What is it about?
Compositions are permutations of partitions. A partition may be visualised by a triangle of rows of decreasing length. An important geometrical concept is the size of the largest square (the Durfee square) contained in this triangle. In the permuted object (compositions), one loses the notion of the contained square but we have reinterpreted the concept for compositions so that it still makes geometrical sense.
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Why is it important?
This helps to unify the disparate theories of compositions and partitions. Just as a whole class of partitions is characterised by each having the same Durfee square, so a whole class of compositions is also characterised by each having the same Durfee square (in the composition sense).
Perspectives
I guess I have an obsession with an overall unifying theory and the invention of this concept by my research group gave me a chance to indulge my obsession, have a bit of fun and unify the disparate theories in a not immediately obvious way. In further papers we managed to tease out the ramifications of this idea and this led to a further notion of the depth of compositions.
Aubrey Blecher
University of the Witwatersrand
Read the Original
This page is a summary of: Durfee squares in compositions, Discrete Mathematics and Applications, December 2018, De Gruyter,
DOI: 10.1515/dma-2018-0032.
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