Fillomino solutions

Consider a polyomino \(S\). We want to divide \(S\) into several (may be one) smaller polyominoes. The division is called a Fillomino solution if no two polyominoes of equal size share a side. (They may touch at a point.) In other words, the division is a valid solution of a Fillomino puzzle.

Determine the number of distinct Fillomino solutions of the \(2 \times 3\) rectangle.

Try here for a harder version, or here for a much harder version.
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