Given a set of $N$ non-commutative variables $x_k$. Is there a special name for symmetric polynomials of homogenous degree $d$ of the form that all $x_k$s appear with exponent at most $1$ at a time? So the appearance of $x_k^2$ is forbidden.
EDIT I mean, for every term subsequent factors $x_k$ and $x_l$ are never equal.
Some examples for $N=3$ and $d=2$:
- $P_0(x)=x_0x_1+x_1x_0+x_0x_2+x_2x_0+x_1x_2+x_2x_1$
and $d=3$:
- $P_1(x)=x_0x_1x_0+x_0x_2x_0+x_1x_0x_1+x_1x_2x_1+x_2x_0x_2+x_2x_1x_2$
- $P_2(x)=x_0x_1x_2+x_0x_2x_1+x_1x_0x_2+x_1x_2x_0+x_2x_0x_1+x_2x_1x_0$
- $P_3(x)=nP_1(x) +mP_2(x)$
The total number of possible elements in a polynomial is $N(N-1)^{d-1}$.