Abstract
The size of maximum antichains in the product of n linear orders is known when the n linear orders have the same length. We present an exact expression for the size of maximum antichains when the linear orders have (possibly) different lengths. From this, we derive an exact expression for the size of maximum antichains in the product of n linear orders with the same length. This expression is equivalent to but different from the existing expression. It allows us to present an asymptotic result for the size of maximum antichains of n linear orders with the same length m going to infinity.
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Notes
This example was suggested to us by an anonymous referee.
Most definitions about posets are taken from Proctor et al. (1980).
When we submitted the first version of this paper, we were not aware of Scott’s post and our proof used generating functions. We thank an anonymous reviewer for pointing to us this post on StatExchange.
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Bouyssou, D., Marchant, T. & Pirlot, M. The size of the maximum antichains in products of linear orders. TOP 29, 648–659 (2021). https://doi.org/10.1007/s11750-020-00587-6
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DOI: https://doi.org/10.1007/s11750-020-00587-6