Related papers: Splitting Kronecker squares, 2-decomposition numbe…
We introduce the concepts of an amazing hypercube decomposition and a double shortcut for it, and use these new ideas to formulate a conjecture implying the Combinatorial Invariance Conjecture of the Kazhdan--Lusztig polynomials for the…
Compound matrices play an important role in many fields of mathematics and have recently found new applications in systems and control theory. However, the explicit formulas for these compounds are non-trivial and not always easy to use.…
By prepending zeros to a given sequence Hankel determinants of backward shifts of this sequence become meaningful. We obtain some results for the sequences of Catalan numbers and of some numbers and polynomials which are related to Catalan…
This note presents some results about Hankel determinants of backwards shifted Catalan-like numbers and related sequences.
We investigate the iterated Kronecker product of a square matrix with itself and prove an invariance property for symmetric subspaces. This motivates the definition of an iterated symmetric Kronecker product and the derivation of an…
In this paper, constructions of multimagic squares are investigated. Diagonal Latin squares and Kronecker products are used to get some constructions of multimagic squares. Consequently, some new families of compound multimagic squares are…
We present a study of three families of Kronecker coefficients, which we describe in terms of reduced Kronecker coefficients. This study is grounded on the generating function of the coefficients, proved by a bijection between two…
Numerous congruences for partitions with designated summands have been proven since first being introduced and studied by Andrews, Lewis, and Lovejoy. This paper explicitly characterizes the number of partitions with designated summands…
In (Haglund, Remmel, Wilson 2018) Haglund, Remmel and Wilson introduced their Delta conjectures, which give two different combinatorial interpretations of the symmetric function $\Delta'_{e_{n-k-1}} e_n$ in terms of rise-decorated or…
We give an algorithm for working out the indecomposable direct summands in a Krull--Schmidt decomposition of a tensor product of two simple modules for G=SL_3 in characteristics 2 and 3. It is shown that there is a finite family of modules…
We provide new formulas for the coefficients in the partial fraction decomposition of the restricted partition generating function. These techniques allow us to partially resolve a recent conjecture of Sills and Zeilberger. We also describe…
We introduce a new approach for generating combinatorial identities and formulas by the application of Kronecker substitution to polynomial expansions within quotient rings. Our main result enables the derivation of elementary arithmetic…
We develop a higher order generalization of the LQ decomposition and show that this decomposition plays an important role in likelihood-based estimation and testing for separable, or Kronecker structured, covariance models, such as the…
It is well known that the Catalan number C_n counts dissections of a regular (n+2)-gon into triangles. Here we count such dissections by number of triangles that contain two sides of the polygon among their three edges, leading to a…
We present new proofs and generalizations of unimodality of the q-binomial coefficients \binom{n}{k}_q as polynomials in q. We use an algebraic approach by interpreting the differences between numbers of certain partitions as Kronecker…
We study certain series with Catalan numbers and reciprocal Catalan numbers, respectively, and provide seemingly new closed form evaluations of these series with Fibonacci (Lucas) entries. In addition, we state some combinatorial sums that…
We study the properties of the odd Catalan numbers, C_n, modulo 2^k for k >= 2. We show that there exist only k - 1 different congruences of the odd Catalan numbers modulo 2^k. Moreover, these congruences can be obtained by C_{2^m - 1} (mod…
We formulate a series of conjectures on the stable tensor product of irreducible representations of symmetric groups, which are closely related to the reduced Kronecker coefficients. These conjectures are certain generalizations of…
For each integer $k\ge 1$, we define an algorithm which associates to a partition whose maximal value is at most $k$ a certain subset of all partitions. In the case when we begin with a partition $\lambda$ which is square, i.e…
In the paper, with the aid of the series expansions of the square or cubic of the arcsine function, the authors establish several possibly new combinatorial identities containing the ratio of two central binomial coefficients which are…