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Sep 4, 2016, 9:06:11 AM9/4/16

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I am using a multiple variable summation formula, but I cannot seem to

render it in mathematical text.

The principle is as follows: You have five hundred thousand units, they

can be stored in any sized allotments/partitions, so in theory you

could have five hundred thousand allotments with one unit each (easy)

to fifty allotments of variable potential sizes or one allotment of

five hundred thousand units. The number of allotments is a Rand

function, and the number of units in each allotment is a Rand function

where they add up to the total units precisely.

I do realize this will create extremely high numbers, that is not an

issue, just the formula is of interest to me.

I appreciate any help.

Sep 5, 2016, 6:46:56 AM9/5/16

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"Ramanujan partition function". It is the number of collections

(possibly with duplicates) of positive integers that sum to a given

number.

One reference is https://en.wikipedia.org/wiki/Partition_(number_theory)

--

Jeff Barnett

Oct 2, 2016, 7:35:43 AM10/2/16

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allotments.

Suppose Rand chooses 10 allotments. As you said, then there can only be

1 unit in each allotment.

Suppose Rand chooses 9 allotments. Each allotment must contain at least

one unit, so put one unit in each of the 9 allotments. This leaves 10 -

9 = 1 unit to place. Use Rand to choose a number from 1 to 9 and put

that one unit there.

Suppose Rand chooses 8 allotments. Each allotment must contain at least

one unit, so put one unit in each of the 8 allotments. This leaves 10 -

8 = 2 units to place. This can be done in two ways. Either put two

units in a random choice among the 8 allotments or put one unit in a

random choice and the other unit in a second random choice of the 8

allotments. If you want equal probabilities of storage selections, that

second random choice cannot be the same as the first one.

With 7 allotments there will be 10 - 7 = 3 units to place. Here you

have to partition 3:

3 + 0 = 3

2 + 1 = 3

1 + 1 + 1 = 3

and use Rand to choose one of the partitions, then use Rand the number

of times necessary to distribute that partition of units.

One can go on, but you can already see the problem presented by a large

number of units. If you have n units and m allotments, then you must

place n - m = k units. If k is large you must devise a method to

randomly generate one partition out of the many that are possible. That

partition itself may contain many numbers, and those units must be

randomly placed as previously illustrated. That might take a lot of

computer time.

Oct 2, 2016, 1:30:00 PM10/2/16

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Einstein <mich...@gmail.com> wrote:

> The principle is as follows: You have five hundred thousand units, they

> can be stored in any sized allotments/partitions,

> ...
> The principle is as follows: You have five hundred thousand units, they

> can be stored in any sized allotments/partitions,

> issue, just the formula is of interest to me.

Look for Pólya enumeration theorem.
You will get the formula of a cycle index. If you are interested only

in

single coefficients of this cycle index, there are methods to calculate

only

one choosen coefficient.

You could make your Pólya enumeration with the specialized software

SYMMETRICA: http://www.algorithm.uni-bayreuth.de/en/research/SYMMETRICA

SYMMETRICA is also contained in SageMath: http://www.sagemath.org

If you have a formula for your cycle index coefficients, then you could

try

to simplify it with the summation package Sigma:

https://www.risc.jku.at/research/combinat/risc/software/Sigma/index.php

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