A strong coloring on a cardinal $\kappa$ is function $f:[\kappa]^2\to \kappa$ such that for every $A\subseteq of full size $\kappa$, color $\gamma<\kappa$ attained by $f\upharpoonright[A]^2$. The symbol $\kappa\nrightarrow [\kappa]^2_\kappa$ asserts the existence $\kappa$. We introduce $\kappa\nrightarrow_p[\kappa]^2_\kappa$ which over partition $p:[\kappa]^2\to\theta$. $f$ $p$ if $A\in [\kappa...

We prove two partition identities which are dual to the Rogers–Ramanujan identities. These inspired by (and proved using) a correspondence between three kinds of objects: new type partitions (neighborly partitions), monomial ideals and some infinite graphs.

I present a bijection on integer partitions that leads to recursive expressions, closed formulae and generating functions for the cardinality of certain sets of partitions of a positive integer n. The bijection leads also to a product on partitions that is associative with a natural grading thus defining a free associative algebra on the set of integer partitions. As an outcome of the computati...

The problem to determine partitions of a given reet angle which are optimal for segment approximation (e.g. by bivariate pieeewise polynomials) is investigated. We give eriteria for optimal partitions and develop algorithms for eomputing optimal partitions of eertain types. It is shown that there is a surprising relationship between various types of optimal partitions. In this way, we obtain go...