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Power sum symmetric polynomials $p_k$
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Numbers
$n$
$k$ 
1
1:
x1
1
2:
x1^2
1
3:
x1^3
1
4:
x1^4
1
5:
x1^5
1
6:
x1^6
1
7:
x1^7
1
8:
x1^8
1
9:
x1^9
1
10:
x1^10
1
11:
x1^11
1
12:
x1^12
2
1:
x1 + x2
2
2:
x1^2 + x2^2
2
3:
x1^3 + x2^3
2
4:
x1^4 + x2^4
2
5:
x1^5 + x2^5
2
6:
x1^6 + x2^6
2
7:
x1^7 + x2^7
2
8:
x1^8 + x2^8
2
9:
x1^9 + x2^9
2
10:
x1^10 + x2^10
2
11:
x1^11 + x2^11
2
12:
x1^12 + x2^12
3
1:
x1 + x2 + x3
3
2:
x1^2 + x2^2 + x3^2
3
3:
x1^3 + x2^3 + x3^3
3
4:
x1^4 + x2^4 + x3^4
3
5:
x1^5 + x2^5 + x3^5
3
6:
x1^6 + x2^6 + x3^6
3
7:
x1^7 + x2^7 + x3^7
3
8:
x1^8 + x2^8 + x3^8
3
9:
x1^9 + x2^9 + x3^9
3
10:
x1^10 + x2^10 + x3^10
3
11:
x1^11 + x2^11 + x3^11
3
12:
x1^12 + x2^12 + x3^12
4
1:
x1 + x2 + x3 + x4
4
2:
x1^2 + x2^2 + x3^2 + x4^2
4
3:
x1^3 + x2^3 + x3^3 + x4^3
4
4:
x1^4 + x2^4 + x3^4 + x4^4
4
5:
x1^5 + x2^5 + x3^5 + x4^5
4
6:
x1^6 + x2^6 + x3^6 + x4^6
4
7:
x1^7 + x2^7 + x3^7 + x4^7
4
8:
x1^8 + x2^8 + x3^8 + x4^8
4
9:
x1^9 + x2^9 + x3^9 + x4^9
4
10:
x1^10 + x2^10 + x3^10 + x4^10
4
11:
x1^11 + x2^11 + x3^11 + x4^11
4
12:
x1^12 + x2^12 + x3^12 + x4^12
5
1:
x1 + x2 + x3 + x4 + x5
5
2:
x1^2 + x2^2 + x3^2 + x4^2 + x5^2
5
3:
x1^3 + x2^3 + x3^3 + x4^3 + x5^3
5
4:
x1^4 + x2^4 + x3^4 + x4^4 + x5^4
5
5:
x1^5 + x2^5 + x3^5 + x4^5 + x5^5
5
6:
x1^6 + x2^6 + x3^6 + x4^6 + x5^6
5
7:
x1^7 + x2^7 + x3^7 + x4^7 + x5^7
5
8:
x1^8 + x2^8 + x3^8 + x4^8 + x5^8
5
9:
x1^9 + x2^9 + x3^9 + x4^9 + x5^9
5
10:
x1^10 + x2^10 + x3^10 + x4^10 + x5^10
5
11:
x1^11 + x2^11 + x3^11 + x4^11 + x5^11
5
12:
x1^12 + x2^12 + x3^12 + x4^12 + x5^12
6
1:
x1 + x2 + x3 + x4 + x5 + x6
6
2:
x1^2 + x2^2 + x3^2 + x4^2 + x5^2 + x6^2
6
3:
x1^3 + x2^3 + x3^3 + x4^3 + x5^3 + x6^3
6
4:
x1^4 + x2^4 + x3^4 + x4^4 + x5^4 + x6^4
6
5:
x1^5 + x2^5 + x3^5 + x4^5 + x5^5 + x6^5
6
6:
x1^6 + x2^6 + x3^6 + x4^6 + x5^6 + x6^6
6
7:
x1^7 + x2^7 + x3^7 + x4^7 + x5^7 + x6^7
6
8:
x1^8 + x2^8 + x3^8 + x4^8 + x5^8 + x6^8
6
9:
x1^9 + x2^9 + x3^9 + x4^9 + x5^9 + x6^9
6
10:
x1^10 + x2^10 + x3^10 + x4^10 + x5^10 + x6^10
6
11:
x1^11 + x2^11 + x3^11 + x4^11 + x5^11 + x6^11
6
12:
x1^12 + x2^12 + x3^12 + x4^12 + x5^12 + x6^12
Definition
For $k \geq 1$ the power sum symmetric polynomial in $n$ variables is $p_k(x_1, \dots, x_n) = x_1^{k} + x_2^{k} + \cdots + x_n^{k}$.
Parameters
$n$
—   integer ($1 \leq n \leq 6$)
$k$
—   integer ($1 \leq k \leq 12$)
Formulas
(1)
$k\,e_k = \sum_{i=1}^{k} (-1)^{i-1} e_{k-i}\, p_i$, where $e_k$ are the elementary symmetric polynomials Elementary_symmetric_polynomials (Newton's identities). These determine the $p_k$ from the $e_k$ and back again.
(2)
$\sum_{k \geq 1} p_k\, t^{k} = \sum_{i=1}^{n} \frac{x_i t}{1 - x_i t}$.
(3)
$p_k = m_{(k)}$, the monomial symmetric polynomial Monomial_symmetric_polynomials for the one-part partition.
(4)
Over $\mathbb{Q}$ the $p_1, \dots, p_n$ generate the symmetric polynomials freely; over $\mathbb{Z}$ they do not, which is why Newton's identities divide by $k$.
Comments
(5)
These are the shortest of the symmetric families: $p_k$ has $n$ terms whatever $k$ is, so the table runs to $k = 12$ where the others stop much sooner. In six variables the longest entry is 39 characters.
(6)
If $x_1, \dots, x_n$ are the eigenvalues of a matrix $A$, then $p_k = \operatorname{tr}(A^{k})$, which is how these are most often met outside symmetric function theory.
Programs
(P1)
Sage
def power_sum(n, k):
    R = PolynomialRing(ZZ, ['x%d' % (i+1) for i in range(n)])
    return sum(g^k for g in R.gens())

power_sum(6, 13)                 # the next degree after this table
Links
Data properties
Entries are of type: integral polynomial
Table is complete: false
How well the digits are known: exact


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