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%I #2 Feb 27 2009 03:00:00
%S 1,4,864,2579890176
%N Number of maximal (and largest) superset towers with n base elements.
%C On 3 elements ABC, some tower (Halmos, "Naive Set Theory" among many) that begins with the empty set can be written without loss of generality as {0, A, AB, ABC}. But we need to have sets B, C, BC, AC included somewhere too so that the thing is "largest", i.e., includes every subset of {A,B,C}. For ABCD, there are 3^3 ways to include B,C,D into AB,AC,AD,BC,BD,CD and 2^5 ways to include AC,AD,BC,BD,CD into ABC,ABD,ACD,BCD. So a(4) = 3^3*2^5.
%F (n-1)^(n-1) * (n-2)^(nC2 - 1) * (n-3)^(nC3 - 1) *...* 2^(nC(n-2) - 1) * 1^(n-1)
%e a(2) = 4 because:
%e (1) 0->A A->AB B->AB C->AB AB->ABC AC->ABC BC->ABC ABC->ABC maximal
%e (2) 0->A A->AB B->AB C->AC AB->ABC AC->ABC BC->ABC ABC->ABC maximal
%e (3) 0->A A->AB B->AC C->AB AB->ABC AC->ABC BC->ABC ABC->ABC maximal
%e (4) 0->A A->AB B->AC C->AC AB->ABC AC->ABC BC->ABC ABC->ABC maximal
%K nonn
%O 0,2
%A Lee Corbin (lcorbin(AT)tsoft.com), Jan 28 2006
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