Planck intermediate results. VII. Statistical properties of infrared and
  radio extragalactic sources from the Planck Early Release Compact Source
  Catalogue at frequencies between 100 and 857 GHz
  (abridged for arXiv) We make use of the Planck all-sky survey to derive
number counts and spectral indices of extragalactic sources -- infrared and
radio sources -- from the Planck Early Catalogue (ERCSC) at 100 to 857GHz. Our
sample contains, after the 80% completeness cut, between 122 and 452 and
sources, with flux densities above 0.3 and 1.9Jy at 100 and 857GHz, over about
31 to 40% of the sky. Using Planck HFI, all the sources have been classified as
either dust-dominated or synchrotron-dominated on the basis of their spectral
energy distributions (SED). Our sample is thus complete, flux-limited and
color-selected to differentiate between the two populations. We find an
approximately equal number of synchrotron and dusty sources between 217 and
353GHz; at 353GHz or higher (or 217GHz and lower) frequencies, the number is
dominated by dusty (synchrotron) sources, as expected. For most of the sources,
the spectral indices are also derived. We provide for the first time counts of
bright sources from 353 to 857GHz and the contributions from dusty and
synchrotron sources at all HFI frequencies in the key spectral range where
these spectra are crossing. The observed counts are in the Euclidean regime.
The number counts are compared to previously published data (earlier Planck,
Herschel, BLAST, SCUBA, LABOCA, SPT, and ACT) and models taking into account
both radio or infrared galaxies. We derive the multi-frequency Euclidean level
and compare it to WMAP, Spitzer and IRAS results. The submillimetre number
counts are not well reproduced by current evolution models of dusty galaxies,
whereas the millimetre part appears reasonably well fitted by the most recent
model for synchrotron-dominated sources. Finally we provide estimates of the
local luminosity density of dusty galaxies, providing the first such
measurements at 545 and 857GHz.
