[seq-fan] Re: chess tournaments:
Svante Linusson
linusson at matematik.su.se
Thu Apr 29 09:41:36 CEST 1999
I wrote:
> By changing <=2 in the description of A007747 to <=p, you should get the
> same sequences.
I'm very sorry for sounding so vague. I ment that one _will_ get the same
sequences.
A small proof is needed to show that every partition (a_1,...,a_n)
majorised by (pn-p,pn-2p,...,p,0) of length at most n is indeed a score
sequence. It is not very complicated.
Show e.g. that one can always change
(...,a_k,a_{k+1},...) to (...,a_k-1,a_{k+1}+1,...)
as long as a_k-1>=a_{k+1}+1
With majorised I mean,
f_p(n,k):=pn-p+...+(pn-pk+p)-(a_1+a_2+...+a_k) >= 0 for all k.
Now, the sequence 0=f_p(n,0),f_p(n,1),f_p(n,2),...,f_p(n,n)=0
is of the required type in A007747, but with <=2 replaced with <=p,
which has now been computed by David.
This gives a bijection.
/Svante
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