A Note on Covering Young Diagrams with Applications to Local Dimension of Posets

TL;DR

This paper establishes optimal bounds for covering Young diagrams with rectangles, revealing implications for local dimension in posets and answering open questions from prior research.

Contribution

It proves the minimal number of rectangles needed to cover Young diagrams and relates these results to local covering numbers in difference graphs, advancing understanding in poset theory.

Findings

Optimal covering bounds for Young diagrams with rectangles.

Partitioning diagrams into rectangles with limited row and column usage.

Applications to local dimension of partially ordered sets.

Abstract

We prove that in every cover of a Young diagram with $\binom{2k}{k}$ steps with generalized rectangles there is a row or a column in the diagram that is used by at least $k+1$ rectangles. We show that this is best-possible by partitioning any Young diagram with $\binom{2k}{k}-1$ steps into actual rectangles, each row and each column used by at most $k$ rectangles. This answers two questions by Kim et al. (2018). Our results can be rephrased in terms of local covering numbers of difference graphs with complete bipartite graphs, which has applications in the recent notion of local dimension of partially ordered sets.