  Luminous Red Galaxies (LRG) from the Sloan Digital Sky Survey are considered
among the best understood samples of galaxies, and they are employed in a broad
range of cosmological studies. Because they form a relatively homogeneous
population, with high stellar masses and red colors, they are expected to
occupy halos in a relatively simple way. In this paper, we study how LRGs
occupy massive halos via direct counts in clusters and we reveal several
unexpected trends suggesting that the connection between LRGs and dark matter
halos may not be straightforward. Using the redMaPPer cluster catalog, we
derive the central occupation of LRGs as a function richness, Ncen({\lambda}).
Assuming no correlation between cluster mass and central galaxy luminosity at
fixed richness, we show that clusters contain a significantly lower fraction of
central LRGs than predicted from the two-point correlation function. At halo
masses of 10^14.5 Msun, we find Ncen=0.73, compared to Ncen of 0.89 from
correlation studies. Our central occupation function for LRGs converges to 0.95
at large halo masses. A strong anti-correlation between central luminosity and
cluster mass at fixed richness is required to reconcile our results with those
based on clustering studies. We also derive P_BNC, the probability that the
brightest cluster member is not the central galaxy. We find P_BNC ~ 20-30%
which is a factor of ~2 lower than the value found by Skibba et al. 2011.
Finally, we study the radial offsets of bright non-central LRGs from cluster
centers and show that bright non-central LRGs follow a different radial
distribution compared to red cluster members, which follow a
Navarro-Frank-White profile. This work demonstrates that even the most massive
clusters do not always have an LRG at the center, and that the brightest galaxy
in a cluster is not always the central galaxy.
