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<DIV>Let us define the signature of a (finite) multiset to be the partition composed of the multiplicities of the members of the multiset. For example, for the multiset {1,1,1,1,2,2,3,4}, the signature is {1,1,2,4}. The prime signature of a number is then the signature of its prime factor multiset.</DIV>
<DIV> </DIV>
<DIV>(I don't know if anybody has identified this concept before, possibly with a different name.)</DIV>
<DIV> </DIV>
<DIV>Since a partition is also a multiset, the signature operation can be applied repeatedly. Starting with any non-empty multiset, repeated application will eventually lead to {1}. The number of steps required to get there is:</DIV>
<DIV> </DIV>
<DIV>0,1,2,1,3,2,1,3,2,4,2,1,3,3,4,4,4,2,1,3,3,2,4,3,2,4,3,4,2,1,3,3,</DIV>
<DIV>3,4,3,4,4,4,5,4,4,4,4,2,1,3,3,3,2,4,3,3,4,4,4,5,3,5,2,4,5,4,4,4,</DIV>
<DIV>4,2,1,3,3,3,3,4,3,3,4,4,3,2,4,5,5,5,5,4,4,5,4,5,4,4,5,3,4,4,4,2</DIV>
<DIV> </DIV>
<DIV>for partitions in Abramowitz and Stegun order. This is not (yet) in the OEIS.</DIV>
<DIV> </DIV>
<DIV>The smallest partition requiring n steps is:</DIV>
<DIV>1</DIV>
<DIV>2</DIV>
<DIV>1,1</DIV>
<DIV>1,2</DIV>
<DIV>1,1,2</DIV>
<DIV>1,1,2,3</DIV>
<DIV>1,1,1,2,2,3,4</DIV>
<DIV>1,1,1,1,2,2,2,3,3,4,4,5,6,7</DIV>
<DIV>...</DIV>
<DIV> </DIV>
<DIV>which is A012257, with an initial 1,2. ("Smallest" partition is in general an ambiguous concept, but there in this case, any other partition requiring the same number of steps has a larger total.)</DIV>
<DIV> </DIV>
<DIV>Another application is A048996, which counts the number of compositions associated with a partition. (A072811 is the Mathematica version of this sequence, but they do not cross-reference each other, and the description in A048996 does not describe them well.) A048996(P)=Multinomial(Signature(P)), where Multinomial is A036038 (in A&S order, no Mma equivalent is present).</DIV>
<DIV> </DIV>
<DIV>One should probably also add:</DIV>
<DIV>1</DIV>
<DIV>1, 2,</DIV>
<DIV>1, 1,1, 3,</DIV>
<DIV>1, 1,1, 2, 1,2, 4, (or 1, 1,1, 2, 2,1, 4,)</DIV>
<DIV>...</DIV>
<DIV>The sequence of signatures of partitions, with an associated sequence giving their lengths:</DIV>
<DIV>1, 1,1, 1,2,1, 1,2,1,2,1, ...</DIV>
<DIV> </DIV>
<DIV>The number of items on each line of the former sequence is A000070, as indicated by the comment line there beginning "Also the total number of all different ...".</DIV>
<DIV> </DIV>
<DIV>Franklin T. Adams-Watters<BR>16 W. Michigan Ave.<BR>Palatine, IL 60067<BR>847-776-7645</DIV></DIV></DIV>
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