Séminaire Lotharingien de Combinatoire, 78B.72 (2017), 12 pp.

Jason P. Smith

Pattern Posets

Abstract. 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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