substituting fi for (iyields + fi fi- - fi- fi- fi for for these numbers are the fibonacci numbers of order in generalthe fibonacci numbers of order are defined as followsf + (pfi(pf - (pfi- (pf ( ( for for < note that the ordinary fibonacci numbers are those of order we have now seen that the initial numbers of runs for perfect polyphase sort with sequences are the sums of any - - (see table consecutive fibonacci numbers of order - table numbers of runs allowing for perfect distribution this apparently implies that this method is only applicable to inputs whose number of runs is the sum of - such fibonacci sums the important question thus ariseswhat is to be done when the number of initial runs is not such an ideal sumthe answer is simple (and typical for such situations)we simulate the existence of hypothetical empty runssuch that the sum of real and hypothetical runs is perfect sum the empty runs are called dummy runs but this is not really satisfactory answer because it immediately raises the further and more difficult questionhow do we recognize dummy runs during mergingbefore answering this question we must first investigate the prior problem of initial run distribution and decide upon rule for the distribution of actual and dummy runs onto the - tapes in order to find an appropriate rule for distributionhoweverwe must know how actual and dummy runs are merged clearlythe selection of dummy run from sequence imeans precisely that sequence is ignored during this merge resulting in merge from fewer than - sources merging of dummy run from all - sources implies no actual merge operationbut instead the recording of the resulting dummy run on the output sequence from this we conclude that dummy runs should be distributed to the -