Related papers: How Expressive Are Friendly School Partitions?

Motivated by children inventing creatively laws on how to add or multiply fractions, we introduce four fraction operations and discover some of their properties.

In this paper, we introduce a natural geometric extension of the partition function. More precisely, we investigate the problem of counting partitions of a rectangle into rectangular blocks with integer sides. Here, two partitions of a…

We consider an assignment problem that has aspects of fair division as well as social choice. In particular, we investigate the problem of assigning a small subset from a set of indivisible items to multiple players so that the chosen…

A classical method for partition generating functions is developed into a tool with wide applications. New expansions of well-known theorems are derived, and new results for partitions with n copies of n are presented.

In deciding on a student's grade in a class, an instructor generally needs to combine many individual grading judgments into one overall judgment. Two relatively common numerical scales used to specify individual grades are the 4-point…

When two patterns occur equally often in a set of permutations, we say that these patterns are equipopular. Using both structural and analytic tools, we classify the equipopular patterns in the set of separable permutations. In particular,…

We consider a one-sided matching problem where agents who are partitioned into disjoint classes and each class must receive fair treatment in a desired matching. This model, proposed by Benabbou et al. [2019], aims to address various…

Fair graph partition of social networks is a crucial step toward ensuring fair and non-discriminatory treatments in unsupervised user analysis. Current fair partition methods typically consider node balance, a notion pursuing a…

Fair division with unequal shares is an intensively studied recourse allocation problem. For $ i\in [n] $, let $ \mu_i $ be an atomless probability measure on the measurable space $(C,\mathcal{S}) $ and let $ t_i $ be positive numbers…

In many reserve-based affirmative action systems, admission cutoffs are publicly disclosed to enable students to verify whether reserved seats are correctly assigned. We introduce a verifiability criterion: each student must be able to…

Drawing appropriate diagrams is a useful problem solving heuristic that can transform a given problem into a representation that is easier to exploit for solving it. A major focus while helping introductory physics students learn problem…

A topological graph is a graph drawn in the plane. A topological graph is $k$-plane, $k>0$, if each edge is crossed at most $k$ times. We study the problem of partitioning the edges of a $k$-plane graph such that each partite set forms a…

Unfair work distribution in student teams is a common issue in project-based learning. One contributing factor is that students are differently skilled developers. In a course with group work intertwining engineering and business aspects,…

Teamwork is integral to higher education, fostering students' interpersonal skills, improving learning outcomes, and preparing them for professional collaboration later in their careers. While team formation has traditionally been managed…

Multiphysics problems involving two or more coupled physical phenomena are ubiquitous in science and engineering. This work develops a new partitioned exponential approach for the time integration of multiphysics problems. After a possible…

We show that cylindric partitions are in one-to-one correspondence with a pair which has an ordinary partition and a colored partition into distinct parts. Then, we show the general form of the generating function for cylindric partitions…

We give a series of recursive identities for the number of partitions with exactly $k$ parts and with constraints on both the minimal difference among the parts and the minimal part. Using these results we demonstrate that the number of…

We define and study obvious strategy-proofness with respect to a partition of the set of agents. It encompasses strategy-proofness as a special case when the partition is the coarsest one and obvious strategy-proofness when the partition is…

Collaboration skills are important for future software engineers. In computer science education, these skills are often practiced through group assignments, where students develop software collaboratively. The approach that students take in…

Let $p_n$ be the number of partitions of an integer $n$. For each of the partition statistics of counting their parts, ranks, or cranks, there is a natural family of integer polynomials. We investigate their asymptotics and the limiting…