a variation of this method) and so determine the number of free polyominoes by Burnside's lemma. Transfer-matrix method The most modern algorithm for enumerating the fixed polyominoes was discovered by Iwan Jensen. An improvement on Andrew Conway's method, it is exponentially faster than the previous methods (however, its running time is still exponential in n). Both Conway's and Jensen's versions of the transfer-matrix method involve counting the number of polyominoes that have a certain width. Computing the number for all widths gives the total number of polyominoes. The basic idea behind the method is that possible beginning rows are considered, and then to determine the minimum number of squares needed to complete the polyomino of the given width. Combined with the use of generating functions, this technique is able to count many polyominoes at once, thus enabling it to run many times faster than methods that have to generate every polyomino. Although it has excellent running time, the tradeoff is that this algorithm uses exponential amounts of memory (many gigabytes of memory are needed for n above 50), is much harder to program than the other methods, and can't currently be used to count free polyominoes. Asymptotic growth of the number of polyominoes Fixed polyominoes Theoretical arguments and numerical calculations support the estimate for the number of fixed polyominoes of size n where λ = 4.0626 and c = 0.3169. However, this result is not proven and the values of λ and c are only estimates. The known theoretical results are not nearly as specific as this estimate. It has been proven that exists. In other words, An grows exponentially. The best known lower bound for λ, found in 2016, is 4.00253. The best known upper bound, not improved since the 1970s, is . To establish a lower bound, a simple but highly effective method is concatenation of polyominoes. Define the upper-right square to be the rightmost square in the uppermost row of the polyomino. Define the bottom-left square similarly. Then, the upper-right square of any polyomino of size n can be attached to the bottom-left square of any polyomino of size m to produce a unique (n+m)-omino. This proves . Using this equation, one can show for all n. Refinements of this procedure combined with data for An produce the lower bound given above. The upper bound is attained by generalizing the inductive method of enumerating polyominoes. Instead of