homogenous equations & partitions & catalans -> Witt vector ??

[David W. Wilson]

"Meeussen Wouter (bkarnd)" wrote:
> 
> hi,
> 
> a recent posting (again) pointed out that the partitions and the catalans
> have surprising underground connections.
> This time, it pointed me towards solutions of homogenous equations where the
> powers of the variables are given by partitions of the degree of the
> equation.
> 
> A very simple homogeneous equation can be generated by using (as exponents)
> the partitions of n, as in:
> PadRight[#,6]&/@Partitions[6]
>  {{6, 0, 0, 0, 0, 0},
>   {5, 1, 0, 0, 0, 0},
>   {4, 2, 0, 0, 0, 0},
>   {4, 1, 1, 0, 0, 0},
>   {3, 3, 0, 0, 0, 0},
>   {3, 2, 1, 0, 0, 0},
>   {3, 1, 1, 1, 0, 0},
>   {2, 2, 2, 0, 0, 0},
>   {2, 2, 1, 1, 0, 0},
>   {2, 1, 1, 1, 1, 0},
>   {1, 1, 1, 1, 1, 1}}
> 
> where {3,2,1,0,0,0} can be read as (z_1)^3 * (z_2)^2 * (z_3)^1  leads to the
> following degree 6 equation, homogenous in the z[..]
> Writing z[k] for "z_sub_k", or "zk"   (* don't be tricked in taking k as an
> argument for 'function z', no, it's just a subscript *)
> 
> (Times@@(Array[z,6]^PadRight[#,6])&)/@Partitions[6]
> 
> {z[1]^6, z[1]^5*z[2], z[1]^4*z[2]^2, z[1]^4*z[2]*z[3],
>   z[1]^3*z[2]^3, z[1]^3*z[2]^2*z[3],
>   z[1]^3*z[2]*z[3]*z[4], z[1]^2*z[2]^2*z[3]^2,
>   z[1]^2*z[2]^2*z[3]*z[4], z[1]^2*z[2]*z[3]*z[4]*z[5],
>   z[1]*z[2]*z[3]*z[4]*z[5]*z[6]}
> 
> call this "hom_6".
> Remark that the coefficients in hom_6 are all 1.
> 
> Now, look at the system of equations hom_k == catalan[k]
> This can be solved for k=1..24 to give the solutions in z[k] ::
> 
> {1, 1, 3, 8/3, 25/8, 72/25, 245/72, 772/245, 673/193, 8925/2692, 6413/1785,
> 1998/583, 44447/12210, 1402723/400023, 5165327/1402723,
>   18484746/5165327, 4037381/1087338, 248339122/68635477,
> 930138521/248339122, 3406231198/930138521, 12810761323/3406231198,
>   47306348881/12810761323, 178987624513/47306348881,
> 665627041157/178987624513}
> 
> numerator={1, 1, 3, 8, 25, 72, 245, 772, 673, 8925, 6413, 1998, 44447,
> 1402723,
> 5165327, 18484746, 4037381, 248339122, 930138521, 3406231198, 12810761323,
> 47306348881, 178987624513, 665627041157}
> 
> denominator={1, 1, 1, 3, 8, 25, 72, 245, 193, 2692, 1785, 583, 12210,
> 400023, 1402723,
> 5165327, 1087338, 68635477, 248339122, 930138521, 3406231198, 12810761323,
> 47306348881, 178987624513}
> 
> Now, this is getting curious, the only entry in EIS that starts
> "1,3,8,25,72,.." is  the strange A006177 :
> 
>                 ID Number: A006177 (Formerly M2750)
>                 Sequence:
> 1,1,3,8,25,72,245,772,2692,8925,32065,109890,400023,1402723,
> 
> 5165327,18484746,68635477,248339122,930138521,3406231198
>                 Name:      Witt vector *2!/2!.
>                 References H. Gaudier, Relevement des coefficients binomiaux
> dans les vecteurs
>                               de Witt, S\'{e}minaire Lotharingien de
> Combinatoire. Institut de
>                               Recherche Math. Avanc\'{e}e, Universit\'{e}
> Louis Pasteur,
>                               Strasbourg, Actes 16 (1988), 358/S-18, pp.
> 93-108.
>                 Keywords:  nonn
>                 Offset:    1
>                 Author(s): SP
> 
> and can you believe that the numerator /(denominator shifted right) is
> simply :
> {1, 1, 1, 1, 1, 1, 1, 1/4, 4, 1/5, 1/11, 55/9, 9, 1, 1, 1/17, 17, 1, 1, 1,
> 1, 1, 1}
> 
> weird, no ??

I suspect your numerator and denominators are just the reduced numerators
and denominators of the fractions A006177(n+1)/A006177(n).