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A multiplicative deformation of the Möbius function for the poset of partitions of a multiset. (English) Zbl 1210.05180

Chow, Timothy Y. (ed.) et al., Communicating mathematics. A conference in honor of Joseph A. Gallian’s 65th birthday, Duluth, MN, USA, July 16–19, 2007. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4345-1/pbk). Contemporary Mathematics 479, 113-118 (2009).
Summary: The Möbius function of a partially ordered set is a very convenient formalism for counting by inclusion-exclusion. An example of central importance is the partition lattice, namely the partial order by refinement on partitions of a set \(\{1,\dots ,n\}\). It seems quite natural to generalize this to partitions of a multiset, i.e. to allow repetition of the letters. However, the Möbius function is not nearly so well-behaved. We introduce a multiplicative deformation, denoted \(\mu'\), for the Möbius function of the poset of partitions of a multiset and show that it possesses much more elegant formulas than the usual Möbius function in this case.
For the entire collection see [Zbl 1157.90002].

MSC:

05E18 Group actions on combinatorial structures
05E05 Symmetric functions and generalizations
06A06 Partial orders, general
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