Jan 29, 2002
From: Richard Guy <rkg(AT)cpsc.ucalgary.ca>

I got interested in the number of ways
of packing calissons (diamonds, rhombuses) in a
hexagon.  I laboriously calculated 1, 2, 20, 980
and sent this to `lookup', only to find that
this was well known to those who well know it as
A008793.  But I'm also interested in the number
of ways, not counting rotations and reflexions
as different.

n=0.   1 way, by a usual convention.

n=1.   1 way (rotate thru pi/6 to get the other)

n=2.   6 ways:
    ________         ________         ________
   /   /\   \       /\   \   \       /\   \   \
  /___/  \___\     /  \___\___\     /  \___\___\
 /\   \  /   /\   /\  /\   \   \   /\  /   /\   \
/  \___\/___/  \ /  \/  \___\___\ /  \/___/  \___\  
\  /   /\   \  / \  /\  /   /   / \  /\   \  /   /
 \/___/  \___\/   \/  \/___/___/   \/  \___\/___/
  \   \  /   /     \  /   /   /     \  /   /   /
   \___\/___/       \/___/___/       \/___/___/

   1 of these           2 of each of these
    ________         ________         ________
   /   /\   \       /\   \   \       /   /\   \
  /___/  \___\     /  \___\___\     /___/  \___\
 /   /\  /\   \   /\  /   /\   \   /\   \  /   /\
/___/  \/  \___\ /  \/___/  \___\ /  \___\/___/  \
\   \  /\  /   / \  /   /\  /   / \  /\   \   \  /
 \___\/  \/___/   \/___/  \/___/   \/  \___\___\/
  \   \  /   /     \   \  /   /     \  /   /   /
   \___\/___/       \___\/___/       \/___/___/ 

   3 of these           6 of each of these

makes 20 in all.

n=3.  113 ways.  I won't draw them all, but here
is a check:
                           ____________
    8 x  2 =  16          /   /\   \   \
    1 x  4 =   4         /___/  \___\___\
   48 x  6 = 288        /   /\  /   /   /\
   56 x 12 = 672       /___/  \/___/___/  \
  ----      -----     /\   \  /   /\   \  /\
  113        980     /  \___\/___/  \___\/  \
                     \  /\   \   \  /\   \  /
                      \/  \___\___\/  \___\/
                       \  /   /   /\  /   /
                        \/___/___/  \/___/
                         \   \   \  /   /
                          \___\___\/___/

                       Here's the unique one.



###################################################################

Postscript

Date: Thu, 07 Feb 2002 10:23:10 +0000
From: Don Reble <djr(AT)nk.ca>

For n=4, 20174 ways. Similarly,

      2 x  1 =      2
     33 x  2 =     66
     18 x  3 =     54
     16 x  4 =     64
   1433 x  6 =   8598
  18672 x 12 = 224064
  -----        ------
  20174        232848

Here are both fully-symmetric ones.

          ________________
         /   /   /\   \   \
        /___/___/  \___\___\
       /   /   /\  /\   \   \
      /___/___/  \/  \___\___\
     /\   \   \  /\  /   /   /\
    /  \___\___\/  \/___/___/  \
   /\  /\   \   \  /   /   /\  /\
  /  \/  \___\___\/___/___/  \/  \
  \  /\  /   /   /\   \   \  /\  /
   \/  \/___/___/  \___\___\/  \/
    \  /   /   /\  /\   \   \  /
     \/___/___/  \/  \___\___\/
      \   \   \  /\  /   /   /
       \ __\___\/  \/___/___/
        \   \   \  /   /   /
         \___\___\/___/___/


          ________________
         /   /   /\   \   \
        /___/___/  \___\___\
       /   /\   \  /   /\   \
      /___/  \___\/___/  \___\
     /\   \  /   /\   \  /   /\
    /  \___\/___/  \___\/___/  \
   /\  /   /\   \  /   /\   \  /\
  /  \/___/  \___\/___/  \___\/  \
  \  /\   \  /   /\   \  /   /\  /
   \/  \___\/___/  \___\/___/  \/
    \  /   /\   \  /   /\   \  /
     \/___/  \___\/___/  \___\/
      \   \  /   /\   \  /   /
       \___\/___/  \___\/___/
        \   \   \  /   /   /
         \___\___\/___/___/