A362258 Triangle read by rows: T(n,k) is the maximum number of ways in which a set of integer-sided squares can tile an n X k rectangle, up to rotations and reflections, 0 <= k <= n.
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 4, 1, 1, 2, 4, 13, 20, 1, 1, 4, 8, 33, 125, 277, 1, 1, 6, 12, 72, 403, 2505, 7855, 1, 1, 9, 22, 204, 1438, 12069, 101587, 487662
Offset: 0
Examples
Triangle begins:
n\k| 0 1 2 3 4 5 6 7 8
---+----------------------------------------
0 | 1
1 | 1 1
2 | 1 1 1
3 | 1 1 1 1
4 | 1 1 2 2 4
5 | 1 1 2 4 13 20
6 | 1 1 4 8 33 125 277
7 | 1 1 6 12 72 403 2505 7855
8 | 1 1 9 22 204 1438 12069 101587 487662
See A362142 for an illustration of T(5,4) = 13.
The following table shows which sets of squares can tile the n X k rectangle in T(n,k) ways. A list x_1, ..., x_j represents a set of x_1 squares of side 1, ..., x_j squares of side j. When there are multiple solutions they are shown on separate lines. For (n,k) = (4,3), for example, the maximum number T(4,3) = 2 of tilings is obtained both for the set of 8 squares of side 1 and 1 square of side 2, and for the set of 4 squares of side 1 and 2 squares of side 2.
n\k| 1 2 3 4 5 6 7 8
---+------------------------------------------------
1 | 1
2 | 2 4
| 0,1
3 | 3 6 9
| 2,1 5,1
| 0,0,1
4 | 4 4,1 8,1 8,2
| 4,2
5 | 5 6,1 7,2 12,2 13,3
| 2,2
6 | 6 4,2 10,2 12,3 14,4 20,4
7 | 7 6,2 13,2 12,4 19,4 22,5 25,6
8 | 8 8,2 12,3 16,4 20,5 24,6 23,6,1 27,7,1