homogenous equations & partitions & catalans -> Witt vector ??
Meeussen Wouter (bkarnd)
wouter.meeussen at vandemoortele.com
Thu Aug 23 18:17:40 CEST 2001
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 ??
The whole thing could be viewed as yet another transform : any sequence
could be substituted into the z[k], and
the hom_k then gives the transform.
For instance, make z[..] = 1, -1, 1, -1, ... and you get the obnoxious
1, 0, -1, 1, 3, 1, -3, 0, 6, 4, -6, -3, 11, 11, -10, -11, 17, 25, -14, -27,
22, 50, -15, -55
(not in EIS, and not being missed by anyone!, it even look quite random!)
and, of course , the all_ones z[..]= 1, 1, 1, ... gives the partition
numbers =1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77..
Would a real mathematician call the "Partition-generated homogenous
equations" interesting? Do they arise naturally?
If so, then these sequences might have interest too.
If they don't, then not.
Wouter Meeussen
tel +32 (0)51 33 21 24
fax +32 (0)51 33 21 75
wouter.meeussen at vandemoortele.com
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