Related papers: Recurrence Relations and Determinants
We evaluate four families of determinants of matrices, where the entries are sums or differences of generating functions for paths consisting of up-steps, down-steps and level steps. By specialisation, these determinant evaluations have…
Let n\geq3 and J_{n}:=circ(J_{1},J_{2},...,J_{n}) and j_{n}:=\circ(j_{0},j_{1},...,j_{n-1}) be the n\timesn circulant matrices, associated with the nth Jacobsthal number J_{n} and the nth Jacobsthal-Lucas number j_{n}, respectively. The…
By studying Cameron's operator in terms of determinants, two kinds of "integer" sequences of incomplete numbers were introduced. One was the sequence of restricted numbers, including $s$-step Fibonacci sequences. Another was the sequence of…
We prove that $\det A\leq 6^\frac{n}{6}$ whenever $A\in\{0,1\}^{n\times n}$ contains at most $2n$ ones. We also prove an upper bound on the determinant of matrices with the $k$-consecutive ones property, a generalisation of the consecutive…
Introduced by Okounkov and Reshetikhin, the Schur process is known to be a determinantal point process, meaning that its correlation functions are minors of a single correlation kernel matrix. Previously, this was derived using…
We study the largest values of the $r$th row of Stern's diatomic array. In particular, we prove some conjectures of Lansing. Our main tool is the connection between the Stern sequence, alternating binary expansions and continuants. This…
We consider the numbers $\mathcal{B}_{r,s} = (\mathbf{B}+1)^r \mathbf{B}^s$ (in umbral notation $\mathbf{B}^n = \mathbf{B}_n$ with the Bernoulli numbers) that have a well-known reciprocity relation, which is frequently found in the…
A new matrix operation based on inserting columns and rows, similarly to the mediant operation between fractions, gives rise to the Farey determinants matrix or, equivalently, the matrix of the numerators of the differences of Farey…
Some new relations on skew Schur function differences are established both combinatorially using Sch\"utzenberger's jeu de taquin, and algebraically using Jacobi-Trudi determinants. These relations lead to the conclusion that certain…
We fix three natural numbers $k, n, N$, such that $n+k+1=N$, and introduce the notion of two dual arrangements of hyperplanes. One of the arrangements is an arrangement of $N$ hyperplanes in a $k$-dimensional affine space, the other is an…
This paper presents a theorem which solves the problem of reduction of the determinant order by means of a transformation of it, into other determinant whose each element are a determinant of second order. This implies that, if the process…
We compute the generating function for the characters of the irreducible representations of SU(n) whose associated Young diagrams have only two rows with the same number of boxes. The result is a rational determinantal expression in which…
We propose a generalization of meanders, i.e., configurations of non-selfintersecting loops crossing a line through a given number of points, to SU(N). This uses the reformulation of meanders as pairs of reduced elements of the…
In the space of square matrices, we characterize row-generated subspaces, on which the determinant is an irreducible polynomial. As a corollary, we characterize square systems of polynomial equations with indeterminate coefficients, whose…
We establish some identities relating two sequences that are, as explained, related to the Tribonacci sequence. One of these sequences bears the same resemblance to the Tribonacci sequence as the Lucas sequence does to the Fibonacci…
In this note we investigate the solutions of certain meta-Fibonacci recurrences of the form $f(n)=f(n-f(n-1))+f(n-2)$ for various sets of initial conditions. In the case when $f(n)=1$ for $n\leq 1$, we prove that the resulting integer…
In 1971, by induction on $n$ and using a two-term linear recurrence relation, Graham and Pollak got a beautiful formula $$\det(D_n)=-(n-1)(-2)^{n-2}$$ on the determinant of distance matrix $D_n$ of a tree $T_n$ on $n$ vertices. The…
Nested recurrence relations are highly sensitive to their initial conditions. The best-known nested recurrence, the Hofstadter $Q$-recurrence, generates sequences displaying a wide variety of behaviors. Most famous among these is the…
Motivated by complexity questions in integer programming, this paper aims to contribute to the understanding of combinatorial properties of integer matrices of row rank $r$ and with bounded subdeterminants. In particular, we study the…
In this (mostly expository) paper I want to share some observations prompted by a class of matrices whose determinants are Catalan numbers. Considering different methods of proof we obtain some generalizations and q-analogues and…