[seqfan] Re: A108081 as a combinatorial enumeration

[Joerg Arndt]

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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