Geodesic
A note on covering Young diagrams with applications to local dimension of posets
Stefan Felsner1 ; Torsten Ueckerdt2 ; Stefan Felsner ; Torsten Ueckerdt
1Technische Universität Berlin, Berlin, Germany
2Karlsruhe Institute of Technology, Karlsruhe, Germany
Acta mathematica Universitatis Comenianae, Tome 88 (2019) no. 3, pp. 673-678 
Stefan Felsner; Torsten Ueckerdt; Stefan Felsner; Torsten Ueckerdt. A note on covering Young diagrams with applications to local  dimension of posets. Acta mathematica Universitatis Comenianae, Tome 88 (2019) no. 3, pp. 673-678. http://geodesic.mathdoc.fr/item/AMUC_2019_88_3_a48/
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     author = {Stefan Felsner and Torsten Ueckerdt and Stefan Felsner and Torsten Ueckerdt},
     title = { A note on covering {Young} diagrams with applications to local  dimension of posets},
     journal = {Acta mathematica Universitatis Comenianae},
     pages = {673--678},
     year = {2019},
     volume = {88},
     number = {3},
     url = {http://geodesic.mathdoc.fr/item/AMUC_2019_88_3_a48/}
}
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Voir la notice de l'article provenant de la source Comenius University

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~\textit{et al.} [arXiv 1803.08641]. 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.