Alexander R. Its, Craig A. Tracy, Harold Widom
It is proved that the limiting distribution of the length of the longest weakly increasing subsequence in an inhomogeneous random word is related to the distribution function for the eigenvalues of a certain direct sum of Gaussian unitary ensembles subject to an overall constraint that the eigenvalues lie in a hyperplane.
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In this paper, we evaluate determinants of some families of Toeplitz-Hessenberg matrices having tribonacci number entries. These determinant formulas may also be expressed equivalently as identities that involve sums of products of multinomial coefficients and tribonacci numbers. In particular, we establish a connection between the tribonacci and the Fibonacci and Padovan sequences via Toeplitz-Hessenberg determinants. We then obtain, by combinatorial arguments, extensions of our determinant formulas in terms of generalized tribonacci sequences satisfying an r-th order recurrence of a more gen…
We present a proof of determinant of special nonsymmetric Toeplitz matrices conjectured by Anđelić and Fonseca in \cite{andjelic2020some}. A proof is also demonstrated for a more general theorem. The two conjectures are therefore just two possible results, under two specific settings. Numerical examples validating the theorem are provided.
The Toeplitz determinants (of increasing size) associated with the symbols $exp{t(z+z^{-1})}$ or $(1-ξ{z})^α (1-ξ{z^{-1}})^β$ satisfy recursion relations, thus expressing all the Toeplitz determinants as a rational function of the first few determinants. A. Borodin found these relations using Riemann-Hilbert methods. The nature of Borodin's relations pointed towards the Toeplitz lattice and its Virasoro algebra, as developed by the authors. In this paper, we take the Toeplitz and Virasoro approach for a fairly large class of symbols, leading to a systematic and simple way of generating such re…
In this paper, we find determinant formulas of several Hessenberg-Toeplitz matrices whose nonzero entries are derived from the small and large Schroder and Fine number sequences. Algebraic proofs of these results can be given which make use of Trudi's formula and the generating function of the associated sequence of determinants. We also provide direct arguments of our results that utilize various counting techniques, among them sign-changing involutions, on combinatorial structures related to classes of lattice paths enumerated by the Schroder and Fine numbers. As a consequence of our results…
This paper, a continuation of math.CO/9909169, connects the analysis of the length of the longest weakly increasing subsequence of inhomogeneous random words to a Riemann-Hilbert problem and an associated system of integrable PDEs. In particular, we show that the Poissonization of the distribution function of this length can be identified as the Jimbo-Miwa-Ueno tau function.
In this note we prove positivity of Maclaurin coefficients of polynomials written in terms of rising factorials and arbitrary log-concave sequences. These polynomials arise naturally when studying log-concavity of rising factorial series. We propose several conjectures concerning zeros and coefficients of a generalized form of those polynomials. We also consider polynomials whose generating functions are higher order Toeplitz determinants formed by rising factorial series. We make three conjectures about these polynomials. All proposed conjectures are supported by numerical evidence.
We evaluate the Geiss-Leclerc-Schroer $ϕ$-map for shape modules over the preprojective algebra $Λ$ of type $\hat{A_1}$ in terms of matrix minors arising from the block-Toeplitz representation of the loop group $\SL_2(\mathcal{L})$. Conjecturally these minors are among the cluster variables for coordinate rings of unipotent cells within $\SL_2(\mathcal{L})$. In so doing we compute the Euler characteristic of any generalized flag variety attached to a shape module by counting standard tableaux of requisite shape and parity; alternatively by counting chess tableaux of requisite shape and content.
Symmetry reductions and recurrence degrees for banded Toeplitz determinants and permanents
本文研究带状Toeplitz行列式与积和式的对称约化，提出三种机制并给出对称与斜对称情形下的精确递归度界，同时证明五对角带的积和式-行列式转换。
Under binary matrices we mean matrices whose entries take one of two values. In this paper, explicit formulae for calculating the determinant of some type of binary Toeplitz matrices are obtained. Examples of the application of the determinant of binary Toeplitz matrices for the enumeration of even and odd permutations of different types are given.
Many discrete integrable systems exhibit the Laurent phenomenon. In this paper, we investigate three integrable systems: the Somos-4 recurrence, the Somos-5 recurrence and a system related to so-called $A_1$ $Q$-system, whose general solutions are derived in terms of Hankel determinant. As a result, we directly confirm that they satisfy the Laurent property. Additionally, it is shown that the Somos-5 recurrence can be viewed as a specified Bäcklund transformation of the Somos-4 recurrence. The related topics about Somos polynomials are also studied.
In this paper, we are going to calculate the determinant of a certain type of square matrices, which are related to the well-known Cauchy and Toeplitz matrices. Then, we will use the results to determine the rank of special non-square matrices.
Constructive recurrences for determinants and permanents of banded Toeplitz matrices
针对带状Toeplitz矩阵，提出两种构造性Laplace展开方法，得到行列式与积和式的极小阶递推，并精确分类状态图，推广至循环情形。
We obtain an identity between Fredholm determinants of two kinds of operators, one acting on functions on the unit circle and the other acting on functions on a subset of the integers. This identity is a generalization of an identity between a Toeplitz determinant and a Fredholm determinant that has appeared in the random permutation context. Using this identity, we prove, in particular, convergence of moments for arbitrary rows of a random Young diagram under Plancherel measure.
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