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In this paper, we study the matrix period and the competition period of Toeplitz matrices over a binary Boolean ring $\mathbb{B} = \{0,1\}$. Given subsets $S$ and $T$ of $\{1,\ldots,n-1\}$, an $n\times n$ Toeplitz matrix $A=T_n\langle S ; T \rangle$ is defined to have $1$ as the $(i,j)$-entry if and only if $j-i \in S$ or $i-j \in T$. We show that if $\max S+\min T \le n$ and $\min S+\max T \le n$, then $A$ has the matrix period $d/d'$ and the competition period $1$ where $d = \gcd (s+t \mid s \in S, t \in T)$ and $d' = \gcd(d, \min S)$. Moreover, it is shown that the limit of the matrix sequence $\{A^m(A^T)^m\}_{m=1}^\infty$ is a directed sum of matrices of all ones except zero diagonal. In many literatures we see that graph theoretic method can be used to prove strong structural properties about matrices. Likewise, we develop our work from a graph theoretic point of view.

In this paper, we study structural properties of Toeplitz graphs. We characterize $K_q$-free Toeplitz graphs for an integer $q \ge 3$ and give equivalent conditions for a Toeplitz graph $G_n\langle t_1, t_2,\ldots, t_k\rangle$ with $t_1, Comment: 20 pages, 2 figures

Asymptotic behavior of the Mertens function M ( n ) = ∑ k = 0 n μ ( k ) which is equal to the determinant of the n × n Redheffer matrix, is known to be closely related to the Riemann hypothesis. An infinite matrix whose nth leading principal minor is equal to M ( n ) for all n ≥ 1 is called a Mertens equimodular matrix. We use Riordan matrices to find a large class of Mertens equimodular matrices, each element of which is called by us a Riordan-Redheffer matrix, briefly an R-R matrix. We also give the generating function for the characteristic polynomials of R-R matrices. As a result, we introduce several examples of R-R matrices that reveal interesting spectral properties. Further, we pose two conjectures on the eigenvalues of those R-R matrices. Finally, we find a sufficient condition for the Riemann hypothesis using the smallest singular value of a R-R matrix.

In this paper, we generalize the Laguerre polynomials in terms of q-analogue for Riordan matrices. To be more specific, for α ∈ N 0 , we introduce new q-Laguerre polynomials L n ( α ) ( x ; q ) by defining the Eulerian generating function for L n ( α ) ( x ; q ) as ( ∏ j = 0 α 1 1 + q j + 1 z ) e q [ x z 1 + z ] . Interestingly, it turns out that L n ( α ) ( x ; q ) have combinatorial descriptions in the aspect of the inversions of factorized permutations and q-rook numbers. We locate their zeros and develop their algebraic properties as well.

We present combinatorial and analytical results concerning a Sheffer sequence with a generating function of the form $G(x,z)=Q(z)^{x}Q(-z)^{1-x}$, where $Q$ is a quadratic polynomial with real zeros. By using the properties of Riordan matrices we address combinatorial properties and interpretations of our Sheffer sequence of polynomials and their coefficients. We also show that apart from two exceptional zeros, the zeros of polynomials with large enough degree in such a Sheffer sequence lie on the line $x=1/2+it$.

Riordan matrices are infinite lower triangular matrices determined by a pair of formal power series over the real or complex field. These matrices have been mainly studied as combinatorial objects with an emphasis placed on the algebraic or combinatorial structure. The present paper contributes to the linear algebraic discussion with an analysis of Riordan matrices by means of the interaction of the properties of formal power series with the linear algebra. Specifically, it is shown that if a Riordan matrix $A$ is an $n\times n$ pseudo-involution then the singular values of $A$ must come in reciprocal pairs. Moreover, we give a complete analysis of existence and nonexistence of the eigenvectors of Riordan matrices. This leads to a surprising partition of the group of Riordan matrices into matrices with three different types of eigenvectors. Finally, given a nonzero vector $v$, we investigate the Riordan matrices $A$ that stabilize the vector $v$, t i.e. $Av=v$., Comment: 16 pages

Distinct letters x and y alternate in a word w if after deleting in w all letters but the copies of x and y we either obtain a word of the form xyxy... (of even or odd length) or a word of the form yxyx... (of even or odd length). A simple graph G=(V,E) is word-representable if there exists a word w over the alphabet V such that letters x and y alternate in w if and only if xy is an edge in E. Thus, edges of G are defined by avoiding the consecutive pattern 11 in a word representing G, that is, by avoiding xx and yy. In 2017, Jeff Remmel introduced the notion of a k-11-representable graph for a non-negative integer k, which generalizes the notion of a word-representable graph. Under this representation, edges of G are defined by containing at most k occurrences of the consecutive pattern 11 in a word representing G. Thus, word-representable graphs are precisely 0-11-representable graphs. Our key result in this paper is showing that every graph is 2-11-representable by a concatenation of permutations, which is rather surprising taking into account that concatenation of permutations has limited power in the case of 0-11-representation. Also, we show that the class of word-representable graphs, studied intensively in the literature, is contained strictly in the class of 1-11-representable graphs. Another result that we prove is the fact that the class of interval graphs is precisely the class of 1-11-representable graphs that can be represented by uniform words containing two copies of each letter. This result can be compared with the known fact that the class of circle graphs is precisely the class of 0-11-representable graphs that can be represented by uniform words containing two copies of each letter.

In this paper, we show that an infinite lower triangular matrix A = [ a i j ] i , j ∈ N 0 is an exponential Riordan matrix A = E ( g , f ) given by ∑ i ≥ j a i j z i / i ! = g f j / j ! if and only if there exist both a horizontal pair { h n ; h ˜ n } n ≥ 0 and a vertical pair { v n ; v ˜ n } n ≥ 0 of sequences that represent all the elements in the matrix. As a consequence, we obtain that if the horizontal and vertical pairs of an exponential Riordan matrix are identical then the matrix is an involution. In addition, this concept can be applied to obtain the determinants of the production matrix and some conditions for the d -orthogonality of the Sheffer polynomial sequences.

We introduce a Frechet Lie group structure on the Riordan group. We give a description of the corresponding Lie algebra as a vector space of infinite lower triangular matrices. We describe a natural linear action induced on the Frechet space K N by any element in the Lie algebra. We relate this to a certain family of bivariate linear partial differential equations. We obtain the solutions of such equations using one-parameter groups in the Riordan group. We show how a particular semidirect product decomposition in the Riordan group is reflected in the Lie algebra. We study the stabilizer of a formal power series under the action induced by Riordan matrices. We get stabilizers in the fraction field K ( ( x ) ) using bi-infinite representations. We provide some examples. The main tool to get our results is the paper [18] where the Riordan group was described using inverse sequences of groups of finite matrices.

We first consider two groups, F 0 = { g ∈ C [ [ z ] ] | g ( 0 ) ≠ 0 } under multiplication and F 1 = z F 0 under composition, where C [ [ z ] ] is the ring of formal power series over the complex field. It is known that the Riordan group R is isomorphic to the semidirect product F 0 ⋊ F 1 . It may be viewed as a group extension of F 1 by F 0 . In this paper, the group of three-dimensional Riordan arrays is obtained from an extension of the group R by F 0 . This concept extends to the group of multi-dimensional Riordan arrays. As an application, we illustrate the use of the three-dimensional Riordan array in multiple combinatorial sums.

Riordan bases in a power series ring of several variables are introduced as a special type of Schauder bases. As a part of the method of generating differentials, interplay of Riordan bases generalizes Lagrange inversion formulas. The set of Riordan bases has a group law with a specified set of variables as the unit element. Riordan arrays appear in a representation of the group.

Distinct letters $x$ and $y$ alternate in a word $w$ if after deleting in $w$ all letters but the copies of $x$ and $y$ we either obtain a word of the form $xyxy\cdots$ (of even or odd length) or a word of the form $yxyx\cdots$ (of even or odd length). A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $xy$ is an edge in $E$. In this paper we initiate the study of word-representable Toeplitz graphs, which are Riordan graphs of the Appell type. We prove that several general classes of Toeplitz graphs are word-representable, and we also provide a way to construct non-word-representable Toeplitz graphs. Our work not only merges the theories of Riordan matrices and word-representable graphs via the notion of a Riordan graph, but also it provides the first systematic study of word-representability of graphs defined via patterns in adjacency matrices. Moreover, our paper introduces the notion of an infinite word-representable Riordan graph and gives several general examples of such graphs. It is the first time in the literature when the word-representability of infinite graphs is discussed., Comment: To appear in Discrete Applied Mathematics

We consider ordered trees with a distinguished vertex which we call a mutator. There are many situations where this model arises. An ordered tree could represent a river network, supply lines, an employee organization chart, a phylogenetic tree, or a family tree. The mutator could be a dam or a source of pollution, a break in a supply line, a corrupt employee, and so on. We look at such things as the number of affected vertices, distance from the root, and the effect of various succession rules. Among the types of trees these results apply to are ordered trees defined by a uniform updegree requirement. We find some profiles of trees with a mutation in terms of Riordan matrices. The asymptotic formulas for the ratios of vertices of the new type to all vertices are also derived by using singularity analysis of generating functions.

In this paper we examine an equivalence relation on the set of formal power series with nonzero constant term. This is done both in terms of functional equations and also by interlacing two concepts from Riordan group theory, the A-sequence and the Bell subgroup. The best known example gives an equivalence class { 1 + z , 1 / ( 1 − z ) , C ( z ) , T ( z ) , Q ( z ) , … } where C ( z ) , T ( z ) and Q ( z ) are generating functions of the Catalan numbers, ternary numbers and quaternary numbers, respectively. A power series for one member of an equivalence class can be transformed into power series for the rest of members in the equivalence class and interpretations in terms of weighted lattice paths can also be given.

The notion of a Riordan graph was introduced recently, and it is a far-reaching generalization of the well-known Pascal graphs and Toeplitz graphs. However, apart from a certain subclass of Toeplitz graphs, nothing was known on independent sets in Riordan graphs. In this paper, we give exact enumeration and lower and upper bounds for the number of independent sets for various classes of Riordan graphs. Remarkably, we offer a variety of methods to solve the problems that range from the structural decomposition theorem to methods in combinatorics on words. Some of our results are valid for any graph., Comment: Accepted for publication in Discrete Mathematics

The r-ary number sequences given by(b (r)n ) n≥0 =1(r−1)n+ 1 rnn are analogs of the sequence of the Catalan numbers 1n+1 2nn . Their his-tory goes back at least to Lambert [8] in 1758 and they are of considerableinterest in sequential testing. Usually, the sequences are considered sep-arately and the generalizations can go in several directions. Here we linkthe various r first by introducing a new combinatorial structure relatedto GR trees and then algebraically as well. This GR transition gener-alizes to give r-ary analogs of many sequences of combinatorial interest.It also lets us find infinite numbers of combinatorially defined sequencesthat lie between the Catalan numbers and the Ternary numbers, or moregenerally, between b (r)n and b (r+1)n . 1. IntroductionThe Catalan numbers C n = 1n+12nn are well known to occur in enumera-tion of a great variety of combinatorial objects. Some good resources for theCatalan numbers are provided by Stanley [15, 14], which include more than200 combinatorial interpretations.One of the important generalizations of the Catalan numbers are the r-arynumbers b

Recently, the authors developed a q-analogue for Riordan matrices by means of Eulerian generating functions of the form g ( z ) = ∑ n ≥ 0 g n z n / n ! q where n ! q is the q-factorial. We apply this concept to give q-analogues of some familiar objects from the set partitions with double restrictions on blocks, namely the ( r , s ) -Bessel numbers of both types. By setting r = 0 and letting s → ∞ , these numbers may be reduced to the q-Stirling numbers of both kinds. Several algebraic formulas for the q-analogues are also derived using combinatorial methods together with the concept of q-Riordan matrices. In particular, q-analogues of the classical Bessel numbers of both kinds and their combinatorial interpretations are obtained.

The Stieltjes transform S A of an infinite lower triangular matrix A with nonzero diagonal entries is defined by S A = A − 1 A ¯ where A ¯ is the matrix obtained from A by deleting its initial row. In this paper, we express a sequence of polynomials as the characteristic polynomials of the Stieltjes transforms using a highly structured infinite lower triangular matrix called a Riordan matrix. As a result, computation of the zeros of such polynomials becomes amenable to iterative methods for computing eigenvalues, or to eigenvalue location theorems such as the Gersgorin theorem. We also describe a finite analog of the polynomial correspondence and its relationship to eigenvalue regions. As an application, the recurrence relations for several polynomial sequences are obtained using the Stieltjes transform.

The authors of this paper have used the theory of Riordan matrices to introduce the notion of a Riordan graph in \cite{CJKM}. Riordan graphs are proved to have a number of interesting (fractal) properties, and they are a far-reaching generalization of the well known and well studied Pascal graphs and Toeplitz graphs, and also some other families of graphs. The main focus in \cite{CJKM} is the study of structural properties of families of Riordan graphs obtained from certain infinite Riordan graphs. In this paper, we use a number of results in~\cite{CJKM} to study spectral properties of Riordan graphs. Our studies include, but are not limited to the spectral graph invariants for Riordan graphs such as the adjacency eigenvalues, (signless) Laplacian eigenvalues, nullity, positive and negative inertias, and rank. We also study determinants of Riordan graphs, in particular, giving results about determinants of Catalan graphs., Comment: 33 pages

In this paper, we use the theory of Riordan matrices to introduce the notion of a Riordan graph. The Riordan graphs are a far-reaching generalization of the well known and well studied Pascal graphs and Toeplitz graphs, and also some other families of graphs. The Riordan graphs are proved to have a number of interesting (fractal) properties, which can be useful in creating computer networks with certain desirable features, or in obtaining useful information when designing algorithms to compute values of graph invariants. The main focus in this paper is the study of structural properties of families of Riordan graphs obtained from infinite Riordan graphs, which includes a fundamental decomposition theorem and certain conditions on Riordan graphs to have an Eulerian trail/cycle or a Hamiltonian cycle. We will study spectral properties of the Riordan graphs in a follow up paper., 45 pages, 12 figures

An element of finite order in the Riordan group over the real field must have order 1 or 2. If we extend all the entries to be complex numbers then it may have any finite order. In the present paper, we investigate the elements of finite order of the Riordan group over the complex field. This notion leads us to define an element of pseudo order k ⩾ 2 and k-pseudo involution, respectively. It turns out that the inverse of a k-pseudo involution only differs from it in signs. We clarify some relationship between the elements of pseudo order k and k-pseudo involutions. In particular, k-pseudo involutions for k ≡ 0 ( mod 4 ) are characterized by a single sequence. The subgroups of the Riordan group formed by the elements having pseudo order of a prime power p r are also introduced.

The Riordan group consisting of Riordan matrices shows up naturally in a variety of combinatorial settings. In this paper, we define a q-Riordan matrix to be a q-analogue of the (exponential) Riordan matrix by using the Eulerian generating functions of the form ∑ n ⩾ 0 f n z n / n ! q . We first prove that the set of q-Riordan matrices forms a loop (a quasigroup with an identity element) and find its loop structures. Next, it is shown that q-Riordan matrices associated to the counting functions may be applied to the enumeration problem on set partitions by block inversions. This notion leads us to find q-analogues of the composition formula and the exponential formula, respectively.

Stirling numbers and Bessel numbers have a long history, and both have been generalized in a variety of directions. Here, we present a second level generalization that has both as special cases. This generalization often preserves the inverse relation between the first and second kind, and has simple combinatorial interpretations. We also frame the discussion in terms of the exponential Riordan group. Then the inverse relation is just the group inverse, and factoring inside the group leads to many results connecting the various Stirling and Bessel numbers.

In this paper we study a recursive system of integers { χ ( n , k ) : n > k ⩾ 0 } ; χ ( n + 2 , k ) = χ ( n + 1 , k ) - χ ( n + 1 , k - 1 ) + 2 χ ( n , k - 1 ) χ ( k + 1 , k ) = - χ ( k , k - 1 ) which is uniquely determined by the initial values { χ ( n , 0 ) } n = 1 ∞ . We show under the constant initial dates χ ( n , 0 ) = χ ( 1 , 0 ) for all n that the polynomial χ n ( x ) = ∑ k = 0 n - 1 χ ( n , k ) x k of degree n - 1 is (anti) palindromic. Several explicit formulae for χ ( n , k ) via Vandermonde matrix, mirrored Γ -matrix, weighed Delannoy number, Riordan array, hypergeometric function, Jacobi polynomial, and some combinatorial identities are derived.

An investigation is made of the Krylov matrices associated to the companion matrix C of a monic polynomial p ( x ) of degree n . In this paper, we propose a new approach to the study of Hankel matrices as Krylov matrices of C T . Firstly we focus on algebraic structures of Krylov matrices of C over a field F . We then discuss an equivalent condition for the Hankel matrix to be a Krylov matrix of C T by the notion of compatibility. As a result, we derive new determinant formulas for such Hankel matrices in terms of eigenvalues and eigenvectors of C respectively.

Let G be a finite group of order m ≥ 1 . A Dowling lattice Q n ( G ) is the geometric lattice of rank n over G . In this paper, we define the r -Whitney numbers of the first and second kind over Q n ( G ) , respectively. This concept is a common generalization of the Whitney numbers and the r -Stirling numbers of both kinds. We give their combinatorial interpretations over the Dowling lattice and we obtain various new algebraic identities. In addition, we develop the r -Whitney–Lah numbers and the r -Dowling polynomials associated with the Dowling lattice.

In theory, Riordan arrays can have any A -sequence and any Z -sequence. For examples of combinatorial interest they tend to be related. Here we look at the case that they are identical or nearly so. We provide a combinatorial interpretation in terms of weighted Łukasiewicz paths and then look at several large classes of examples.

In this paper, we describe the uplift principle for ordered trees which lets us solve a variety of combinatorial problems in two simple steps. The first step is to find the appropriate generating function at the root of the tree, the second is to lift the result to an arbitrary vertex by multiplying by the leaf generating function. This paper, though self contained, is a companion piece to Cheon and Shapiro (2008) [2] though with many more possible applications. It also may be viewed as an invitation, via the symbolic method, to the authoritative 800 page book of Flajolet and Sedgewick (2009) [8] . Our examples, with one exception, are different from those in this excellent reference.

We consider an infinite lower triangular matrix L = [ l n , k ] n , k ∈ N 0 and a sequence Ω = ( ω n ) n ∈ N 0 called the (a,b)-sequence such that every element l n + 1 , k + 1 except lying in column 0 can be expressed as l n + 1 , k + 1 = ∑ i = 0 ⌊ ( n - k ) / m ⌋ ω i l n - ai , k + bi , ω 0 ∉ 0 where a and b are integers with a + b = m > 0 and b ⩾ 0 . This concept generalizes the A-sequence of a Riordan matrix. As a result, we explore several structural properties of Riordan matrices by means of (a,b)-sequences. In particular, if b

Let Σ be the set of functions, convergent for all | z | > 1 , with a Laurent series of the form f ( z ) = z + ∑ n ⩾ 0 a n z - n . In this paper, we prove that the set of Faber polynomial sequences over Σ and the set of their normalized kth derivative sequences form groups which are isomorphic to the hitting time subgroup and the Bell(k) subgroup of the Riordan group, respectively. Further, a relationship between such Faber polynomial sequences and Lucas and Sheffer polynomial sequences is derived.

In this paper, more generalized q-factorial coecients are examined by a natural extension of the q-factorial on a sequence of any numbers. This immediately leads to the notions of the extended q-Stirling numbers of both kinds and the extended q-Lah numbers. All results described in this paper may be reduced to well-known results when we set q = 1 or use special sequences.

We give a short combinatorial proof of a Fine number generating function identity and then explore some of the ramifications in terms of random walks, friendly walkers, and ordered trees. The results are also generalized to obtain similar results including those in Motzkin and Schröder settings.

Several important combinatorial arrays, after inserting some minus signs, turn out to be involutions when considered as lower triangular matrices. Among these are the Pascal, RNA, and directed animal matrices. These examples and many others are in the Bell subgroup of the Riordan group. We characterize all such pseudo-involutions by means of a single sequence called the Δ-sequence. Finally we compute the Δ-sequences for the powers of a pseudo-involution in the Bell subgroup.

In this paper, we obtain a generalized Lucas polynomial sequence from the lattice paths for the Delannoy numbers by allowing weights on the steps (1,0),(0,1) and (1,1). These weighted lattice paths lead us to a combinatorial interpretation for such a Lucas polynomial sequence. The concept of Riordan arrays is extensively used throughout this paper.

Using the arithmetic–geometric mean inequality, we give bounds for k-subpermanents of nonnegative n×n matrices F. In the case k=n, we exhibit an n2-set S whose arithmetic and geometric means constitute upper and lower bounds for per(F)/n!. We offer sharpened versions of these bounds when F has zero-valued entries.

The spanning column rank of an matrix A over a semiring is the minimal number of columns that span all columns of A. We characterize linear operators that preserve the sets of matrix ordered pairs which satisfy multiplicative properties with respect to spanning column rank of matrices over semirings.

In this note we start by computing the average number of protected points in all ordered trees with n edges. This can serve as a guide in various organizational schemes where it may be desirable to have a large or small number of protected points. We will also look a few subclasses with a view to increasing or decreasing the proportion of protected points.

By observing that the infinite triangle obtained from some generalized harmonic numbers follows a Riordan array, we obtain very simple connections between the Stirling numbers of both kinds and other generalized harmonic numbers. Further, we suggest that Riordan arrays associated with such generalized harmonic numbers allow us to find new generating functions of many combinatorial sums and many generalized harmonic number identities.

We prove that if D = ( g ( x ) , f ( x ) ) is an element of order 2 in the Riordan group then g ( x ) = ± exp [ Φ ( x , xf ( x ) ] for some antisymmetric function Φ ( x , z ) . Also we prove that every element of order 2 in the Riordan group can be written as BMB - 1 for some element B and M = ( 1 , - 1 ) in the Riordan group. These proofs provide solutions to two open problems presented by L. Shapiro [L.W. Shapiro, Some open questions about random walks, involutions, limiting distributions and generating functions, Adv. Appl. Math. 27 (2001) 585–596].

We study involutions in the Riordan group, especially those with combinatorial meaning. We give a new determinantal criterion for a matrix to be a Riordan involution and examine several classes of examples. A complete characterization of involutions in the Appell subgroup is developed. In another direction we find several examples that generalize the RNA matrix but are of independent interest.

Let G be Kn,n with non-negative edge weights and let U and V be the two colour classes of vertices in G. We define a k-semimatching in G to be a set of k edges such that the edges either have distinct ends in U or distinct ends in V. Semimatchings are to be counted according to the product of the weights on the edges in the semimatching. The Dittert conjecture is a longstanding open problem involving matrix permanents. Here we show that it is equivalent to the following assertion: For a fixed total weight, the number of n-semimatchings in G is maximised by weighting all edges of G equally. We also introduce sub-Dittert functions which count k-semimatchings and are analogous to the subpermanent functions which count k-matchings. We prove some results about the extremal values of our sub-Dittert functions, and also that the Dittert conjecture cannot be disproved by means of unweighted graphs.

In this paper, we obtain important combinatorial identities of generalized harmonic numbers using symmetric polynomials. We also obtain the matrix representation for the generalized harmonic numbers whose inverse matrix can be computed recursively.

We summarise the progress which has been made since 1986 on the conjectures and open problems listed in H. Minc’s survey articles on the theory of permanents.

For a Boolean rank-1 matrix A = ab t , we define the perimeter of A as the number of nonzero entries in both a and b. We characterize the Boolean linear operators that preserve rank and perimeter of Boolean rank-1 matrices.

The n × n generalized Pascal matrix P ( t ) whose elements are related to the hypergeometric function 2 F 1 (a, b; c; x) is presented and the Cholesky decomposition of P ( t ) is obtained. As a result, it is shown that 2 F 1 (−a, −b; 1; x) = ∑ k=0 min (a,b) ( a k )( b k )x k is the solution of the Gauss's hypergeometric differential equation, x(1 − x)y″ + [1 + (a + b − 1)x]y′ − aby = 0 . where a and b are any nonnegative integers. Moreover, a recurrence relation for generating the elements of P ( t ) is given.

A matrix A =[ a ij ] is called a 7-matrix if its entries satisfy the recurrence relation αa i −1, j −1 + βa i −1, j = a ij where α , β are fixed numbers. A 7-matrix is completely determined by its first row and first column. In this paper we determine the structure of 7-matrices and investigate the sequences represented by columns of infinite 7-matrices.

In this paper we investigate the values of permanents according to the positions of zeros over m × n matrices with k zeros. We determine the maximum and minimum values of the permanent function and find the matrices which attain these extreme values.

A woven matrix, W, is a type of block matrix constructed from an m by n (0,1)-matrix D with row sums r1,r2,…,rm and column sums c1,c2,…,cn, ri by ri matrices Ri (i=1,2,…,m), and cj by cj matrices, Cj (j=1,2,…,n). Several properties of the determinant and the spectrum of woven matrices are known. In particular, the determinant of a woven matrix is ±(∏i=1mdetRi)(∏j=1ndetCj). In this paper it is shown that in general the permanent of W is not determined by the permanents of the Ri and Cj. However, there are instances when (1) per W=± ∏ i=1 m per R i ∏ j=1 n per C j . For example, it is shown that (I) holds if at least m−1 of the Ri are diagonal matrices. The main result of the paper is a characterization of the D’s for which each woven matrix, W, using D satisfies (I). As an application, we determine families of matrices whose permanents can be efficiently computed using determinants.

In this paper, we obtain a simple property of the Bernoulli polynomials Bn(x) and the Euler polynomials En(x). As a consequence, the relationship between two polynomials is obtained from bn(ξ)=∑k=0 k≠1nnkBk(0)En−k(ξ).