Let A=(a_(ij)) be the generic n by n circulant matrix given by a_(ij)=x_(i+j), with subscripts on x interpreted mod n. Define d(n) (resp. p(n)) to be the number of terms in the determinant (resp. permanent) of A. The function p(n) is well-known and has several combinatorial interpretations. The function d(n), on the other hand, has not been studied previously. We show that when n is a prime power, d(n)=p(n). The proof uses symmetric functions.
In this paper we give a generalization created by the author of the theory of rook polynomials and permanents of circulants, Toeplits matrices and their submatrices.
Calculating the permanent of a (0,1) matrix is a #P-complete problem but there are some classes of structured matrices for which the permanent is calculable in polynomial time. The most well-known example is the fixed-jump (0,1) circulant matrix which, using algebraic techniques, was shown by Minc to satisfy a constant-coefficient fixed-order recurrence relation. In this note we show how, by interpreting the problem as calculating the number of cycle-covers in a directed circulant graph, it is straightforward to reprove Minc's result using combinatorial methods. This is a two step process: the…
This note provides formula for determinant and inverse of r-circulant matrices with general sequences of third order. In other words, the study combines many papers in the literature.
We study the determinant of the pxp circulant matrix whose first row is (1,-x,0,...,0,-y,0,...,0), the -y being in position q+1. The coefficients of this polynomial are integers that count certain classes of permutations. We show that all of the permutations that contribute to a fixed monomial x^ry^s have the same sign, and we determine that sign. We prove that a monomial x^ry^s appears if and only if p divides r+sq. Finally, we show that the size of the largest coefficient of the monomials that appear grows exponentially with p. We do this by proving that the permanent of the circulant whose …
Let $d(N )$ (resp. $p(N )$) be the number of summands in the determinant (resp. permanent) of an $N\times N$ circulant matrix $A = (a_{ij} )$ given by $a_{ij} = X_{i+j}$ where $i + j$ should be considered $\mod N$ . This short note is devoted to prove that $d(N ) = p(N )$ if and only if $N$ is a prime power. We then give an application to homogeneous monomial ideals failing the Weak Lefschetz property.
Due to their rich algebraic structures and various applications, circulant matrices have been of interest and continuously studied. In this paper, the notions of Binomial-related matrices have been introduced. Such matrices are circulant matrices whose the first row is the coefficients of $(x+zy)^n$, where $z$ is a complex number of norm $1$ and $n$ is a positive integer. In the case where $z\in \{1,-1,i,-i\}$, the explicit formula for the determinant of such matrices are completely determined. Known results on the determinants of binomial circulant matrices can be viewed as special cases. Fin…
We describe algorithms for computing maximal determinants of binary circulant matrices of small orders. Here "binary matrix" means a matrix whose elements are drawn from $\{0,1\}$ or $\{-1,1\}$. We describe efficient parallel algorithms for the search, using Duval's algorithm for generation of necklaces and the well-known representation of the determinant of a circulant in terms of roots of unity. Tables of maximal determinants are given for orders $\le 53$. Our computations extend earlier results and disprove two plausible conjectures.
In this paper, we give upper and lower bounds for the spectral norms of r-circulant matrices with the generalized bi-periodic Fibonacci numbers. Moreover, we investigate the eigenvalues and determinants of these matrices.
We view the determinant and permanent as functions on directed weighted graphs and introduce their analogues for the undirected graphs. We prove that the task of computing the undirected determinants as well as permanents for planar graphs, whose vertices have degree at most 4, is \#P-complete. In the case of planar graphs whose vertices have degree at most 3, the computation of the undirected determinant remains \#P-complete while the permanent can be reduced to the FKT algorithm, and therefore is polynomial. The undirected permanent is a Holant problem and its complexity can be deduced from …
We prove an identity relating the permanent of a rank $2$ matrix and the determinants of its Hadamard powers. When viewed in the right way, the resulting formula looks strikingly similar to an identity of Carlitz and Levine, suggesting the possibility that these are actually special cases of some more general identity (or class of identities) connecting permanents and determinants. The proof combines some basic facts from the theory of symmetric functions with an application of a famous theorem of Binet and Cauchy in linear algebra.
The permanent-determinant method and its generalization, the Hafnian-Pfaffian method, are methods to enumerate perfect matchings of plane graphs that was discovered by P. W. Kasteleyn. We present several new techniques and arguments related to the permanent-determinant with consequences in enumerative combinatorics. Here are some of the results that follow from these techniques: 1. If a bipartite graph on the sphere with 4n vertices is invariant under the antipodal map, the number of matchings is the square of the number of matchings of the quotient graph. 2. The number of matchings of the edg…
We prove that the maximum determinant of an $n \times n $ matrix, with entries in $\{0,1\}$ and at most $n+k$ non-zero entries, is at most $2^{k/3}$, which is best possible when $k$ is a multiple of 3. This result solves a conjecture of Bruhn and Rautenbach. We also obtain an upper bound on the number of perfect matchings in $C_4$-free bipartite graphs based on the number of edges, which, in the sparse case, improves on the classical Bregman's inequality for permanents. This bound is tight, as equality is achieved by the graph formed by vertex disjoint union of 6-vertex cycles.
Let $\FF$ be a finite field of characteristics different from two. We show that no bijective map transforms permanent into determinant when the cardinality of $\FF$ is sufficiently large. We also give an example of non-bijective map when $\FF$ is arbitrary and an example of a bijective map when $\FF$ is infinite which do transform permanent into determinant. The developed technique allows us to estimate the probability of the permanent and the determinant of matrices over finite fields to have a given value. Our results are also true over finite rings without zero divisors.
The goal of this article is to compare the coefficients in the expansion of the permanent with those in the expansion of the determinant of a three-lines circulant matrix. As an application we prove a conjecture concerning the minimality of Galois-Togliatti systems.
In the paper [J. Combin. Theory Ser. A 43 (1986), 103--113], Stanley gives formulas for the number of plane partitions in each of ten symmetry classes. This paper together with results by Andrews [J. Combin. Theory Ser. A 66 (1994), 28-39] and Stembridge [Adv. Math 111 (1995), 227-243] completes the project of proving all ten formulas. We enumerate cyclically symmetric, self-complementary plane partitions. We first convert plane partitions to tilings of a hexagon in the plane by rhombuses, or equivalently to matchings in a certain planar graph. We can then use the permanent-determinant method …
The tensor rank and border rank of the $3 \times 3$ determinant tensor is known to be $5$ if characteristic is not two. In this paper, we show that the tensor rank remains $5$ for fields of characteristic two as well. We also include an analysis of $5 \times 5$ and $7 \times 7$ determinant and permanent tensors, as well as the symmetric $3 \times 3$ permanent and determinant tensors. We end with some remarks on binary tensors.
In this paper, we prove a conjecture raised by Zhi-Wei Sun in 2018 by the eigenvectors-eigenvalues identity found by Denton, Parke, Tao and X. Zhang in 2019.
Let $A$ be a set in a prime field $\mathbb{F}_p$. In this paper, we prove that $d\times d$ matrices with entries in $A$ determine almost $|A|^{3+\frac{1}{45}}$ distinct determinants and almost $|A|^{2-\frac{1}{6}}$ distinct permanents when $|A|$ is small enough.
In this paper we obtain precise asymptotics for certain families of graphs, namely circulant graphs and degenerating discrete tori. The asymptotics contain interesting constants from number theory among which some can be interpreted as corresponding values for continuous limiting objects. We answer one question formulated in a paper from Atajan, Yong and Inaba in [1] and formulate a conjecture in relation to the paper from Zhang, Yong and Golin [21]. A crucial ingredient in the proof is to use the matrix tree theorem and express the combinatorial laplacian determinant in terms of Bessel functi…
In this paper we give the absolutely new proof of a conjecture of R.F.Scott(1881) on the permanent of a Cauchy matrix $\ls \frac{1}{x_i-y_j} \rs_{1 \leqslant i,j \leqslant n},$ where $x_1, ..., x_n$ and $y_1, ..., y_n$ are the distinct roots of the polynomials $x^n-1$ and $y^n +1,$ respectively. The simple formula is given for the permanent of the Cauchy matrix $A= \ls \frac{1}{x_i-y_j} \rs_{1 \leqslant i,j \leqslant n},$ where $x_1, ..., x_n$ and $y_1, ..., y_n$ are the distinct roots of the polynomials $x^n+a$ and $y^n +b$, respectively: \begin{gather*} \per (A) =\frac{n}{(b-a)^n} \prod_{k=1…
We introduce a new notion of the determinant, called symmetrized determinant, for a square matrix with the entries in an associative algebra $Å$. The monomial expansion of the symmetrized determinant is obtained from the standard expansion of the commutative determinant by averaging the products of entries of the matrix in all possible orders. We show that for any fixed finite-dimensional associative algebra $Å$, the symmetrized determinant of an $n\times n$ matrix with the entries in $Å$ can be computed in polynomial in $n$ time (the degree of the polynomial is linear in the dimension of $Å$)…
We show that the permanent of a matrix is a linear combination of determinants of block diagonal matrices which are simple functions of the original matrix. To prove this, we first show a more general identity involving α-permanents: for arbitrary complex numbers αand β, we show that the α-permanent of any matrix can be expressed as a linear combination of β-permanents of related matrices. Some other identities for the α-permanent of sums and products of matrices are shown, as well as a relationship between the α-permanent and general immanants. We conclude with a discussion of the computation…
We show that the weight enumerator of any binary linear code is equal to the permanent of a 3-dimensional hypermatrix (3-matrix). We also show that each permanent is a determinant of a 3-matrix. As an application we write the dimer partition function of a finite 3-dimensional cubic lattice as the determinant of the vertex-adjacency 3-matrix of a 2-dimensional simplicial complex which preserves the natural embedding of the cubic lattice.
A determinantal approximation is obtained for the permanent of a doubly stochastic matrix. For moderate-deviation matrix sequences, the asymptotic relative error is of order $O(n^{-1})$.
Celebrated work of Jerrum, Sinclair, and Vigoda has established that the permanent of a {0,1} matrix can be approximated in randomized polynomial time by using a rapidly mixing Markov chain. A separate strand of the literature has pursued the possibility of an alternate, purely algebraic, polynomial-time approximation scheme. These schemes work by replacing each 1 with a random element of an algebra A, and considering the determinant of the resulting matrix. When A is noncommutative, this determinant can be defined in several ways. We show that for estimators based on the conventional determin…
Scott considered the determinant of 1/(y-z)^2, with y,z running over two sets X,Y of size n, and determined its specialisation when Y and Z are the roots of y^n-a and z^n-b. We give the same specialisation for the determinant 1/\prod_x(xy-z), where {x} is an arbitrary set of indeterminates. The case of the Gaudin-Izergin-Korepin is for {x}={q,1/q}.
Expressions involving the product of the permanent with the (n-1)th power of the determinant of a matrix of indeterminates, and of (0,1)-matrices, are shown to be related to two conjectures that extend the Alon-Tarsi Latin square conjecture to odd dimensions. An identity involving an alternating sum of permanents of (0,1)-matrices is obtained.
Let $A=(a_{ij})$ be an $n$-by-$n$ matrix. For any real number $μ$, we define the polynomial $$P_μ(A)=\sum_{σ\in S_n} a_{1σ(1)}\cdots a_{nσ(n)}\,μ^{\ell(σ)}\; ,$$ as the $μ$-permanent of $A$, where $\ell(σ)$ is the number of inversions of the permutation $σ$ in the symmetric group $S_n$. In this note, we review several less known results of the $μ$-permanent, recalling some of its interesting properties. Some determinantal conjectures are considered and extended to that polynomial. A correction to a previous note is presented as well.
R. Redheffer described an $n\times n$ matrix of 0's and 1's the size of whose determinant is connected to the Riemann Hypothesis. We describe the permutations that contribute to its determinant and evaluate its permanent in terms of integer factorizations. We generalize the Redheffer matrix to finite posets that have a 0 element and find the analogous results in the more general situation.
In this note we settle two open problems in the theory of permanents by using recent results from other areas of mathematics. Bapat conjectured that certain quotients of permanents, which generalize symmetric function means, are concave. We prove this conjecture by using concavity properties of hyperbolic polynomials. Motivated by problems on random point processes, Shirai and Takahashi raised the problem: Determine all real numbers $α$ for which the $α$-permanent (or $α$-determinant) is nonnegative for all positive semidefinite matrices. We give a complete solution to this problem by using re…
Given a 0-1 square matrix A, when can some of the 1's be changed to -1's in such a way that the permanent of A equals the determinant of the modified matrix? When does a real square matrix have the property that every real matrix with the same sign pattern (that is, the corresponding entries either have the same sign or are both zero) is nonsingular? When is a hypergraph with n vertices and n hyperedges minimally nonbipartite? When does a bipartite graph have a "Pfaffian orientation"? Given a digraph, does it have no directed circuit of even length? Given a digraph, does it have a subdivision …
The permanent of a square matrix is defined in a way similar to the determinant, but without using signs. The exact computation of the permanent is hard, but there are Monte-Carlo algorithms that can estimate general permanents. Given a planar diagram of a link L with $n$ crossings, we define a 7n by 7n matrix whose permanent equals to the Jones polynomial of L. This result accompanied with recent work of Freedman, Kitaev, Larson and Wang provides a Monte-Carlo algorithm to any decision problem belonging to the class BQP, i.e. such that it can be computed with bounded error in polynomial time …
I conjecture three identities for the determinant of adjacency matrices of graphene triangles and trapezia with Bloch (and more general) boundary conditions. For triangles, the parametric determinant is equal to the characteristic polynomial of the symmetric Pascal matrix. For trapezia it is equal to the determinant of a sub-matrix. Finally, the determinant of the tight binding matrix equals its permanent. The conjectures are supported by analytic evaluations and Mathematica, for moderate sizes. They establish connections with counting problems of partitions, lozenge tilings of hexagons, dense…
The objective of this paper is to develop a general algebraic theory of supertropical matrix algebra, extending [11]. Our main results are as follows: * The tropical determinant (i.e., permanent) is multiplicative when all the determinants involved are tangible. * There exists an adjoint matrix $\adj{A}$ such that the matrix $A \adj{A}$ behaves much like the identity matrix (times $|A|$). * Every matrix $A$ is a supertropical root of its Hamilton-Cayley polynomial $f_A$. If these roots are distinct, then $A$ is conjugate (in a certain supertropical sense) to a diagonal matrix. * The tropical d…
