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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. 2

%I

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

%C 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 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 Alejandro Erickson, <a href="/A182110/b182110.txt">Table of n, a(n) for n = 0..9999</a>

%H Alejandro Erickson, <a href="/A182110/a182110.txt">Table of coefficients of T_n(z)</a>

%H 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 G.f.: T_n(z) = Sum_{k>=0} T(n,k)*z^k is equal to

%F 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 T_5(z) = 1 + 2*z + 3*z^2 + 6*z^3 + 4*z^4 + 2*z^5 + 2*z^6;

%e T(5,2) = 3, and the tilings are as follows:

%e ._ _ _ _ _.

%e |_|_ _| |_|

%e |_ _| |_| |

%e |_| |_| |_|

%e | |_| |_| |

%e |_|_|_|_|_|

%e .

%e ._ _ _ _ _.

%e |_| |_ _|_|

%e | |_| |_ _|

%e |_| |_| |_|

%e | |_| |_| |

%e |_|_|_|_|_|

%e .

%e ._ _ _ _ _.

%e |_| |_| |_|

%e | |_| |_| |

%e |_| |_| |_|

%e |_|_| |_|_|

%e |_ _|_|_ _|

%e The triangle begins:

%e 1

%e 1,2

%e 1,2,3,2

%e 1,2,3,6,4,2,2

%e 1,2,3,6,9,8,7,6,2,2,2

%e 1,2,3,6,9,14,15,14,14,10,8,6,4,2,2,2

%e 1,2,3,6,9,14,22,24,25,28,25,22,19,14,10,10,8,4,4,2,2,2

%e 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 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 ...

%o (Sage)

%o @cached_function

%o def S(n,z):

%o out = 1

%o for i in [j+1 for j in range(n)]:

%o out = out*(1+z^i)

%o return out

%o 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 ZP.<x> = PolynomialRing(ZZ)

%o #call T(n,x) for the g.f. T_n(x)

%Y S_k(z) is entry A053632.

%Y T_n(z) is a partition of A001787(n)/4.

%Y Tatami tilings with the same number of vertical and horizontal dimers is A182107.

%K nonn,tabf

%O 0,3

%A _Alejandro Erickson_, Apr 12 2012

%E Entry revised by _N. J. A. Sloane_, Jun 06 2013

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Last modified April 14 00:07 EDT 2021. Contains 342941 sequences. (Running on oeis4.)