Setup:A block is any rectangular array of squares, specified by its dimensions \$(w,h)\$. A grid is any finite ordered list of blocks. For example, \$\lambda = ((3,2),(3,1),(1,2))\$ defines a grid.
Let \$\lambda\$ and \$\mu\$ be two grids with equal area.
A tiling of \$\lambda\$ by \$\mu\$ is any rearrangement of the squares of \$\mu\$ into the shape of \$\lambda\$ satisfying two properties:
- horizontally adjacent squares of \$\mu\$ remain horizontally adjacent in \$\lambda\$, and
- vertically adjacent squares of \$\lambda\$ come from vertically adjacent squares of \$\mu\$.
In other words, while rearranging one is allowed to make horizontal cuts to the blocks of \$\mu\$ but not vertical cuts, and one is allowed to place blocks into \$\lambda\$ side-by-side, but not on top of one another.
Two tilings of \$\lambda\$ by \$\mu\$ are considered equivalent if they can be rearranged into one another by any combination of either permuting squares within a column or reordering the columns of a block.
Problem: Write a function \$T(\mu,\lambda)\$ which computes the number of inequivalent tilings of a grid \$\lambda\$ by another grid \$\mu\$ of equal area.
Specifications: You may use any data type you would like to specify a grid.
Examples:
- The grid \$\lambda=((1,2),(1,2),(1,1),(2,1))\$ admits a tiling by \$\mu=((1,3),(1,2),(2,1))\$ given by
There is exactly one other inequivalent tiling given by
(Since the two differently colored columns of height \$2\$ are not part of the same block, they cannot be permuted.)
- The three displayed tilings of \$\lambda=((3,1))\$ by \$\mu=((1,2),(1,1))\$ are equivalent:
Let \$\lambda\$ be an arbitrary grid of area \$n\$ and let \$\lambda[(w,h)]\$ denote the number of blocks of \$\lambda\$ of dimension \$w \times h\$. Then \$T(\lambda,\lambda) = \prod_{w,h\geq 1} \lambda[(w,h)]!\$ and \$T(\lambda,((n,1))) = 1\$.
The matrix of values of \$T(\mu,\lambda)\$ for all pairs of grids of area \$3\$ (row is \$\mu\$, column is \$\lambda\$):
| ((1,3)) | ((1,2),(1,1)) | ((1,1),(1,1),(1,1)) | ((2,1),(1,1)) | ((3,1)) | |
|---|---|---|---|---|---|
| ((1,3)) | 1 | 1 | 1 | 1 | 1 |
| ((1,2),(1,1)) | 0 | 1 | 3 | 2 | 1 |
| ((1,1),(1,1),(1,1)) | 0 | 0 | 6 | 3 | 1 |
| ((2,1),(1,1)) | 0 | 0 | 0 | 1 | 1 |
| ((3,1)) | 0 | 0 | 0 | 0 | 1 |