[seqfan] Re: Tatami
Richard J. Mathar
mathar at mpia-hd.mpg.de
Thu Mar 24 17:06:42 CET 2016
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The formula in http://list.seqfan.eu/pipermail/seqfan/2016-February/016160.html
seems to be correct. I computed (by brute force enumeration with a C++ program)
the arrangements of tatami tilings of the 3 X (2k+1) floor, k>=0,
with one monomer at an arbitrary position and 3k+1 dimers. Arrangements
which are symmetry-related by rotations or flips are counted with
multiplicity. The numerical result is
2, 10, 18, 38, 72, 136, 250, 454, 814, 1446, 2548...
This seems to have the g.f. -2*(x-1)*(2*x^2+4*x+1) ) / (x^2+x-1)^2 .
Then we have in the more general setup the number of tatami tilings of
the (2m+1) X(2k+1) floor with one monomer at an arbitrary position
and 2*(2*m*k+m+k) dimers. The result is symmetric in m <-> k and it
suffices to tabulate the counts as a triangle m>=0, 0<=k<=m:
1
2 10
3 18 10
4 38 8 10
5 72 18 4 x
6 136 24 x x x
7 250 32 x x x x
The second column (or row) is the sequence already shown above.
The first column are the integers because in the 1 X (2k+1) case
the monomer can be placed basically at any position which leaves
an even number of free spaces at both sides for the dimers.
Richard
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