The determinants of matrices with recursive entries

Moghaddamfar, A. R.; Salehy, S. Navid; Salehy, S. Nima

doi:10.1016/j.laa.2007.11.021

Cited by 9 publications

(7 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Evaluating the determinant of so-called Pascal-like matrices (whose entries satisfy some specific 3-term recurrence) has been discussed before; see for instance [1], [13] and [23]. Many useful and efficient tools to evaluate determinants are listed in the survey papers by Krattenthaler ([12] and [14]).…”

Section: Pfaffian Calculationsmentioning

confidence: 99%

Off-diagonally symmetric domino tilings of the Aztec diamond

Lee¹

2023

Preprint

View full text Add to dashboard Cite

We introduce a new symmetry class of domino tilings of the Aztec diamond, called the off-diagonal symmetry class, which is motivated by the off-diagonally symmetric alternating sign matrices introduced by Kuperberg in 2002. We use the method of non-intersecting lattice paths and a modification of Stembridge's Pfaffian formula for families of non-intersecting lattice paths to enumerate our new symmetry class. The number of off-diagonally symmetric domino tilings of the Aztec diamond can be expressed as a Pfaffian of a matrix whose entries satisfy a nice and simple recurrence relation. INTRODUCTION Symmetry classesConsider a finite group G acting on a set of combinatorial objects X. Let H be a subgroup of G. A symmetry class is a collection of H-invariant objects of X. In enumerative combinatorics, it is quite challenging to enumerate each symmetry class because the structure of each class varies with different subgroups H. These usually require different methods to enumerate them.The study of symmetry classes of plane partitions goes back to MacMahon [21], but gained more attention in the 1970s and 80s. Stanley [31] identified ten symmetry classes of plane partitions, all the symmetry classes can be enumerated by nice product formulas. We refer the interested reader to the survey paper by Krattenthaler [15, Section 6] for a modern update to Stanley's paper.An alternating sign matrix (ASM) of order n is an n × n matrix with entries 0, 1 or −1 such that all row and column sums are equal to 1 and the non-zero entries alternate in sign in each row and column. They were introduced by Mills, Robbins and Rumsey [22] in the early 1980s. The symmetry classes of ASMs under the action of the dihedral group of order 8 were proposed and summarized by Stanley [30] and Robbins [27,28]. There are eight symmetry classes of ASMs; five of them were fully solved, one was partially solved, while for the remaining two there are no known or conjectured formula. We refer the interested reader to the detailed account written by Behrend, Fischer and Konvalinka [3, Section 1.2].In the 1990s, Kuperberg's seminal paper [17] brought the statistical mechanical six-vertex model into the study of ASMs. Later, in [18], he successfully provided a unified framework using the six-vertex model to solve some of the symmetry classes of ASMs. He also introduced several new types of ASMs, such as, off-diagonally symmetric, vertically and horizontally perverse, with U-turn sides, and combined them with the original eight symmetry classes. The enumerative results of these new types of ASMs were summarized in [3, Section 1.2].The Aztec diamond of order n, denoted by AD(n), is the union of all unit squares in the region x + y ≤ n + 1 which was introduced by Elkies, Larsen, Kuperberg and Propp [7,8] in the early 1990s. The symmetry classes of domino tilings of AD(n) 1 under the action of the 2020 Mathematics Subject Classification. 05A15, 05B20, 05B45. Key words and phrases. Aztec diamonds, domino tilings, method of non-intersecting lattice paths, Pfaffians,...

show abstract

Section: Pfaffian Calculationsmentioning

confidence: 99%

Off-diagonally symmetric domino tilings of the Aztec diamond

Lee¹

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…We start with the definition of matrices whose elements are given by a recurrence relation as stated in [15]. We will refer to such a matrix as a recursively defined matrix.…”

Section: Ldu-decomposition Of the Bi-moment Matrixmentioning

confidence: 99%

“…In the case of the three parameter model the corresponding determinant was evaluated using theorems from [13] and [14]. In this five parameter case we have been unable to find any similar theorems and thus attempted an LDU decomposition directly.…”

Section: Bi-orthogonal Pair Of Polynomial Sequencesmentioning

confidence: 99%

See 1 more Smart Citation

Bi-orthogonal polynomial sequences and the asymmetric simple exclusion process

Brak¹,

Moore²

2015

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

We apply the bi-moment determinant method to compute a representation of the matrix product algebra -a quadratic algebra satisfied by the operators d and e -for the five parameter (α, β, γ, δ and q) Asymmetric Simple Exclusion Process. This method requires an LDU decomposition of the "bi-moment matrix". The decomposition defines a new pair of basis vectors sets, the 'boundary basis'. This basis is defined by the action of polynomials {P n } and {Q n } on the quantum oscillator basis (and its dual). Theses polynomials are orthogonal to themselves (ie. each satisfy a three term recurrence relation) and are orthogonal to each other (with respect to the same linear functional defining the stationary state). Hence termed 'bi-orthogonal'. With respect to the boundary basis the bi-moment matrix is diagonal and the representation of the operator d + e is tri-diagonal. This tri-diagonal matrix defines another set of orthogonal polynomials very closely related to the the Askey-Wilson polynomials (they have the same moments).

show abstract

“…Furthermore, when one is faced with matrices whose entries obey a recursive relation, inversion can be complicated. Such matrices have been studied e.g., in [1], [6], [8], [10], [11], [12] and [14]. The research done in these papers is mostly related to the determinants; rarely are the inverses discussed.…”

Section: Introductionmentioning

confidence: 99%

Certain matrices related to Fibonacci sequence having recursive entries

Moghaddamfar¹,

Salehy²,

Salehy³

2008

ELA

View full text Add to dashboard Cite

Let φ = (φ i ) i≥1 and ψ = (ψ i ) i≥1 be two arbitrary sequences with φ 1 = ψ 1 . Let A φ,ψ (n) denote the matrix of order n with entries a i,j , 1 ≤ i, j ≤ n, where a 1,j = φ j and a i,1 = ψ i for 1 ≤ i ≤ n, and where, where one of the sequences φ or ψ is the Fibonacci sequence (i.e., 1, 1, 2, 3, 5, 8, . . .) and the other is one of the following sequences:1, 1, . . . , 1, 0, 0, 0, . . .) ,For some sequences of the above type the inverse of A φ,ψ (n) is found. In the final part of this paper, the determinant of a generalized Pascal triangle associated to the Fibonacci sequence is found.

show abstract

The determinants of matrices with recursive entries

Cited by 9 publications

References 3 publications

Off-diagonally symmetric domino tilings of the Aztec diamond

Off-diagonally symmetric domino tilings of the Aztec diamond

Bi-orthogonal polynomial sequences and the asymmetric simple exclusion process

Certain matrices related to Fibonacci sequence having recursive entries

Contact Info

Product

Resources

About