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Consider the set of simplicial complexes on $n$ vertices, with partial ordering by containment. What is the Mobius function for this poset?

Are other combinatorial facts known about it (e.g. the number of chains of a given length from one simplicial complex to another)?

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    $\begingroup$ Isn't it $0$ if to get from the larger to the smaller you have to remove any simplex that is a face of another simplex, and $(-1)$ to the number of simplices removed otherwise? $\endgroup$
    – Will Sawin
    Jan 14, 2014 at 17:05
  • $\begingroup$ @WillSawin: I think I agree on the second case, $(-1)$ to the number of simplices removed if the removed simplices are all incomparable to one another (by comparison to a powerset or boolean algebra), but how does the 0 show up? $\endgroup$
    – Jake Levinson
    Jan 20, 2014 at 4:23
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    $\begingroup$ Same way it does in the divisibility poset of the natural numbers for non-squarefree numbers. $\endgroup$
    – Will Sawin
    Jan 20, 2014 at 4:57

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