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Complete homogeneous symmetric polynomials $h_k$
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Numbers
$n$
$k$ 
1
1:
x1
1
2:
x1^2
1
3:
x1^3
2
1:
x1 + x2
2
2:
x1^2 + x1*x2 + x2^2
2
3:
x1^3 + x1^2*x2 + x1*x2^2 + x2^3
3
1:
x1 + x2 + x3
3
2:
x1^2 + x1*x2 + x2^2 + x1*x3 + x2*x3 + x3^2
3
3:
x1^3 + x1^2*x2 + x1*x2^2 + x2^3 + x1^2*x3 + x1*x2*x3 + x2^2*x3 + x1*x3^2 + x2*x3^2 + x3^3
4
1:
x1 + x2 + x3 + x4
4
2:
x1^2 + x1*x2 + x2^2 + x1*x3 + x2*x3 + x3^2 + x1*x4 + x2*x4 + x3*x4 + x4^2
4
3:
x1^3 + x1^2*x2 + x1*x2^2 + x2^3 + x1^2*x3 + x1*x2*x3 + x2^2*x3 + x1*x3^2 + x2*x3^2 + x3^3 + x1^2*x4 + x1*x2*x4 + x2^2*x4 + x1*x3*x4 + x2*x3*x4 + x3^2*x4 + x1*x4^2 + x2*x4^2 + x3*x4^2 + x4^3
5
1:
x1 + x2 + x3 + x4 + x5
5
2:
x1^2 + x1*x2 + x2^2 + x1*x3 + x2*x3 + x3^2 + x1*x4 + x2*x4 + x3*x4 + x4^2 + x1*x5 + x2*x5 + x3*x5 + x4*x5 + x5^2
5
3:
x1^3 + x1^2*x2 + x1*x2^2 + x2^3 + x1^2*x3 + x1*x2*x3 + x2^2*x3 + x1*x3^2 + x2*x3^2 + x3^3 + x1^2*x4 + x1*x2*x4 + x2^2*x4 + x1*x3*x4 + x2*x3*x4 + x3^2*x4 + x1*x4^2 + x2*x4^2 + x3*x4^2 + x4^3 + x1^2*x5 + x1*x2*x5 + x2^2*x5 + x1*x3*x5 + x2*x3*x5 + x3^2*x5 + x1*x4*x5 + x2*x4*x5 + x3*x4*x5 + x4^2*x5 + x1*x5^2 + x2*x5^2 + x3*x5^2 + x4*x5^2 + x5^3
6
1:
x1 + x2 + x3 + x4 + x5 + x6
6
2:
x1^2 + x1*x2 + x2^2 + x1*x3 + x2*x3 + x3^2 + x1*x4 + x2*x4 + x3*x4 + x4^2 + x1*x5 + x2*x5 + x3*x5 + x4*x5 + x5^2 + x1*x6 + x2*x6 + x3*x6 + x4*x6 + x5*x6 + x6^2
6
3:
x1^3 + x1^2*x2 + x1*x2^2 + x2^3 + x1^2*x3 + x1*x2*x3 + x2^2*x3 + x1*x3^2 + x2*x3^2 + x3^3 + x1^2*x4 + x1*x2*x4 + x2^2*x4 + x1*x3*x4 + x2*x3*x4 + x3^2*x4 + x1*x4^2 + x2*x4^2 + x3*x4^2 + x4^3 + x1^2*x5 + x1*x2*x5 + x2^2*x5 + x1*x3*x5 + x2*x3*x5 + x3^2*x5 + x1*x4*x5 + x2*x4*x5 + x3*x4*x5 + x4^2*x5 + x1*x5^2 + x2*x5^2 + x3*x5^2 + x4*x5^2 + x5^3 + x1^2*x6 + x1*x2*x6 + x2^2*x6 + x1*x3*x6 + x2*x3*x6 + x3^2*x6 + x1*x4*x6 + x2*x4*x6 + x3*x4*x6 + x4^2*x6 + x1*x5*x6 + x2*x5*x6 + x3*x5*x6 + x4*x5*x6 + x5^2*x6 + x1*x6^2 + x2*x6^2 + x3*x6^2 + x4*x6^2 + x5*x6^2 + x6^3
Definition
For $k \geq 1$ the complete homogeneous symmetric polynomial in $n$ variables is $h_k(x_1, \dots, x_n) = \sum_{i_1 \leq i_2 \leq \cdots \leq i_k} x_{i_1} x_{i_2} \cdots x_{i_k}$, the sum of every monomial of degree $k$, each with coefficient one.
Parameters
$n$
—   integer ($1 \leq n \leq 6$)
$k$
—   integer ($1 \leq k \leq 3$)
Formulas
(1)
$\prod_{i=1}^{n} \frac{1}{1 - x_i t} = \sum_{k \geq 0} h_k(x_1,\dots,x_n)\, t^{k}$, with $h_0 = 1$.
(2)
$\sum_{k=0}^{d} (-1)^{k} e_k\, h_{d-k} = 0$ for $d \geq 1$, where $e_k$ are the elementary symmetric polynomials Elementary_symmetric_polynomials. The two generating functions are reciprocal.
(3)
$h_d = \sum_{|\lambda| = d} m_\lambda$, the sum over all partitions of $d$ of the monomial symmetric polynomials Monomial_symmetric_polynomials.
(4)
$h_k = s_{(k)}$, the Schur polynomial Schur_polynomials for the one-row partition.
(5)
$h_k$ in $n$ variables has $\binom{n+k-1}{k}$ terms.
Comments
(6)
This table stops at $k = 3$ where the others go further, because these grow fastest: $h_3$ in six variables is 559 characters written out and $h_6$ would be 6969. The rule across these five tables is the largest complete rectangle whose longest entry stays under about 600 characters.
(7)
Swapping $e$ and $h$ is an involution on the symmetric functions, so every identity between them has a partner with the two exchanged.
Programs
(P1)
Sage
from itertools import combinations_with_replacement
def homogeneous(n, k):
    R = PolynomialRing(ZZ, ['x%d' % (i+1) for i in range(n)])
    return sum(prod(c) for c in combinations_with_replacement(R.gens(), k))

homogeneous(6, 4)                # 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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