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%I #7 Mar 11 2023 08:38:06
%S 1,1,1,1,5,8,2,12,95,719,2,31,682,20600,315107
%N Triangle read by rows: T(n,k) is the maximum number of ways in which a set of integer-sided rectangular pieces can tile an n X k rectangle, up to rotations and reflections.
%e Triangle begins:
%e n\k| 1 2 3 4 5
%e ---+------------------------
%e 1 | 1
%e 2 | 1 1
%e 3 | 1 5 8
%e 4 | 2 12 95 719
%e 5 | 2 31 682 20600 315107
%e 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:
%e +---+---+---+ +---+---+---+ +---+---+---+
%e | | | | | | | | | |
%e +---+---+ + +---+---+---+ +---+---+---+
%e | | | | | | | | |
%e +---+---+---+ +---+---+---+ +---+---+---+
%e | | | | | |
%e +---+---+---+ +---+---+---+ +---+---+---+
%e .
%e +---+---+---+ +---+---+---+ +---+---+---+
%e | | | | | | | | | | |
%e + + +---+ + +---+ + + +---+---+
%e | | | | | | | | | | | |
%e +---+---+---+ +---+---+---+ +---+---+---+
%e | | | | | |
%e +---+---+---+ +---+---+---+ +---+---+---+
%e .
%e +---+---+---+ +---+---+---+
%e | | | | | |
%e +---+---+---+ +---+---+---+
%e | | | |
%e +---+---+---+ +---+---+---+
%e | | | | | |
%e +---+---+---+ +---+---+---+
%e 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).
%e The following table shows all sets of pieces that give the maximum number of tilings for 1 <= k <= n <= 5:
%e \ Number of pieces of size
%e (n,k)\ 1 X 1 | 1 X 2 | 1 X 3 | 2 X 2
%e ------+-------+-------+-------+------
%e (1,1) | 1 | 0 | 0 | 0
%e (2,1) | 2 | 0 | 0 | 0
%e (2,1) | 0 | 1 | 0 | 0
%e (2,2) | 4 | 0 | 0 | 0
%e (2,2) | 2 | 1 | 0 | 0
%e (2,2) | 0 | 2 | 0 | 0
%e (2,2) | 0 | 0 | 0 | 1
%e (3,1) | 3 | 0 | 0 | 0
%e (3,1) | 1 | 1 | 0 | 0
%e (3,1) | 0 | 0 | 1 | 0
%e (3,2) | 2 | 2 | 0 | 0
%e (3,3) | 3 | 3 | 0 | 0
%e (3,3) | 2 | 2 | 1 | 0
%e (4,1) | 2 | 1 | 0 | 0
%e (4,2) | 4 | 2 | 0 | 0
%e (4,3) | 3 | 3 | 1 | 0
%e (4,4) | 5 | 4 | 1 | 0
%e (5,1) | 3 | 1 | 0 | 0
%e (5,1) | 2 | 0 | 1 | 0
%e (5,1) | 1 | 2 | 0 | 0
%e (5,2) | 4 | 3 | 0 | 0
%e (5,3) | 4 | 4 | 1 | 0
%e (5,4) | 7 | 5 | 1 | 0
%e (5,5) | 7 | 6 | 2 | 0
%e 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.
%Y Main diagonal: A361222.
%Y Columns: A361223 (k = 1), A361224 (k = 2), A361225 (k = 3).
%Y Cf. A360629, A361216.
%K nonn,tabl,more
%O 1,5
%A _Pontus von Brömssen_, Mar 05 2023