Séminaire Lotharingien de Combinatoire, 78B.72 (2017), 12 pp.
Jason P. Smith
We introduce a formal definition of a pattern poset. This generalises
many of the existing posets defined in terms of patterns on different
combinatorial objects. We introduce a poset fibration on intervals of
these posets. Applying this fibration gives some general results on
pattern posets, that unify and generalise many of the existing results
on these posets, such as Björner's results on subword order. We
present a formula for the Möbius function of intervals of pattern
posets, which provides an explanation as to why the various
definitions of normal embeddings play such an important role in many
of the existing results for such posets. Moreover, we characterise
when these posets are disconnected and show that Cohen-Macaulayness is
preserved by our fibration. We also conjecture that fibrations
preserve shellability under certain conditions.
Received: November 14, 2016.
Accepted: February 17, 2017.
Final version: April 1, 2017.
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