On Numbers Related to Partitions of Unlike Objects and Occupancy Problems

Judicious partitions and related problems

Many classical partitioning problems in combinatorics ask for a single quantity to be maximized or minimized over a set of partitions of a combinatorial object. For instance, Max Cut asks for the largest bipartite subgraph of a graphG, while Min Bisection asks for the minimum size of a cut into two equal pieces. In judicious partitioning problems, we seek to maximize or minimize a number of qua...

Coding objects related to Catalan numbers

A coding method using binary sequences is presented for different computation problems related to Catalan numbers. This method proves in a very easy way the equivalence of these problems.

On Untouchable Numbers and Related Problems

In 1973, Erdős proved that a positive proportion of numbers are untouchable, that is, not of the form σ(n) − n, the sum of the proper divisors of n. We investigate the analogous question where σ is replaced with the sum-of-unitary-divisors function σ∗ (which sums divisors d of n such that (d, n/d) = 1), thus solving a problem of te Riele from 1976. We also describe a fast algorithm for enumerat...

On partitions, surjections, and Stirling numbers

the evaluation of language related engagment and task related engagment with the purpose of investigating the effect of metatalk and task typology

abstract while task-based instruction is considered as the most effective way to learn a language in the related literature, it is oversimplified on various grounds. different variables may affect how students are engaged with not only the language but also with the task itself. the present study was conducted to investigate language and task related engagement on the basis of the task typolog...