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A182110 Irregular triangle read by rows: generating function counting rotationally distinct n X n tatami tilings with n monomers and exactly k vertical dimers.

%I A182110 #34 Nov 04 2019 02:24:50
%S A182110 1,1,2,1,2,3,2,1,2,3,6,4,2,2,1,2,3,6,9,8,7,6,2,2,2,1,2,3,6,9,14,15,14,
%T A182110 14,10,8,6,4,2,2,2,1,2,3,6,9,14,22,24,25,28,25,22,19,14,10,10,8,4,4,2,
%U A182110 2,2,1,2,3,6,9,14,22,32,37,42,49,48,49,46,38,34,30,24,20,16,12,12,10,6,4,4,2,2,2
%N A182110 Irregular triangle read by rows: generating function counting rotationally distinct n X n tatami tilings with n monomers and exactly k vertical dimers.
%C A182110 Monomer-dimer tatami tilings are arrangements of 1 X 1 monomers, 2 X 1 vertical dimers and 1 X 2 horizontal dimers on subsets of the integer grid, with the property that no four tiles meet at any point.  a(n) applies to tilings of this type which have monomers in their top corners.
%C A182110 a(n) is the table T(2,0); T(3,0), T(3,1); T(4,0), T(4,1), T(4,2), T(4,3); T(5,0), T(5,1) ... where T(n,k) is the number of n X n tilings of the type described above with exactly k vertical dimers when n is even and exactly k horizontal dimers when n is odd.
%H A182110 Alejandro Erickson, <a href="/A182110/b182110.txt">Table of n, a(n) for n = 0..9999</a>
%H A182110 Alejandro Erickson, <a href="/A182110/a182110.txt">Table of coefficients of T_n(z)</a>
%H A182110 Alejandro Erickson, Frank Ruskey, <a href="http://arxiv.org/abs/1304.0070">Enumerating maximal tatami mat coverings of square grids with v vertical dominoes</a>, arXiv:1304.0070 [math.CO], 2013.
%F A182110 G.f.: T_n(z) = Sum_{k>=0} T(n,k)*z^k is equal to
%F A182110 T_n(z) = 2*Sum_{i=1..floor((n-1)/2)} S_{n-i-2}(z)*S_{i-1}(z)*z^{n-i-1} + (S_{floor((n-2)/2))^2, where S_k(z) = Product_{i=1..k} (1+z^i).  Note that deg(T_n(z)) = binomial(n-1,2).
%e A182110 T_5(z) = 1 + 2*z + 3*z^2 + 6*z^3 + 4*z^4 + 2*z^5 + 2*z^6;
%e A182110 T(5,2) = 3, and the tilings are as follows:
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%e A182110 The triangle begins:
%e A182110 1
%e A182110 1,2
%e A182110 1,2,3,2
%e A182110 1,2,3,6,4,2,2
%e A182110 1,2,3,6,9,8,7,6,2,2,2
%e A182110 1,2,3,6,9,14,15,14,14,10,8,6,4,2,2,2
%e A182110 1,2,3,6,9,14,22,24,25,28,25,22,19,14,10,10,8,4,4,2,2,2
%e A182110 1,2,3,6,9,14,22,32,37,42,49,48,49,46,38,34,30,24,20,16,12,12,10,6,4,4,2,2,2
%e A182110 1,2,3,6,9,14,22,32,46,56,66,78,84,90,92,88,81,76,69,58,51,44,38,34,28,22,20,16,14,12,8,6,4,4,2,2,2
%e A182110 ...
%o A182110 (Sage)
%o A182110 @cached_function
%o A182110 def S(n,z):
%o A182110     out = 1
%o A182110     for i in [j+1 for j in range(n)]:
%o A182110         out = out*(1+z^i)
%o A182110     return out
%o A182110 T = lambda n,z: 2*sum([S(n-i-2,z)*S(i-1,z)*z^(n-i-1) for i in range(1,floor((n-1)/2)+1)]) + S(floor((n-2)/2),z)^2
%o A182110 ZP.<x> = PolynomialRing(ZZ)
%o A182110 #call T(n,x) for the g.f. T_n(x)
%Y A182110 S_k(z) is entry A053632.
%Y A182110 T_n(z) is a partition of A001787(n)/4.
%Y A182110 Tatami tilings with the same number of vertical and horizontal dimers is A182107.
%K A182110 nonn,tabf
%O A182110 0,3
%A A182110 _Alejandro Erickson_, Apr 12 2012
%E A182110 Entry revised by _N. J. A. Sloane_, Jun 06 2013