A cover of the tetrominos with each integer between 1 and 5 being the count of a tetromino:
It is interesting to consider the relative frequencies of each polyomino within a pattern. Our definition of a cover requires that each polyomino occur at least once, but we might want to look for other qualities in the set of polyomino frequencies. Erich Friedman suggests this generalization: an n-cover is defined as one where each polyomino occurs at least n times.
A Succinct cover is one where each polyomino occurs exactly once. Note that it may not be possible to create a succinct cover of a set that is in one unbroken piece. When making a succinct cover of the pentominos, for example, the x pentomino, because of its symmetry, cannot be added to without creating two of the same pentomino. Erich Friedman has suggested the term n-succinct cover for a cover where each polyomino occurs exactly n times. I like this concept, but I'm less sure about the term; I chose "succinct" because metaphorically, a succinct cover says everything it needs to say exactly once, and no more. If a 4-succinct cover says things exactly 4 times, is it really being succinct? However, I can't think of a better term, and metaphors are made to be broken anyway.
Polyominoes that have fewer symmetries, or are more compact, tend generally to occur in higher frequencies in a pattern than the others. This is illustrated in the problem of finding a contiguous cover of the tetrominoes where the frequency of each tetromino's occurance is an unique integer between 1 and 5 inclusive. There are many different solutions, the one shown, which Erich Friedman found, is probably the smallest.