Random Partitioning of a List

The list of integers from 1 to is successively partitioned at random points until the list is totally partitioned. At each iteration the maximum segment lengths of the lists are analyzed statistically over the sample of runs and the quartiles are plotted. This sequence is plotted from right to left on the abscissa. Quartiles (plotted on the ordinate) are marked in different colors. For longer runs the plot exhibits phases of behavior. Over the first few partitions the maximum segment lengths drop quickly. Then there is a stable region where the maximums drop slowly. Near the end the maximum steps down to 1 as the process is constrained to terminate at a full partition.
This is a simple view of filling in a list of spaces with names. The first few name choices leave most of the name spaces free. As the list fills up the list of names dwindles and the blocks of spaces shrink. Over a range of parameters the graphs indicate that the blocks of free spaces do not shrink uniformly, but that the behavior is different at stages in the process.

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