  We introduce a remarkable new family of norms on the space of $n \times n$
complex matrices. These norms arise from the combinatorial properties of
symmetric functions, and their construction and validation involve probability
theory, partition combinatorics, and trace polynomials in noncommuting
variables. Our norms enjoy many desirable analytic and algebraic properties,
such as an elegant determinantal interpretation and the ability to distinguish
certain graphs that other matrix norms cannot. Furthermore, they give rise to
new dimension-independent tracial inequalities. Their potential merits further
investigation.
