[seqfan] Re: A108081 as a combinatorial enumeration
Joerg Arndt
arndt at jjj.de
Tue Oct 27 10:03:44 CET 2015
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
>
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