
EXAMPLE

Triangle begins:
n\k 1 2 3 4 5
+
1  1
2  1 1
3  1 5 8
4  2 12 95 719
5  2 31 682 20600 315107
A 3 X 3 square can be tiled by one 1 X 3 piece, two 1 X 2 pieces and two 1 X 1 pieces in the following 8 ways:
++++ ++++ ++++
         
+++ + ++++ ++++
        
++++ ++++ ++++
     
++++ ++++ ++++
.
++++ ++++ ++++
          
+ + ++ + ++ + + +++
           
++++ ++++ ++++
     
++++ ++++ ++++
.
++++ ++++
     
++++ ++++
   
++++ ++++
     
++++ ++++
This is the maximum for a 3 X 3 square, so T(3,3) = 8. There is one other set of pieces that also can tile the 3 X 3 square in 8 ways: three 1 X 2 pieces and three 1 X 1 pieces (see illustration in A361216).
The following table shows all sets of pieces that give the maximum number of tilings for 1 <= k <= n <= 5:
\ Number of pieces of size
(n,k)\ 1 X 1  1 X 2  1 X 3  2 X 2
++++
(1,1)  1  0  0  0
(2,1)  2  0  0  0
(2,1)  0  1  0  0
(2,2)  4  0  0  0
(2,2)  2  1  0  0
(2,2)  0  2  0  0
(2,2)  0  0  0  1
(3,1)  3  0  0  0
(3,1)  1  1  0  0
(3,1)  0  0  1  0
(3,2)  2  2  0  0
(3,3)  3  3  0  0
(3,3)  2  2  1  0
(4,1)  2  1  0  0
(4,2)  4  2  0  0
(4,3)  3  3  1  0
(4,4)  5  4  1  0
(5,1)  3  1  0  0
(5,1)  2  0  1  0
(5,1)  1  2  0  0
(5,2)  4  3  0  0
(5,3)  4  4  1  0
(5,4)  7  5  1  0
(5,5)  7  6  2  0
It seems that all optimal solutions for A361216 are also optimal here, but occasionally there are other optimal solutions, e.g. for n = k = 3.
