[seqfan] Re: Prime signature of 1, and second signature
David Wilson
davidwwilson at comcast.net
Mon Jul 2 04:34:14 CEST 2012
So that is interesting. For multiset A, define sig(A) as the multiset of
the number of times each element of A appears in A.
For example, for
A = {1,2,2,3,3,3,4,4,4,5,5,5,5,5}
we have
sig(A) = {1,2,3,3,5}
sig^2(A) = sig(sig(A)) = {1,1,1,2}
sig^3(A) = {1,3}
sig^4(A) = {1,1}
sig^5(A) = {2}
sig^6(A) = {1}
and higher compositions continue to yield {1}.
I am guessing that trajectories of the sig function on finite multiset
have fixed points {} and {1}, that there are no other oscillators, that
{} is reached only from the empty starting multiset, and {1} from every
nonempty starting multiset. Since for any multiset of nonnegative
integers B, it is easy enough to construct multiset A with sig(A) = B,
it would follow that there are nonempty multisets whose trajectory
requires an arbitrary number of iterations before reaching {1}.
On 7/1/2012 4:49 PM, franktaw at netscape.net wrote:
> To me, it's clear that the best definition of the prime signature is
> that it is a multiset. But then, it is equally clear to me that
> partitions should be defined as multisets, although the traditional
> definition is otherwise.
>
> Note, by the way, my definition of the signature of a partition in
> A115621. (Since I submitted this sequence, I have come to realize that
> this signature definition applies to any finite multiset, not just
> partitions.) If we regard the factorization of a number as a multiset
> of primes (another definition that seems obvious to me), the prime
> signature is then the signature of the prime factorization.
>
> Franklin T. Adams-Watters
>
>
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