Number of monomial symmetric polynomials in three variables

Question

I am trying to find a reference for a formula regarding the number of monomial symmetric polynomials of degree $m$, in three variables. I believe that this number is given by $1+\left \lfloor{\frac{m^2+6m}{12}}\right \rfloor $, where $\lfloor{..}\rfloor$ denotes the integer part.

This problem is equivalent, I think, to the number of ways one can select three non-negative integers such that their sum is equal to $m$, ignoring permutations.

Any help appreciated.

Answer 1

The number of monomial symmetric polynomials of degree $m$ in $3$ variables equals the number of partitions of $m$ into at most $3$ parts; the parts stand for the nonzero exponents. Transposing the partition, this also equals the number of partitions of $m$ into parts that are at most $3$ each, in other words, the coefficient of $X^m$ in the power series expansion of $$\frac{1}{(1-X)(1-X^2)(1-X^3)}$$ which, according to OEIS A001399, is $\left\lfloor\frac{(m+3)^2+6}{12}\right\rfloor$ whereas your formula equals $\left\lfloor\frac{(m+3)^2+3}{12}\right\rfloor$. These are equal because the square in the numerator is congruent to one of $0,1\pmod{4}$ and to one of $0,1\pmod{3}$, so is never one of $6,7,8\pmod{12}$ where the truncations would differ. In fact, the Maple example there uses your version.

Regarding references: The above OEIS entry refers to

• L. Comtet, Advanced Combinatorics, Reidel 1974, p. 110, $D(n)$

Indeed there we find formula [6g'] for $D(n;1,2,3)$, the number of partitions of $n$ where each part is in $\{1,2,3\}$, as a corollary of more general theorems.

Answer 2

Another expression for these numbers is $$ \frac{1}{12}\,{m}^{2}+\frac 1 2\,m+ \frac 1 8\,\cos \left( \pi \,m \right) +\frac 2 9\,\cos \left( \frac 2 3\,\pi \,m \right) +{\frac {47}{72}}. $$