Re: Equal sums of parts

From: David Eppstein (eppstein_at_ics.uci.edu)
Date: 06/07/04 

Date: Mon, 07 Jun 2004 11:18:51 -0700

In article <5227cf44.0406070918.3a49e39d@posting.google.com>,
 zellerg@wanadoo.fr (georgesZ) wrote:

> For n>=3.
> Set m(n) =the least integer m>=3 s.t.,
> for any increasing sequence (bi)_(1<=i <=m) in [[1, n]]
> (1) there are 2 distincts parts I and J of [[1, m]] s.t.
> Sum(bi, i in I)=Sum(bi, i in J)?
>
> Note that we can replace (1) by
> (1') there are 2 non -empty disjoint parts I and J of [[1, m]] s.t.
> Sum(bi, i in I)=Sum(bi, i in J).
>
> We check m(3)=3,
> m(i)=4, for i=4..6,
> I am almost sure that
> m(i)=5, for i=7..13,
> m(i)=6, for i=14..23.
>
> Take {a, a-1,a-2, a-4,a-8,a-16,a-32}, we can't have (1') for parts I
> and J with the same cardinal and, if a+(a-1)+(a-2) < (a-4)+(a-8)
> +(a-16)+(a-32),
> id est a>57, we can't have (1').
>
> We get that m(i)>=8 if i>=58. (Note that 2^6=64).
>
> Any comments?

The following look relevant:

http://www.math.princeton.edu/~matdevos/open/subsumdist.html
http://www.ams.org/proc/1996-124-12/S0002-9939-96-03653-2/home.html
http://portal.acm.org/citation.cfm?id=763391
http://www.combinatorics.org/Volume_5/PDF/v5i1r3.pdf
http://portal.acm.org/citation.cfm?id=7941

and no doubt many more references... 

-- 
David Eppstein                      http://www.ics.uci.edu/~eppstein/
Univ. of California, Irvine, School of Information & Computer Science

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