[seqfan] Re: A108081 as a combinatorial enumeration

[Neil Sloane]

I added Li-yao's comments (slightly edited) to A108081. This is a very nice
problem!

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com


On Tue, Oct 27, 2015 at 5:03 AM, Joerg Arndt <arndt at jjj.de> wrote:

> I cannot offer a proof.
> Please add a comment to A108081!
>
> Best regards,   jj
>
>
> * Li-yao Xia <li-yao.xia at ens.fr> [Oct 23. 2015 08:21]:
> > Hello seqfans,
> >
> > Consider the smallest set X of finite sequences of integer (words), such
> > that
> > - 0 belongs to it;
> > - if a and b are two words in X, let L(a) be the word obtained by
> reversing
> > a and subtracting one to every element, and R(b) be the word obtained by
> > reversing b and adding one to every element, then the concatenations
> L(a).b
> > and a.R(b) belong to X.
> >
> > Examples of L and R values:
> > L(10,30,20) = 19, 29, 9
> > R(10,30,20) = 21, 31, 11
> >
> > Words of X of lengths 1, 2, 3:
> >
> >  0
> >
> >  0,  1
> > -1,  0
> >
> > -1,  0,  1 = L(0), 0, 1 = -1, 0, R(0)
> >  0,  2,  1 = 0, R(0, 1)
> >  1, -1,  0 = L(0), -1, 0
> >  0,  1,  0 = 0, R(-1, 0)
> >  0, -1,  0 = L(0, 1), 0
> >  0,  1,  1 = 0, 1, R(0)
> > -1, -2,  0 = L(-1, 0), 0
> >
> > The sequence of words of X of length n=1,... starts:
> > 1,2,7,25,92,344,1300,4950,18955,72905,281403,1089343
> >
> > that matches (up to a shift of indices) A108081(n) = sum(i = 0 .. n, C(2
> * n
> > - i, n + i)) but I am at a loss as to how to prove or disprove the
> validity
> > of this formula.
> >
> > The operations L(a).b and a.R(b) in the definition of X come up in the
> study
> > of something called pregroup types, somewhere in the intersection of
> > linguistics and category theory--I don't know any more than that about
> their
> > origins. The question of the enumeration of X seems to be only
> > recreationally motivated, but I found the shortness of the conjectured
> > formula quite odd.
> >
> > Any ideas?
> >
> > Li-yao
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
>
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