Linear Recurrence Relations Research Articles

Explicit enumeration of 321, hexagon-avoiding permutations

The 321, hexagon-avoiding ( 321-hex) permutations were introduced and studied by Billey and Warrington (J. Alg. Comb. 13 (2001) 111–136). as a class of elements of S n whose Kazhdan–Lusztig polynomials and the singular loci of whose Schubert varieties have certain fairly simple and explicit descriptions. This paper provides a 7-term linear recurrence relation leading to an explicit enumeration of the 321-hex permutations. A complete description of the corresponding generating tree is obtained as a by-product of enumeration techniques used in the paper, including Schensted's 321-subsequences decomposition, a 5-parameter generating function and the symmetries of the octagonal patterns avoided by the 321-hex permutations.

Basic matrix algebra with algorithms and applications

SYSTEMS OF LINEAR EQUATIONS AND THEIR SOLUTION Recognizing Linear Systems and Solutions Matrices, Equivalence and Row Operations Echelon Forms and Gaussian Elimination Free Variables and General Solutions The Vector Form of the General Solution Geometric Vectors and Linear Functions Polynomial Interpolation MATRIX NUMBER SYSTEMS Complex Numbers Matrix Multiplication Auxiliary Matrices and Matrix Inverses Symmetric Projectors, Resolving Vectors Least Squares Approximation Changing Plane Coordinates The Fast Fourier Transform and the Euclidean Algorithm. DIAGONALIZABLE MATRICES Eigenvectors and Eigenvalues The Minimal Polynomial Algorithm Linear Recurrence Relations Properties of the Minimal Polynomial The Sequence {Ak} Discrete dynamical systems Matrix compression with components DETERMINANTS Area and Composition of Linear Functions Computing Determinants Fundamental Properties of Determinants Further Applications Appendix: The abstract setting Selected practice problem answers Index

A cubic analogue of the RSA cryptosystem

In this paper, we investigate a public key cryptosystem which is derived from a third order linear recurrence relation and is analogous to the RSA and LUC cryptosystems. The explicit formulation involves a generalisation of the rule for composition of powers and of the calculus of the Euler totient function which underlie the algebra of the RSA cryptosystem. The security of all these systems appears to be comparable and to depend on the intractability of factorization but the systems do not seem to be mathematically equivalent.

Recursively constructible families of graphs

An infinite sequence of graphs { G n } n⩾0 is called recursive if the Tutte polynomials T( G n ; x, y) satisfy a linear recurrence relation whose coefficients are polynomials in x and y. In this paper we introduce a general method based on transfer matrices for proving that a family is recursive that covers all examples known to us. As an application we show that, for fixed s, the sequence of complete bipartite graphs { K s, n } is recursive and satisfies a linear recurrence whose degree is the number of partitions of the integer s.

More Power to Fibonacci

You should never judge a book by its title - but I nearly always do. A mathematical colleague recommended a book to me Concrete Mathematics [1]. Just recently, someone else also referred me to the same book. Well a wink’s as good as a nod to a blind man and I now realise what a mistake I’d made; concrete is a conflation of ‘continuous’ and ‘discrete’. Like all good textbooks in mathematics, it distinguishes itself not just by the quality of its contents and presentation, but also by the superb collection of problems that it includes. One such problem - question 58 in Chapter 6 - inspired this article. There was a hint towards a generalised result, but there were no details. It did give a reference, [2, 3], but despite some efforts I found this unobtainable. Hence this article and my attempt to cover the same ground. The question posed in the book is concerned with the linear recurrence relations satisfied by the successive powers of Fibonacci numbers - so we start with the Fibonacci sequence itself.

The Specker–Blatter theorem does not hold for quaternary relations

Let C be a class of relational structures. We denote by f C (n) the number of structures in C over the labeled set {0,…, n−1}. For any C definable in monadic second-order logic with unary and binary relation symbols only, E. Specker and C. Blatter showed that for every m∈ N , the function f C satisfies a linear recurrence relation modulo m, and hence it is ultimately periodic modulo m. The case of ternary relation symbols, and more generally of arity k symbols for k⩾3, was left open. In this paper we show that for every m there is a class of structures C m , which is definable even in first-order logic with one quaternary (arity four) relation symbol, such that f C m is not ultimately periodic modulo m. This shows that the Specker–Blatter Theorem does not hold for quaternary relations, leaving only the ternary case open.

The Solution of Linear Probabilistic Recurrence Relations

Linear probabilistic divide-and-conquer recurrence relations arise when analyzing the running time of divide-and-conquer randomized algorithms. We consider first the problem of finding the expected value of the random process T(x) , described as the output of a randomized recursive algorithm T . On input x , T generates a sample (h1,···,hk) from a given probability distribution on [0,1]k and recurses by returning g(x) + Σi=1kciT(hi x) until a constant is returned when x becomes less than a given number. Under some minor assumptions on the problem parameters, we present a closed-form asymptotic solution of the expected value of T(x) . We show that E[T(x)] = Θ( xp + xp∈t1x(g(u)/ up+1 ) du) , where p is the nonnegative unique solution of the equation Σi=1kciE[hip] = 1 . This generalizes the result in [1] where we considered the deterministic version of the recurrence. Then, following [2], we argue that the solution holds under a broad class of perturbations including floors and ceilings that usually accompany the recurrences that arise when analyzing randomized divide-and-conquer algorithms.

Single-Boundary Iteration for Polynomial Invariants

In an earlier paper [1] we show that if a family of oriented links obey an iterative pattern, their HOMFLY polynomials satisfy a homogeneous linear recurrence relation. This also holds for Kauffman polynomials. Here we prove the result using a different construction which in some cases yields a recurrence relation of much lower order, thereby drastically reducing the amount of computation required to obtain explicit polynomials for individual members or a general formula for the entire family.

Linear recurring sequence subgroups in finite fields

Given a finite field F and a linear recurrence relation over F it is possible to find, in a fairly “obvious” way, a finite extension L of F and a subgroup M of the multiplicative group of L such that the elements of M may be written, without repetition, so as to form a cyclically closed sequence which obeys the recurrence. Here we investigate this phenomenon for second-order recurrences; the situation in which F has prime order and the characteristic polynomial of the relation is irreducible over F is described.

Matrices determined by a linear recurrence relation among entries

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.

A unified rounding error bound for polynomial evaluation

We present a unified rounding error bound for polynomial evaluation. The bound presented here takes the same general form for the evaluation of a polynomial written in any polynomial basis when the evaluation algorithm can be expressed as a linear recurrence or a first-order linear matrix recurrence relation. Examples of these situations are: Horner's algorithm in the evaluation of power series, Clenshaw's and Forsythe's algorithms in the evaluation of orthogonal polynomial series, de-Casteljau's algorithm for Bernstein polynomial series, the modification of Clenshaw's algorithms in the evaluation of Szegő polynomial series, and so on.

A Class of Functions Computable by Index Grammars

The classes of functions IGF and IGF0 are investigated that can be computed by means of special index grammars. The class IGF0 is demonstrated to coincide with the class of strictly increasing functions (of a natural argument), which can be represented by systems of linear recurrence relations with natural coefficients, and to be closed with respect to the operations of addition, multiplication, series summation, and addition of a natural number to their arguments or results.

A Reciprocity Theorem for Monomer-Dimer Coverings

The problem of counting monomer-dimer coverings of a lattice is a longstanding problem in statistical mechanics.It has only been exactly solved for the special case of dimer coverings in two dimensions ([Ka61], [TF61]). In earlier work, Stanley [St85] proved a reciprocity principle governing the number $N(m,n)$ of dimer coverings of an $m$ by $n$ rectangular grid (also known as perfect matchings), where $m$ is fixed and $n$ is allowed to vary. As reinterpreted by Propp [P01], Stanley's result concerns the unique way of extending $N(m,n)$ to $n<0$ so that the resulting bi-infinite sequence, $N(m,n)$ for $n \in \mathbb{Z}$, satisfies a linear recurrence relation with constant coefficients. In particular, Stanley shows that $N(m,n)$ is always an integer satisfying the relation $N(m,-2-n) = \varepsilon_{m,n} N(m,n)$ where $\varepsilon_{m,n}=1$ unless $m \equiv 2(\mod 4)$ and $n$ is odd, in which case $\varepsilon_{m,n}=-1$. Furthermore, Propp's method was applicable to higher-dimensional cases.This paper discusses similar investigations of the numbers $M(m,n)$, of monomer-dimer coverings, or equivalently (not necessarily perfect) matchings of an $m$ by $n$ rectangular grid. We show that for each fixed $m$ there is a unique way of extending $M(m,n)$ to $n<0$ so that the resulting bi-infinite sequence, $M(m,n)$ for $n \in \mathbb{Z}$, satisfies a linear recurrence relation with constant coefficients.We show that $M(m,n)$, a priori a rational number, is always an integer, using a generalization of the combinatorial model offered by Propp. Lastly, we give a new statement of reciprocity in terms of multivariate generating functions from which Stanley's result follows.

Transcendence of Certain Reciprocal Sums of Linear Recurrences

Suppose that {R n } n ⩾ 0 is a sequence of integers satisfying a binary linear recurrence relation with suitable conditions. We prove the transcendency of the numbers $$$$

HOMFLY POLYNOMIALS OF ITERATIVE FAMILIES OF LINKS

We show that if a family of oriented links obey an iterative pattern, their HOMFLY polynomials satisfy a homogeneous linear recurrence relation. The construction embodied in the proof is applied to derive explicit formulas for the HOMFLY polynomials of some iterative families, including the torus links {Tn,3}n≥0.

Backward Stability of Clenshaw's Algorithm

We study numerical properties of Clenshaw's algorithm for summing the series w = ∑n = 0Nbnpn where pn satisfy the linear three-term recurrence relation. We prove that under natural assumptions Clenshaw's algorithm is backward stable with respect to the data bn, n = 0,N.

Linear recurrence relations in the algebra of matrices and applications

We study here some linear recurrence relations in the algebra of square matrices. With the aid of the Cayley–Hamilton Theorem, we derive some explicit formulas for A n (n⩾r) and e tA for every r×r matrix A, in terms of the coefficients of its characteristic polynomial and matrices A j , where 0⩽j⩽r−1.

A Reciprocity Theorem for Domino Tilings

Let $T(m,n)$ denote the number of ways to tile an $m$-by-$n$ rectangle with dominos. For any fixed $m$, the numbers $T(m,n)$ satisfy a linear recurrence relation, and so may be extrapolated to negative values of $n$; these extrapolated values satisfy the relation $$T(m,-2-n)=\epsilon_{m,n}T(m,n),$$ where $\epsilon_{m,n}=-1$ if $m \equiv 2$ (mod 4) and $n$ is odd and where $\epsilon_{m,n}=+1$ otherwise. This is equivalent to a fact demonstrated by Stanley using algebraic methods. Here I give a proof that provides, among other things, a uniform combinatorial interpretation of $T(m,n)$ that applies regardless of the sign of $n$.

Elliptic Anomaly in Constructing Long-Term and Short-Term Dynamical Theories

The techniques of Brumberg and Brumberg (1999) based on the use of elliptic anomaly are specified in this paper in two aspects. The iteration technique (Broucke, 1969) to construct short-term semi-analytical theories of motion in rectangular coordinates in lines of Encke and Hill is reelaborated in terms of elliptic anomaly resulting in extending this technique for high-eccentricity orbits. In constructing long-term semi-analytical theories the key point is to integrate trigonometric functions of several angular arguments related to time by different differential expressions. This problem is reduced in the paper to linear algebraic recurrence relations admitting efficient solution by iterations.

The solutions of a linear homogeneous recurrence relation with unit coefficients

In this paper, three expressions of solutions of linear homogeneous recurrence relations with unit coefficients$$ x_n=x_{n-1}+x_{n-2}+\cdots+x_{n-(k-1)}+x_{n-k},\quad \forall n\ge k, k\ge 3$$are derived.