  For each positive integer n this paper considers a one-parameter family of
complex-valued measures on the symmetric group S_n, depending on a complex
parameter z. For parameter values z=p^f a prime power, this measure describes
splitting probabilities for monic degree n polynomials over the finite field
with p^f elements, conditioned on being square-free. It studies these measure
in the special case z=1, and shows they have an interesting internal structure
having a representation theoretic interpretation. These measures may provide
data relevant to the hypothetical "field with one element ${\mathbb F_1}$". It
additionally studies the case z=-1, which also has a representation-theoretic
interpretation
