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You can do a 2 dimensional table using just a tiny bit of Python, though this is dramatically slow

sage: for n in range(1,8):
....:     d = [0] * (n+1)
....:     for P in Posets(n):
....:         d[P.dimension()] += 1
....:     for i in d:
....:         print "%4s" %i,
....:     print
   0    1
   0    1    1
   0    1    4    0
   0    1   15    0    0
   0    1   62    0    0    0
   0    1  314    3    0    0    0
   0    1 1955   89    0    0    0    0

As you see, up to 7 vertices there is no such lattice.