  We present a sample of Type Icn supernovae (SNe Icn), a newly-discovered
class of transients characterized by their interaction with H- and He-poor
circumstellar material (CSM). This sample is the largest collection of SNe Icn
to date and includes observations of two published objects (SN 2019hgp and SN
2021csp) as well as two objects (SN 2019jc and SN 2021ckj) not yet published in
the literature. The SNe Icn display a range of peak luminosities, rise times,
and decline rates, as well as diverse late-time spectral features. To
investigate their explosion and progenitor properties we fit their bolometric
light curves to a semi-analytical model consisting of luminosity inputs from
circumstellar interaction and radioactive decay of $^{56}$Ni. We infer low
ejecta masses ($\lesssim$ 2 M$_\odot$) and $^{56}$Ni masses ($\lesssim$ 0.04
M$_\odot$) from the light curves, suggesting that normal stripped-envelope
supernova (SESN) explosions within a dense CSM cannot be the underlying
mechanism powering SNe Icn. Additionally, we find that an upper limit on the
star formation rate density at the location of SN 2019jc lies at the lower end
of a distribution of SESNe, in conflict with a massive star progenitor of this
object. Based on the estimated ejecta masses, $^{56}$Ni masses, and explosion
site properties, we favor a low-mass, ultra-stripped star as the progenitor of
some SNe Icn. For others, we suggest that a Wolf-Rayet star progenitor may
better explain their observed properties. This study demonstrates that multiple
progenitor channels may produce SNe Icn and other interaction-powered
transients.
