• Need help deriving a formula

    From Einstein@21:1/5 to All on Sun Sep 4 07:06:08 2016
    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.

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  • From Jeff Barnett@21:1/5 to Einstein on Mon Sep 5 04:46:53 2016
    Einstein wrote on 9/4/2016 7:06 AM:

    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.

    If I understand what you want: Try searching for information on the
    "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

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  • From Robert Delaney@21:1/5 to Einstein on Sun Oct 2 05:35:40 2016
    On 2016-09-04 13:06:08 +0000, Einstein said:

    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.


    Here¹s a possible approach. I¹ll illustrate with 10 units and up to 10 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.

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  • From IV@21:1/5 to Einstein on Sun Oct 2 11:29:58 2016
    Einstein <michaelhh@gmail.com> wrote:

    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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