[seqfan] abstract simplicial complexes of same-dimensional facets

hv at crypt.org hv at crypt.org
Tue Aug 10 13:40:39 CEST 2010

If I understand correctly how they work, with n vertices there are 2^C(n, d)
ASCs of 0 or more d-dimensional facets, and thus sum_0^n{2^C(n,i)}+1-n ASCs
for which all facets have the same dimension:

%I A000000
%S A000000 2,3,6,17,96,2111,1114238,68723671293,1180735735906024030716,
%T A000000 170141183460507917357914971986913657851,
%U A000000 7237005577335553223087828975127304179197147198604070555943173844710572689402
%N A000000 abstract simplicial complexes over n vertices for which all facets have the same dimension
%F A000000 sum_0^n{ 2^C(n,i) } + 1 - n
%Y A000000 A000372(n) is the count over n vertices when we don't restrict to same-dimensional facets
%A A000000 Hugo van der Sanden (hv(AT)crypt.org)
%O A000000 0,1
%e A000000 There are 20 abstract simplicial complexes over 3 vertices; of these, all facets are the same dimension expect for the 3 consisting of a line and a point such as {{1,2}, {3}}, so a(3)=17.
%K A000000 easy,nonn

I don't fully grasp antichains, but I suspect this could easily be recast
as a sequence counting antichains of some type (?? "such that all subsets
have the same cardinality" ??).

(Considering how to calculate more terms of A000372 led me to consider this
sequence as a trivial lower bound.)


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