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  1. - Top - End - #1
    Barbarian in the Playground
     
    NecromancerGuy

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    Default Math Notation Question (Combinatorics)

    Is there some notation for the set of unique permutations of n elements from a set?

    For example:
    Let S be the set of [a1, a2, a3] (from the set of complex numbers, not necessarily unique)

    Then
    n = outcome
    1 a1+a2+a3
    2 a1a2 + a1a3 + a2a3
    3 a1a2a3
    In other words, for n=1, you get the sum of all elements of the set, n=2 produces the sum of all pair-wise products from the set, n=3 produces the sum of all 3-way products from the set, and in general F(S,n) is the sum of all n-element products from the set S.

    I feel like I'm vaguely remembering there being some form of notation for this, but I can't for the life of me remember what it is and my google-fu is failing.

    Edit: It's not quite the power set.
    If P = P(S) for some set S
    and P_n is the collection of all sets in P that have n elements (P_n[1], P_n[2], ... P_n[k]).
    then I'm looking for the sum from i=1 to k of the product of the elements of P_n[i]
    Last edited by Astral Avenger; 2021-01-28 at 11:48 PM.
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    OldWizardGuy

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    Default Re: Math Notation Question (Combinatorics)

    Are you think of the Power Set?

    The Power Set of S is the set of all subsets of S, including S and the empty set.

    Written P(S) or ℙ(S)

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    Ettin in the Playground
     
    Rockphed's Avatar

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    Default Re: Math Notation Question (Combinatorics)

    I remember seeing things like C(5 2) for the combinations of 5 things taken 2 at a time and P(7 3) for the permutations of 7 things taken 3 at a time, but those are normally shorthand for how many things are in the set, not the set itself.
    Quote Originally Posted by Wardog View Post
    Rockphed said it well.
    Quote Originally Posted by Sam Starfall
    When your pants are full of crickets, you don't need mnemonics.
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    PaladinGuy

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    Default Re: Math Notation Question (Combinatorics)

    Quote Originally Posted by Rockphed View Post
    I remember seeing things like C(5 2) for the combinations of 5 things taken 2 at a time and P(7 3) for the permutations of 7 things taken 3 at a time, but those are normally shorthand for how many things are in the set, not the set itself.
    The calculator button has nCr and nPr for that.
    I've also seen Cn for cyclic permutations with Cn*Cm then being more interesting ones.

    I suspect it's something that if you need it you (or your textbook) define the notation in the way that matches what you need (and doesn't clash with whatever else you are doing).

    If you need it, it will be for something that you'll be using on rare occasions but when you do you will be using it a lot that the cost of saying "Let Squiggle be the combinations of the set..." is relatively small compared to forcing a standard notation (that might not match your use case) and taking up a symbol.
    Last edited by jayem; 2021-01-30 at 05:10 PM.

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    Default Re: Math Notation Question (Combinatorics)

    If you're looking for the actual permutations, and not just counting the number of them, then what you're looking at is subgroups or subsets of the permutation group over n elements.

    I'm not sure what they're all called, but the second one on your list is all the permutations that are a single transposition.

    Hopefully this will help you find the term you're looking for. I'm away from my textbooks at the moment, so I'm not sure if a name for this is given in those.
    Quote Originally Posted by Harnel View Post
    where is the atropal? and does it have a listed LA?

  6. - Top - End - #6
    Titan in the Playground
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    Default Re: Math Notation Question (Combinatorics)

    First of all, you're not talking about permutations, which are ways that things can be ordered. The permutations of a1, a2, and a3 are:

    a1, a2, a3
    a1, a3, a2
    a2, a1, a3
    a2, a3, a1
    a3, a1, a2
    a3, a2, a1

    I'm not sure what you are talking about. If you could tell us what you're using it for, we might be more able to help you, but at present, I have no idea.

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    Troll in the Playground
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    Default Re: Math Notation Question (Combinatorics)

    Quote Originally Posted by Sermil View Post
    Are you think of the Power Set?

    The Power Set of S is the set of all subsets of S, including S and the empty set.

    Written P(S) or ℙ(S)
    Not exactly a power set, but the structure it builds does organize things well, if you sort its elements by well... number of elements inside (strictly speaking cardinality). So from the power set P(S) you take the n-element sets An and perform the following operation:
    SumAn in P(S) Productx in Anx

    You can even define the set of all n-element subsets of S as
    Pn(S) = {A: A in P(S) and card(A)=n}

    With such a notation you get for example
    SumA in Pn(S) Productx in A x
    In a war it doesn't matter who's right, only who's left.

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    Barbarian in the Playground
     
    NecromancerGuy

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    Default Re: Math Notation Question (Combinatorics)

    Thanks for the responses everyone. It's for the coefficients of the polynomial form of det(sI-A) = (s-L1)(s-L2)(s-L3)...(s-Ln) = A0 + A1s + A2s^2 + ... + An-1s^(n-1) + s^n.
    The eigenvalues of (sI-A), (L1,..., Ln) are known in this specific case, but for finding an associated transfer function we need the polynomial coefficients, not the factored form.

    For the nxn case, there are n eigenvalue terms for the factored form (not necessarily unique). There is some relation to combinatorics, since I'm looking for all combinations of x elements from the set of eigenvalues without replacement for each term, but probably would have been more useful to mention the exact problem in my initial post.

    I threw together some python code to generate them (see spoiler below).

    Spoiler: Python 3
    Show

    Code:
    # Eigenterms for expanded characteristic polynomial of (sI-A)
    
    eigvals = []
    for n in range(1,6):eigvals.append('x'+str(n))
    print(eigvals)
    # eigvals = ['x1', 'x2', 'x3', 'x4', 'x5']
    
    
    def coef( n, eig ):
        if(n < 0 or type(n) != int): raise ValueError("N is not positive integer")
        if(len(eig)<1): raise ValueError("eig is empty")
        if(n ==0): return [1]
        if(n ==1): return eig
        if(n == len(eig)):
            out = ''
            for e in eig: out += e
            return [out]
        if(n > 1 and n < len(eig) ):
            out = []
            for k in range(len(eig)-n+1):
                lead_term = eig[k]
                for sub_term in coef(n-1,eig[k+1:]): out.append(lead_term+sub_term)
            return out
    
    
    print("coef(1, eigvals)", coef(1, eigvals))
    print("coef(2, eigvals)", coef(2, eigvals))
    print("coef(3, eigvals)", coef(3, eigvals))
    print("coef(4, eigvals)", coef(4, eigvals))
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  9. - Top - End - #9
    Ogre in the Playground
     
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    Default Re: Math Notation Question (Combinatorics)

    You could use a FFT polynomial multiplication algorithm to multiply the factors.
    Quote Originally Posted by Harnel View Post
    where is the atropal? and does it have a listed LA?

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