Math Notation Question (Combinatorics)

[Astral Avenger]

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]

[Sermil]

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)

[Rockphed]

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.

[jayem]

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 C_n for cyclic permutations with C_n*C_m 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.

[gomipile]

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.

[Jay R]

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.

[Radar]

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 A_n and perform the following operation:
Sum_{An in P(S)} Product_{x in An}x

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

With such a notation you get for example
Sum_{A in Pn(S)} Product_{x in A} x

[Astral Avenger]

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 + ... + A_n-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

# 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))

[gomipile]

You could use a FFT polynomial multiplication algorithm to multiply the factors.