cp's OEIS Frontend

A165642 Number of ways to assemble an n-cube from 2n labeled (n-1)-cubes with labeled vertices, where left-handed and right-handed counterparts are considered distinct.

%I A165642 #5 Jan 13 2013 03:55:10
%S A165642 2,96,7864320,5917648890101760,131757106216193173905620336640,
%T A165642 213712134396364787165698675955466240000000000000,
%U A165642 52598387159216969439004693376714880262603303706802782208000000000000000
%N A165642 Number of ways to assemble an n-cube from 2n labeled (n-1)-cubes with labeled vertices, where left-handed and right-handed counterparts are considered distinct.
%F A165642 a(n) = 2 * ((2n-2)!!)^(2n-1) * (2n-1)!
%e A165642 For n=2, we are constructing a square from 4 labeled line-segments with labeled endpoints. Solutions which differ by a rotation are considered equivalent, but solutions which are a reflection of each other are considered distinct (assume the square we are constructing is embedded in a plane, so we cannot flip it over to convert a left-handed solution to right-handed solution). There are 6 ways to order the line-segments, and each line-segment can be oriented in 2 ways, so the total number of solutions is 6 * 2^4 = 96. For n=3, we are constructing a cube from 6 labeled squares with labeled vertices (assume we are confined to 3-space, so we consider reflections of the cube to be distinct). Without loss of generality, we can pick one labeled square to serve as our face of reference. For this face, we must decide which side of the square will face the interior of the cube, but we do not care about which rotation we pick as these just translate into rotations of the cube. From this reference square, there are 5! ways to assign the remaining squares to the faces of the cube, and each square can be oriented in 8 ways (we can pick which side of the square will face the interior of the cube, and we can pick from 4 rotations). This gives 2 * 8^5 * 5! solutions. The factor of "2" comes from the choice of which side of the reference square will face the interior of the cube (a choice which would go away if we considered reflections to be equivalent).
%Y A165642 Cf. A165643 (same idea, but reflections are equivalent). A165644 and A091868 are the corresponding sequences for simplices instead of cubes.
%K A165642 nonn
%O A165642 1,1
%A A165642 _Andrew Weimholt_, Sep 23 2009
%E A165642 Example reformatted by _Andrew Weimholt_, Sep 25 2009