Some new exact van der Waerden numbers.
Some new facts about group 𝒢 generated by the family of convergent permutations
The aim of this paper is to present some new and essential facts about group 𝒢 generated by the family of convergent permutations, i.e. the permutations on ℕ preserving the convergence of series of real terms. We prove that there exist permutations preserving the sum of series which do not belong to 𝒢. Additionally, we show that there exists a family G (possessing the cardinality equal to continuum) of groups of permutations on ℕ such that each one of these groups is different than 𝒢 and is composed...
Some new good characterizations for directed graphs
Some new infinite families of congruences modulo 3 for overpartitions into odd parts
Let denote the number of overpartitions of n in which only odd parts are used. Some congruences modulo 3 and powers of 2 for the function have been derived by Hirschhorn and Sellers, and Lovejoy and Osburn. In this paper, employing 2-dissections of certain quotients of theta functions due to Ramanujan, we prove some new infinite families of Ramanujan-type congruences for modulo 3. For example, we prove that for n, α ≥ 0, .
Some new methods in the theory of -quasi-invariants.
Some new optimum Golomb rectangles.
Some new partially symmetric designs and their resolution
Some new Ramsey colorings.
Some new results about the shortness exponent in polyhedral graphs
Some New Tilings of the Sphere with Congruent Triangles
Some new transformations for Bailey pairs and WP-Bailey pairs
We derive several new transformations relating WP-Bailey pairs. We also consider the corresponding transformations relating standard Bailey pairs, and as a consequence, derive some quite general expansions for products of theta functions which can also be expressed as certain types of Lambert series.
Some news about oblique graphs
A k-gon α of a polyhedral graph G(V,E,F) is of type ⟨b₁,b₂,...,bₖ⟩ if the vertices incident with α in cyclic order have degrees b₁,b₂,...,bₖ and ⟨b₁,b₂,...,bₖ⟩ is the lexicographic minimum of all such sequences available for α. A polyhedral graph G is oblique if it has no two faces of the same type. Among others it is shown that an oblique graph contains vertices of degree 3.
Some news about the independence number of a graph
For a finite undirected graph G on n vertices some continuous optimization problems taken over the n-dimensional cube are presented and it is proved that their optimum values equal the independence number of G.
Some non-normal Cayley digraphs of the generalized quaternion group of certain orders.
Some observations and solutions to short and long global Nim.
Some observations on the constructions of chemical balance weighing designs
The construction of some optimum chemical balance weighing designs from affine μ-resolvable balanced incomplete block (BIB) designs are discussed in the light of a characterization theorem on the parameters of affine μ-resolvable BIB designs as given by Mohan and Kageyama (1982), for the sake of practical use of researchers who need some selective designs for the construction of chemical balance weighing designs.
Some observations on the Lah and Laguerre transforms of integer sequences.
Some pairwise balanced designs.
Some Parity Statistics in Integer Partitions
We study integer partitions with respect to the classical word statistics of levels and descents subject to prescribed parity conditions. For instance, a partition with summands may be enumerated according to descents while tracking the individual parities of and . There are two types of parity levels, E = E and O = O, and four types of parity-descents, E > E, E > O, O > E and O > O, where E and O represent arbitrary even and odd summands. We obtain functional equations and explicit...
Some plethysm results related to Foulkes' conjecture.