OFFSET
0,3
COMMENTS
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.
LINKS
Alejandro Erickson, Table of n, a(n) for n = 0..9999
Alejandro Erickson, Table of coefficients of T_n(z)
Alejandro Erickson, Frank Ruskey, Enumerating maximal tatami mat coverings of square grids with v vertical dominoes, arXiv:1304.0070 [math.CO], 2013.
FORMULA
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).
EXAMPLE
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:
._ _ _ _ _.
|_|_ _| |_|
|_ _| |_| |
|_| |_| |_|
| |_| |_| |
|_|_|_|_|_|
.
._ _ _ _ _.
|_| |_ _|_|
| |_| |_ _|
|_| |_| |_|
| |_| |_| |
|_|_|_|_|_|
.
._ _ _ _ _.
|_| |_| |_|
| |_| |_| |
|_| |_| |_|
|_|_| |_|_|
|_ _|_|_ _|
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
...
PROG
(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)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Alejandro Erickson, Apr 12 2012
EXTENSIONS
Entry revised by N. J. A. Sloane, Jun 06 2013
STATUS
approved