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We investigate paths in the hexagonal circle packing and enumerate them with respect to width, height, number of steps, area, and kissing number. Functional equations and the kernel method yield closed bivariate generating functions…
A recent paper of Church, Ellenberg, and Farb uses topology and representation theory of the symmetric group to prove enumerative results about square-free polynomials and F-stable maximal tori of the general linear group over the algebraic…
In this paper, we introduce the generating functions of partition sequences. Partition sequences have a one-to-one correspondence with partitions. Therefore, the generating function has no multiplicity and appears meaningless initially.…
In this paper we focus on two new families of polynomials which are connected with exponential polynomials and geometric polynomials. We discuss their generalizations and show that these new families of polynomials and their generalizations…
We study linear divisibility sequences of order 4, providing a characterization by means of their characteristic polynomials and finding their factorization as a product of linear divisibility sequences of order 2. Moreover, we show a new…
We determine, in a polynomial ring over a field, the arithmetical rank of certain ideals generated by a set of monomials and one binomial.
We build a class of polynomial problems with not polynomial certificates. The parameter concerning which are defined efficiency of corresponding algorithms is the number $n$ of elements of the set has used at construction of combinatory…
We introduce a remarkable new family of norms on the space of $n \times n$ complex matrices. These norms arise from the combinatorial properties of symmetric functions, and their construction and validation involve probability theory,…
A vertex subset $W\subseteq V$ of the graph $G=(V,E)$ is an independent dominating set if every vertex in $V\backslash W$ is adjacent to at least one vertex in $W$ and the vertices of $W$ are pairwise non-adjacent. The independent…
We study universal polynomials of characteristic classes associated to the $\mathcal{A}$-classification (i.e. up to right-left equivalence) of holomorphic map-germs $(\mathbb{C}^2,0) \to (\mathbb{C}^n, 0)$ $(n=2,3)$. That enables us to…
A pattern class is a set of permutations closed under the formation of subpermutations. Such classes can be characterised as those permutations not involving a particular set of forbidden permutations. A simple collection of necessary and…
In this paper we introduce a family of two-variable derivative polynomials for tangent and secant. We study the generating functions for the coefficients of this family of polynomials. In particular, we establish a connection between these…
We consider the class of directed graphs with $N\geq 1$ edges and without loops shorter than $k\geq1$. Using the concept of a labelled graph, we determine graphs from this class that maximize the number of all paths of length $k$. Then we…
A Skolem sequence is a linear arrangement of the multiset, {1, 1, 2, 2, ..., n, n} such that if r in [n] appears in positions i and j, then |i-j| = r. We first translate the problem to a particular set of perfect matchings, then apply the…
We describe a generating tree approach to the enumeration and exhaustive generation of k-nonnesting set partitions and permutations. Unlike previous work in the literature using the connections of these objects to Young tableaux and…
New methods for derivation of Bell polynomials of the second kind are presented. The methods are based on an ordinary generating function and its composita. The relation between a composita and a Bell polynomial is demonstrated. Main…
Let L be a bounded distributive lattice. We give several characterizations of those L^n --> L mappings that are polynomial functions, i.e., functions which can be obtained from projections and constant functions using binary joins and…
We study generating functions of strict and non-strict order polynomials of series-parallel posets, called order series. These order series are closely related to Ehrhart series and h*-polynomials of the associated order polytopes. We…
We construct a class of positive linear maps on matrix algebras. We find conditions when these maps are atomic, decomposable and completely positive. We obtain a large class of atomic positive linear maps. As applications in quantum…
Fix two lattice paths P and Q from (0,0) to (m,r) that use East and North steps with P never going above Q. We show that the lattice paths that go from (0,0) to (m,r) and that remain in the region bounded by P and Q can be identified with…