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arXiv · 1103.2877

Partitioning in the space of antimonotonic functions

Patrick De Causmaecker·Stefan De Wannemacker
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Abstract

This paper studies partitions in the space of antimonotonic boolean functions on sets of n elements. The antimonotonic functions are the antichains of the partially ordered set of subsets. We analyse and characterise a natural partial ordering on this set. We study the inter- vals according to this ordering. We show how intervals of antimonotonic functions, and a fortiori the whole space of antimonotonic functions can be partitioned as disjoint unions of certain classes of intervals. These in- tervals are uniquely determined by antimonotonic functions on smaller sets. This leads to recursive enumeration algorithms and new recursion relations. Using various decompositions, we derive new recursion formu- lae for the number of antimonotonic functions and hence for the number of monotonic functions (i.e. the Dedekind number).

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Patrick De Causmaecker, Stefan De Wannemacker. 2011-03-15. Partitioning in the space of antimonotonic functions. https://arxiv.org/abs/1103.2877

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