# Induction on.docx - Induction on #element ofThe...

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Induction on #element ofThe meet-distributivity hypothesis implies that the numberofL.LSuppose that. Trivial for #t Lcovers= 1. Now let #jelements ofLL2, and set,and letstL≤be a maximal0=tfor whichL− {t}.[s,t]=Bkis equal to. HenceXgk(L)xk=xj+ Xgk(L0)(1 +x)k,andand the proof follows by induction sinceL0 is meet-distributive.Note that in the special caseL=J(P),gk(L) is equal to the number ofk-element antichains ofP.(c) Letx= −1 in (b). This result was first proved (in a different way) fordistributiveJ. Combinatorial Theory, Ser. B26(1979), 295–299. It canlattices by S. K. Das, also be proved using the identityµζ ζ=ζin theincidence algebra of the latticeL{ˆ1}.1
Topological remark.This exercise has an interesting topologicalgeneralization (done in collaboration with G. Kalai). GivenL, definean abstract cubical complex Ω = Ω(L) as follows: the vertices of Ω arethe elements ofL, and the faces of Ω consist of intervals [s,t] ofLisomorphic to boolean algebras. (It follows from Exercise 3.177(a)that Ω is indeed a cubical complex.)Proposition.The geometric realization |Ω| is contractible. In fact, Ω iscollapsible.Sketch of proof.scontractible.Ω(be the meet of elements thatL0)| is obtainedfromLettbe a maximal element of|t

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