Use of exclamation point I'm quite puzzled by the use of an exclamation point in this paper. The authors introduce the following linear constraints to a quadratic program:
$
\sum_k a^{(l)}_k b_j (\mathbf{x}_k) = r_j^{(l)}
$
for $j \in I$,
where $l = 2,3,...,6$, $\mathbf{b}(\mathbf{x})=[1,x,y,x^2,xy,y^2]^T$; $r = [r_1^{(l)}, r_2^{(l)}, ..., r_I^{(l)}]^T = \alpha_l ! \mathbf{e}_l$. Here, $\mathbf{e}_l$ is the $l$-th standard basis for $\mathbb{R}^I$ and $\alpha$'s are further defined (for $I=6$) as $\alpha_1 = (0, 0)$, $\alpha_2 = (1, 0)$, $\alpha_3 = (0, 1)$, $\alpha_4 = (2, 0)$, $\alpha_5 = (1, 1)$, $\alpha_6 = (0, 2)$.
What is the meaning of this exclamation mark in $\alpha_l ! \mathbf{e}_l$? It looks neither like a factorial, nor a derangement, since the result should be in $\mathbb{R}^I$. I suspect that it has to do with the power of a bivariate polynomial they use: $f(x,y) = [1, x, y, x^2, xy, y^2]^T\mathbf{a}$, but can't get my head around it still.

I'll just type in the answer of egreg (thanks!) to get it over with. This fits the context very well.

In the paper $\alpha_i$ is a multi-index, $\alpha_i = (a_1,a_2,\dots,a_d)$, with $a_{j} \in \mathbb{N}$ and $\alpha_i!=a_1! \cdot a_2! \cdot \ldots \cdot a_d!$. This seems clear looking at equations A.6 and A.7.
See en.wikipedia.org/wiki/Multi-index_notation
