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A182110
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Irregular triangle read by rows: generating function counting rotationally distinct n X n tatami tilings with n monomers and exactly k vertical dimers.
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2
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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, 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, 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
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OFFSET
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0,3
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COMMENTS
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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.
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.
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LINKS
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FORMULA
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G.f.: T_n(z) = Sum_{k>=0} T(n,k)*z^k is equal to
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).
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EXAMPLE
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T_5(z) = 1 + 2*z + 3*z^2 + 6*z^3 + 4*z^4 + 2*z^5 + 2*z^6;
T(5,2) = 3, and the tilings are as follows:
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The triangle begins:
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
1,2,3,6,9,14,15,14,14,10,8,6,4,2,2,2
1,2,3,6,9,14,22,24,25,28,25,22,19,14,10,10,8,4,4,2,2,2
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
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
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PROG
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(Sage)
@cached_function
def S(n, z):
out = 1
for i in [j+1 for j in range(n)]:
out = out*(1+z^i)
return out
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
ZP.<x> = PolynomialRing(ZZ)
#call T(n, x) for the g.f. T_n(x)
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CROSSREFS
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T_n(z) is a partition of A001787(n)/4.
Tatami tilings with the same number of vertical and horizontal dimers is A182107.
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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