1. Purpose of arithinva@trinitas.mju.ac.kr     [ 한글 설명 ]
Computation of the value of all the coefficients, called arithmetic invariants, of the expansion of the product of Gaussian polynomials, 

(*)        (1+t)^e(1)*(1+t+t^2)^e(2)*(1+t+t^2+t^3)^e(3)***(1+t+t^2+...+t^(n-1))^e(n)

and immediate return to the e-mail sender of the computed results.

2. Introduction
When the number of Gaussian polynomials 1+t+t^2+...+t^(n-1) in such a product in Eq (*) above is increased, it quickly becomes very difficult to compute all the coefficients of the product. Recently, Prof. Sun T. Soh discovered new recursive formulas, absolute and (quasi-) recursive, for the (graded) partition function p_M(n) over a multiset M, whose successive application provides us with an algorithm for computing them efficiently.

Soh's algorithm for q-series for (*)

We use his algorithm for all the coefficients of such a product in this section of our InetCompu. Time efficiency of his algorithm is O(m^2), where m is the total number of coefficients successively from the first one, 1, of the following product (i.e., when all the e(i) =1 in Eq (*) above):

(**)     (1+t)*(1+t+t^2)*(1+t+t^2+t^3)***(1+t+t^2+...+t^n).  

3. Contributor: The program we are using was developed in Reduce commands by Prof. Sun T. Soh, Dept of Math, Myong Ji Univ., Rep. of Korea. 

4. Note
For a trouble-free handling of your e-mail, we strongly recommend you to use MicroSoft Outlook Express, New mail > Alt+O > Alt+X (with No Encryption),  to send out an e-mail to arithinva@trinitas.mju.ac.kr   .   

5. How to do: When the polynomial product under consideration is 

(***)     (1+t+t^2)^3*(1+t+t^2+t^3+t^4+t^5)^4*(1+t+t^2+t^3+t^4+t^5+t^6+t^7+t^8)^5 

send an e-mail with plain text style (for instance, in the case of MicroSoft Outlook Express, New mail > Alt+o > Alt+x (with No Encryption)) to arithinva@trinitas.mju.ac.kr whose main body should consist of 

     input: 
         multiset:={3,3,3,6,6,6,6,9,9,9,9,9}$
         f:=2$
     end input:  

where 
(i) the needed multiset is figured by the rule
          1+t+t^2+t^3+...t^n-1    ->  n
for each Gaussian polynomial in the product, 
(ii) the first line, multiset:={3,3,3,6,6,6,6,9,9,9,9,9}$ , can be replaced with, 
multiset:={{3,3},{6,4},{9,5}}$ ,   
(iii) a diffenent value other than 2 may be chosen for efficiency control parameter f >1 by the sender.

Moreover, if one puts, for instance, 

     input:
         n:=10$
     end input:

in his mail, then all the coefficients of the expansion of the following polynomial are computed:

          (1+t)*(1+t+t^2)***(1+t+t^2+...+t^(10-1))

In this case, the optimal value for f > 1 is automatically determined by the formula in the following Note.

Note: For a complicated or large multiset, a different value other than 2 may be chosen for f > 1 using the formula: 

f:=[exp(sqrt( ln(2)*ln(N) ))]$ 

where [m] means the largest number <= m  and N is the cardinality of the multiset under consideration. (It is usually most efficient when f:=2$, unless multiset is quite large).  

Then upon the arrival of the e-mail, all the coefficients of the product in (*) above are automatically computed and sent back to the e-mail sender immediately.

[Reminder] When the requested job is computationally not very complicated, it should be quite the case that you will receive the result within a few Minutes.  But, our response consisting of the computed results can not be delivered to the sender properly, if either there is a spelling mistake in sender's e-mail address or sender's mail box is already filled with too many of other e-mails. Thus, if you do not receive the results although you have waited for some time, then please check your mail account to correct the above trouble-causing problems.  After that, try again according to the procedure described in Section 5 above.