[seqfan] Re: Prime signature of 1, and second signature

Meeussen Wouter (bkarnd) wouter.meeussen at vandemoortele.com
Tue Jul 3 12:41:37 CEST 2012

the reverse function, starting from {2} and choosing the smallest positive integers, produces multisets with sum = A011784 (Levine's sequence). Its length also equals A011784 with prepending "1". More regularities are cited there.


-----Original Message-----
From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of David Wilson
Sent: maandag 2 juli 2012 4:34
To: Sequence Fanatics Discussion list
Subject: [seqfan] Re: Prime signature of 1, and second signature

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
> _______________________________________________
> Seqfan Mailing list - http://list.seqfan.eu/


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