being the number of items to be sorted and being the degree of the merge operationspolyphase promises significant improvement over balanced merging of coursethe distribution of initial runs was carefully chosen in the above examples in order to find out which initial distributions of runs lead to proper functioningwe work backwardstarting with the final distribution (last line in fig rewriting the tables of the two examples and rotating each row by one position with respect to the prior row yields tables and for six passes and for three and six sequencesrespectively ( (lsum ai( table perfect distribution of runs on two sequences (la (la (la ( (lsum ai( table perfect distribution of runs on five sequences from table we can deduce for the relations ( + (la ( + (la (land ( ( defining fi+ ( )we obtain for fi+ fi fi- these are the recursive rules (or recurrence relationsdefining the fibonacci numbersf each fibonacci number is the sum of its two predecessors as consequencethe numbers of initial runs on the two input sequences must be two consecutive fibonacci numbers in order to make polyphase work properly with three sequences how about the second example (table with six sequencesthe formation rules are easily derived as ( + (la ( + (la (la (la ( - ( + (la (la (la ( - ( - ( + (la (la (la ( - ( - ( - ( + (la (la (la ( - ( - ( - ( -