Subject: Asking for referencies
Date: Sat, 1 Mar 2003 17:28:34 +0100
From: "Boccellari" <boccellari@inwind.it>
To: "Don Davis" <dmd1@Lehigh.EDU>

Dear Professor Davies,

I would like to submit to your kind attention this message.
If you think it is the case, please put it in your mailing list.

Recently I thought to approach the study of partition of vectors with non
negative integer entries by employing algebraic topology.
I don't know if this way was already examined or if it was shown to be
unfruitful, with this mail I would like to ask for informations in this
sense.
In the following I will describe with more detail what I thought to do.


Take a vector v of finite lenght n and with non-negative integer entries.
An r partition of v is a family of vectors (w_1, ..., w_r)  each one of
finite lenght n and with non-negative integer entries such that w_1 + ... +
w_r = v.
A partition of v will be said non degenerate if it does't contain the null
vector.

A CW complex will be built which has one k cell for each k+2 non degenerate
partition of v.

For each non-negative integer k take the set V_k made by all k+2 partitions
(w_1, ..., w_{k+2})  of  v such that bot w_1 and w_{k+2} are not the null
vector.

Define the following degeneracies and facies operators d_i : V_k ----->
V_{k-1}, and s_i : V_k -----> V_{k+1}  with 0 =< i =< k as:

d_i (w_1, ..., w_{k+2}) = (w_1, ..., w_i, w_{i+1} + w_{i+2}, w_{i+3}, ...,
w_{k+2})

s_i (w_1, ..., w_{k+2}) = (w_1, ..., w_{i+1} , 0, w_{i+2}, ..., w_{k+2})

This defines a simplicial complex with one k cell for each k+2 non
degenerate partition of v.

I would like to study this complex to obtain information on the partitioning
of v, in particular I am interested in the case when v is a 2n dimentional
vector with all entries given by 2^n.

Every help will be very welcome.

Thank you for your kind attention.

Yours faithfully.

Tommaso Boccellari