\le j,k\le n},$$
where $n$ ...
- 884
9
votes
1
answer
676
views
Permanent identities
The permanent $\mathrm{per}(A)$ of a matrix $A$ of size $n\times n$ is defined to be:
$$\mathrm{per}(A)=\sum_{\tau\in S_n}\prod_{j=1}^na_{j,\tau(j)}.$$
Let
$$A=\left[\tan\pi\frac{j+k}n\right]_{1\le j,...
- 884
20
votes
2
answers
1k
views
Euler numbers and permanent of matrices
Motivated by Question 402249 of Zhi-Wei Sun, I consider the permanent of matrices
$$e(n)=\mathrm{per}\left[\operatorname{sgn} \left(\tan\pi\frac{j+k}n \right)\right]_{1\le j,k\le n-1},$$
where $n$ is ...
- 884
4
votes
0
answers
187
views
Dyadic distribution of $0/1$ permanents
Fix reals $a,b\in(1,2)$ satisfying $1<b<a<ab<2$.
What fraction of $0/1$ matrices of dimensions $n\times n$ have permanents
in $[b2^m,a2^m]$ at some $m\in\{0,1,2,\dots,\lfloor\log_2n!\...
- 13.9k
2
votes
2
answers
193
views
growth of the permanent of some band matrix
Consider such special band matrix of dimension $n$. It is a $0-1$ matrix, and only the first few diagonals are nonzero. Specifically,
$$ H_{ij} = 1 $$
if and only if $|i-j| \leq 2$.
How does the ...
- 265
1
vote
0
answers
94
views
Number of extremal $\{0,1\}$ matrices having permanent $1$ property
Is there a function which describes the number of $\{0,1\}^{n\times n}\cap\mathbb Z^{n\times n}$ matrices having permanent $1$?
I think it might be $\mathsf{poly}(n!)$ bounded.
Is there a function ...
- 13.9k
3
votes
0
answers
75
views
Bunch of matrices with vanishing permanents
$\DeclareMathOperator{\Per}{Per}$
$\newcommand{\oI}{{\overline I}}$
$\newcommand{\oJ}{{\overline J}}$
Is it possible to classify pairs $(A,B)$ of square, nonsingular matrices over a field of prime ...
- 23k
2
votes
0
answers
111
views
Standard interpretation of permanents (of orthogonal included) over finite fields
Given a $0/1$ matrix in $\mathbb Z^{n\times n}$ the standard interpretation of permanent of the matrix is the number of perfect matchings in the underlying $2n$ vertex balanced bipartite graph with ...
- 13.9k
-1
votes
1
answer
383
views
On the permanent dominance conjecture for symmetric group
The Lieb's permanent dominance conjecture states that the expression $$\frac{d_{\chi}^HA}{\chi(e)}\le per(A)$$ holds for all positive semidefinite matrices $A$, where $d_{\chi}^HA=\sum_\limits{\sigma\...
- 2,089
13
votes
1
answer
311
views
Permanent of the Coxeter matrix of a distributive lattice
Let $L$ be a finite distributive lattice with $n$ elements. Let $C=(c_{x,y})$ be the $n \times n$ matrix with entry 1 in case $x \leq y$ and 0 else.
The Coxeter matrix of $L$ is defined as the matrix $...
- 26.5k
2
votes
0
answers
123
views
Mod $2$ of $\#PM(G)$ for arbitrary G?
Permanent mod $2$ of biadjacency gives polynomial time algorithm of $\#PM(G)\mod 2$ of perfect matchings of bipartite graph. Is there a similar efficient strategy for general graphs?
- 13.9k
8
votes
1
answer
364
views
Is the permanent of the matrix $[(\frac{i+j}{2n+1})]_{0\le i,j\le n}$ always positive?
Recall that the permanent of an $n\times n$ matrix $A=[a_{i,j}]_{1\le i,j\le n}$ is defined by
$$\operatorname{per}A=\sum_{\sigma\in S_n}\prod_{i=1}^n a_{i,\sigma(i)}.$$
In 2004, R. Chapman [Acta ...
- 15.6k
7
votes
3
answers
703
views
Distribution of sum of two permutation matrices
Determinant and permanent of sum of two $n\times n$ permutation matrices can be arbitrarily different.
What is the distribution of determinant of sum and difference of two $n\times n$ permutation ...
- 13.9k
3
votes
1
answer
220
views
On particular sumset properties of permanent?
Denote $\mathcal R_2[n]=\mathcal R[n] + \mathcal R[n]$ to be sumset of integers in $\mathcal R[n]$ where $\mathcal R[n]$ to be set of permanents possible with permanents of $n\times n$ matrices with $...
- 13.9k
2
votes
0
answers
364
views
Permanent of a matrix
Let $n \geq 2$ $a,b$ complex numbers (or in some other ring if you wish).
What is the permanent of the matrix
$$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
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
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
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
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
18
votes
3
answers
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 ...
- 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