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%I #20 Dec 15 2016 02:33:06
%S 1,2,2,2,4,4,2,2,4,6,8,6,4,2,2,4,6,12,12,8,12,12,6,4,2,2,4,6,12,18,20,
%T 18,16,16,18,20,18,12,6,4,2,2,4,6,12,18,28,34,32,32,28,28,28,28,32,32,
%U 34,28,18,12,6,4,2,2,4,6,12,18,28,44,52,54,60,58,52,54,48,40,48,54,52,58,60,54,52,44,28,18,12,6,4,2
%N Number of tatami tilings of an n X n square with exactly k horizontal dimers and n monomers (no restriction on the number of vertical dimers).
%C A tatami tiling consists of dimers (1 X 2) and monomers (1 X 1) where no four meet at a point.
%C The (n, r) entry contains the number of tatami tilings of an n X n square with exactly r horizontal dimers and n monomers and arbitrarily many vertical dimers(n: row number, r: column number).
%C Rows are of length 1 + 1*0/2, 1 + 2*1/2, 1 + 3*2/2, 1 + 4*3/2, ... and in the range [1, 8].
%C Columns are counted from 0.
%C Here is the first three rows of the sequence:
%C 1
%C 2 2
%C 2 4 4 2
%C The sum of all entries in the n-th row is n*2^(n-1) [1].
%C Note that numbers of horizontal dimers and vertical dimers are interchangeable.
%H 1. A. Erickson, F. Ruskey, M. Schurch and J. Woodcock, <a href="http://webhome.cs.uvic.ca/~ruskey/Publications/Tatami/TatamiMonomer.html">Auspicious Tatami Mat Arrangements</a>, The 16th Annual International Computing and Combinatorics Conference (COCOON 2010), July 19-21, Nha Trang, Vietnam. LNCS 6196 (2010) 288-297.
%H 2. A. Erickson, F. Ruskey, M. Schurch and J. Woodcock, <a href="http://www.combinatorics.org/Volume_18/Abstracts/v18i1p109.html">Monomer-Dimer Tatami Tilings of Rectangular Regions</a>, Electronic Journal of Combinatorics, 18(1) (2011) P109, 24 pages.
%e Here are the tatami tilings of the 3 X 3 square with three monomers:
%e No horizontal dimer:
%e _ _ _ _ _ _
%e |_| |_| | |_| |
%e | |_| | |_| |_|
%e |_|_|_| |_|_|_|
%e One horizontal dimer:
%e _ _ _ _ _ _ _ _ _ _ _ _
%e |_ _| | |_| |_| |_| |_| | |_ _|
%e |_| |_| |_|_| | | |_|_| |_| |_|
%e |_|_|_| |_ _|_| |_|_ _| |_|_|_|
%e Two horizontal dimers:
%e _ _ _ _ _ _ _ _ _ _ _ _
%e |_ _|_| |_|_ _| |_|_| | | |_|_|
%e | |_ _| |_ _| | |_ _|_| |_|_ _|
%e |_|_|_| |_|_|_| |_|_ _| |_ _|_|
%e Three horizontal dimers:
%e _ _ _ _ _ _
%e |_ _|_| |_|_ _|
%e |_|_ _| |_ _|_|
%e |_ _|_| |_|_ _|
%K tabf,nonn
%O 1,2
%A _Frank Ruskey_ and Yuji Yamauchi (eugene.uti(AT)gmail.com), Jul 14 2011