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How many symmetric polynomials of n degree with all their coefficients $\ =1 $ are there?Is there a type that computes their number?

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The number of different symmetric monomials of degree $n$ in $k$ variables with all the coefficients 1 is equal to the number $p(n,k)$ of partitions of $n$ in up to $k$ parts, for every such partition say $a_1+a_2+\dots + a_k = n$ with $a_1 \ge a_2 \ge \dots \ge a_k \ge 0$ we have the symmetric "monomial": $$ x_1^{a_1} x_2^{a_2} \dots x_k^{a_k} + \dots $$

The number of different symmetric homogeneous polynomials of degree $n$ in $k$ variables with all the coefficients 1 is as a consequence $ 2^{p(n,k)}-1 $

The number of different symmetric polynomials of degree (exactly) $n$ in $k$ variables with all the coefficients 1 is then $$ (2^{p(n,k)}-1)2^{p(n-1,k)+p(n-2,k)+ \dots+ p(1,k)+p(0,k)} $$

So if you write $$P(n,k) = p(n,k) + p(n-1,k)+ \dots +p(0,k) $$ the number of partitions of numbers up to $n$ in at most $k$ parts then the number you ask for (if I have understood the question and I haven't made a silly mistake is) $$ 2^{P(n,k)} -2^{P(n-1,k)} $$ EDIT: This was wrong. Sorry.

EDIT 2: The answer above is answering the question if we consider the total degree of the symmetric functions, in view of the comments we are looking instead for polynomials with largest degree $n$ in any of the variables then we have the following:

  1. The number of different symmetric monomials with degree up to $n$ in all the variables is $n^k$.
  2. The total number of symmetric polynomials composed with this monomials is $2^{n^k}$.
  3. So the number of symmetric polynomials having at least one term of degree $n$ in one of the variables is $$ 2^{n^k}-2^{(n-1)^k} $$
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It seems that something went wrong with my question.When I said degree n i meant the higher degree of one (so of all) invariant . –  t.k Mar 13 '11 at 9:36
    
I don't understand what you mean by invariant, do you mean homogenous?, all the terms have the same total degree? take this four examples with $n=3$ and $k=2$, a) $x^3+y^3$, b) $x^2y + xy^2$, c) $x^3+y^3 + x + y$ d) $x^2y + xy^2+x+y$, which are you counting as degree 3? –  Esteban Crespi Mar 13 '11 at 9:46
    
i mean the cases a) and c) –  t.k Mar 13 '11 at 16:36
    
many thanks for your time. –  t.k Mar 15 '11 at 18:40
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