Monomial symmetric polynomials $m_\lambda$

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Numbers

$n$

$\lambda$

x1^2

x1^3

x1^4

x1 + x2

x1^2 + x2^2

1, 1:

x1*x2

x1^3 + x2^3

2, 1:

x1^2*x2 + x1*x2^2

x1^4 + x2^4

3, 1:

x1^3*x2 + x1*x2^3

2, 2:

x1^2*x2^2

x1 + x2 + x3

x1^2 + x2^2 + x3^2

1, 1:

x1*x2 + x1*x3 + x2*x3

x1^3 + x2^3 + x3^3

2, 1:

x1^2*x2 + x1*x2^2 + x1^2*x3 + x2^2*x3 + x1*x3^2 + x2*x3^2

1, 1, 1:

x1*x2*x3

x1^4 + x2^4 + x3^4

3, 1:

x1^3*x2 + x1*x2^3 + x1^3*x3 + x2^3*x3 + x1*x3^3 + x2*x3^3

2, 2:

x1^2*x2^2 + x1^2*x3^2 + x2^2*x3^2

2, 1, 1:

x1^2*x2*x3 + x1*x2^2*x3 + x1*x2*x3^2

x1 + x2 + x3 + x4

x1^2 + x2^2 + x3^2 + x4^2

1, 1:

x1*x2 + x1*x3 + x2*x3 + x1*x4 + x2*x4 + x3*x4

x1^3 + x2^3 + x3^3 + x4^3

2, 1:

x1^2*x2 + x1*x2^2 + x1^2*x3 + x2^2*x3 + x1*x3^2 + x2*x3^2 + x1^2*x4 + x2^2*x4 + x3^2*x4 + x1*x4^2 + x2*x4^2 + x3*x4^2

1, 1, 1:

x1*x2*x3 + x1*x2*x4 + x1*x3*x4 + x2*x3*x4

x1^4 + x2^4 + x3^4 + x4^4

3, 1:

x1^3*x2 + x1*x2^3 + x1^3*x3 + x2^3*x3 + x1*x3^3 + x2*x3^3 + x1^3*x4 + x2^3*x4 + x3^3*x4 + x1*x4^3 + x2*x4^3 + x3*x4^3

2, 2:

x1^2*x2^2 + x1^2*x3^2 + x2^2*x3^2 + x1^2*x4^2 + x2^2*x4^2 + x3^2*x4^2

2, 1, 1:

x1^2*x2*x3 + x1*x2^2*x3 + x1*x2*x3^2 + x1^2*x2*x4 + x1*x2^2*x4 + x1^2*x3*x4 + x2^2*x3*x4 + x1*x3^2*x4 + x2*x3^2*x4 + x1*x2*x4^2 + x1*x3*x4^2 + x2*x3*x4^2

1, 1, 1, 1:

x1*x2*x3*x4

x1 + x2 + x3 + x4 + x5

x1^2 + x2^2 + x3^2 + x4^2 + x5^2

1, 1:

x1*x2 + x1*x3 + x2*x3 + x1*x4 + x2*x4 + x3*x4 + x1*x5 + x2*x5 + x3*x5 + x4*x5

x1^3 + x2^3 + x3^3 + x4^3 + x5^3

2, 1:

x1^2*x2 + x1*x2^2 + x1^2*x3 + x2^2*x3 + x1*x3^2 + x2*x3^2 + x1^2*x4 + x2^2*x4 + x3^2*x4 + x1*x4^2 + x2*x4^2 + x3*x4^2 + x1^2*x5 + x2^2*x5 + x3^2*x5 + x4^2*x5 + x1*x5^2 + x2*x5^2 + x3*x5^2 + x4*x5^2

1, 1, 1:

x1*x2*x3 + x1*x2*x4 + x1*x3*x4 + x2*x3*x4 + x1*x2*x5 + x1*x3*x5 + x2*x3*x5 + x1*x4*x5 + x2*x4*x5 + x3*x4*x5

x1^4 + x2^4 + x3^4 + x4^4 + x5^4

3, 1:

x1^3*x2 + x1*x2^3 + x1^3*x3 + x2^3*x3 + x1*x3^3 + x2*x3^3 + x1^3*x4 + x2^3*x4 + x3^3*x4 + x1*x4^3 + x2*x4^3 + x3*x4^3 + x1^3*x5 + x2^3*x5 + x3^3*x5 + x4^3*x5 + x1*x5^3 + x2*x5^3 + x3*x5^3 + x4*x5^3

2, 2:

x1^2*x2^2 + x1^2*x3^2 + x2^2*x3^2 + x1^2*x4^2 + x2^2*x4^2 + x3^2*x4^2 + x1^2*x5^2 + x2^2*x5^2 + x3^2*x5^2 + x4^2*x5^2

2, 1, 1:

x1^2*x2*x3 + x1*x2^2*x3 + x1*x2*x3^2 + x1^2*x2*x4 + x1*x2^2*x4 + x1^2*x3*x4 + x2^2*x3*x4 + x1*x3^2*x4 + x2*x3^2*x4 + x1*x2*x4^2 + x1*x3*x4^2 + x2*x3*x4^2 + x1^2*x2*x5 + x1*x2^2*x5 + x1^2*x3*x5 + x2^2*x3*x5 + x1*x3^2*x5 + x2*x3^2*x5 + x1^2*x4*x5 + x2^2*x4*x5 + x3^2*x4*x5 + x1*x4^2*x5 + x2*x4^2*x5 + x3*x4^2*x5 + x1*x2*x5^2 + x1*x3*x5^2 + x2*x3*x5^2 + x1*x4*x5^2 + x2*x4*x5^2 + x3*x4*x5^2

1, 1, 1, 1:

x1*x2*x3*x4 + x1*x2*x3*x5 + x1*x2*x4*x5 + x1*x3*x4*x5 + x2*x3*x4*x5

Definition

For a partition $\lambda$ with at most $n$ parts, the monomial symmetric polynomial is $m_\lambda(x_1, \dots, x_n) = \sum x^{\alpha}$, summed over the distinct rearrangements $\alpha$ of $\lambda$ padded with zeros to length $n$. Each monomial appears once.

Parameters

$n$

— integer ($1 \leq n \leq 5$)

$\lambda$

— Symbolic (Unknown type) (a partition of at most $4$, with at most $n$ parts)

Formulas

(1)

The $m_\lambda$ with $\lambda$ running over partitions of $d$ into at most $n$ parts are a basis of the symmetric polynomials of degree $d$ in $n$ variables.

(2)

$h_d = \sum_{|\lambda| = d} m_\lambda$, where $h_d$ is the complete homogeneous symmetric polynomial Complete_homogeneous_symmetric_polynomials.

(3)

$e_k = m_{(1^k)}$, the partition of $k$ into $k$ ones, where $e_k$ is the elementary symmetric polynomial Elementary_symmetric_polynomials.

(4)

$p_k = m_{(k)}$, the one-part partition, where $p_k$ is the power sum Power_sum_symmetric_polynomials.

(5)

$m_\lambda$ has as many terms as there are distinct rearrangements of $\lambda$ padded to length $n$: $\binom{n}{m_0, m_1, m_2, \dots}$ where $m_i$ counts the parts equal to $i$.

Comments

(6)

The partition is written as its parts, largest first, separated by commas: $2,1,1$ is the partition $(2,1,1)$ of $4$.

(7)

Partitions of at most $4$ in at most $5$ variables, so that the longest entry is 387 characters. The rule across these five tables is the largest complete rectangle whose longest entry stays under about 600.

(8)

The Schur polynomials Schur_polynomials are indexed by the same partitions and expand in this basis with non-negative integer coefficients, the Kostka numbers.

Programs

(P1)

Sage

from itertools import permutations
def monomial_symmetric(n, lam):
    R = PolynomialRing(ZZ, ['x%d' % (i+1) for i in range(n)])
    exps = list(lam) + [0]*(n - len(lam))
    return sum(prod(g^a for g, a in zip(R.gens(), e))
               for e in sorted(set(permutations(exps))))

monomial_symmetric(5, [3, 2])    # a partition of 5, past this table

Links

[1]

Wikipedia: Symmetric polynomial

Data properties

Entries are of type: integral polynomial

Table is complete: false

How well the digits are known: exact