Tiling with bars under tomographic constraints

TL;DR

This paper investigates the problem of tiling rectangles or tori with vertical and horizontal bars of fixed length, satisfying specific row and column constraints, providing results for certain cases and generalizations.

Contribution

It introduces new results for specific instances and a broader version of the tiling problem under tomographic constraints, while leaving some open questions.

Findings

Results for particular tiling instances

Generalizations of the tiling problem

Open problems in the initial formulation

Abstract

We wish to tile a rectangle or a torus with only vertical and horizontal bars of a given length, such that the number of bars in every column and row equals given numbers. We present results for particular instances and for a more general problem, while leaving open the initial problem.