
COMMENTS

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.


EXAMPLE

a(2) = 4 because:
(1) 0>A A>AB B>AB C>AB AB>ABC AC>ABC BC>ABC ABC>ABC maximal
(2) 0>A A>AB B>AB C>AC AB>ABC AC>ABC BC>ABC ABC>ABC maximal
(3) 0>A A>AB B>AC C>AB AB>ABC AC>ABC BC>ABC ABC>ABC maximal
(4) 0>A A>AB B>AC C>AC AB>ABC AC>ABC BC>ABC ABC>ABC maximal
