On the Mfobius Function of Pointed Partitions and Exponential Pointed Structures
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Date
2015-05-14
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Addis Ababa University
Abstract
This thesis is concerned with partial orders on set and integer partitions and related
structures. The study of set and integer partitions dates back to Euler and Sylvester
[28]. Over the course of time it has become apparent that the combinatorics of partitions
encodes important mathematical structures in a variety of felds. Results on
partially ordered sets of partitions involving Mfobius numbers, homology and homotopy
of order complexes emerged in the works of Rota and Stanley [56]. In this work,
we use pointed partition structures where a part of an integer and set partition is
marked. These have been introduced in the works of Ehrenborg and Readdy [60],
Ehrenborg and Jung [63] and Ziegler [29]. We exhibit new enumerative and geometric
properties of the partial orders of pointed partitions studied by Ehrenborg and
Readdy [60]. In particular, we compute the Mfobius numbers and homotopy types of
lower intervals.
In addition, we investigate their ordered counterparts and provide analogous results
in this setting. Then we use the pointed set partition lattice to introduce exponential
pointed structures which are the pointed analog of exponential structures introduced
by Stanley [55]. We show that this concept encompasses many examples introduced
before. In particular, we introduce pointed decompositions of lattices and study their
enumerative and geometric structure. We also show that exponential pointed structures
satisfy pointed analogs of Stanley's compositional and exponential formulas
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On the Mfobius Function of Pointed Partitions