  The low number density of the Sloan Digital Sky Survey (SDSS) Luminous Red
Galaxies (LRGs) suggests that LRGs occupying the same dark matter halo can be
separated from pairs occupying distinct dark matter halos with high fidelity.
We present a new technique, Counts-in-Cylinders (CiC), to constrain the
parameters of the satellite contribution to the LRG Halo-Occupation
Distribution (HOD). For a fiber collision-corrected SDSS spectroscopic LRG
subsample at 0.16 < z < 0.36, we find the CiC multiplicity function is fit by a
halo model where the average number of satellites in a halo of mass M is
<Nsat(M)> = ((M - Mcut)/M1)^alpha with Mcut = 5.0 +1.5/-1.3 (+2.9/-2.6) X 10^13
Msun, M1 = 4.95 +0.37/-0.26 (+0.79/-0.53) X 10^14 Msun, and alpha = 1.035
+0.10/-0.17 (+0.24/-0.31) at the 68% and 95% confidence levels using a WMAP3
cosmology and z=0.2 halo catalog.
  Our method tightly constrains the fraction of LRGs that are satellite
galaxies, 6.36 +0.38/-0.39, and the combination Mcut/10^{14} Msun + alpha =
1.53 +0.08/-0.09 at the 95% confidence level. We also find that mocks based on
a halo catalog produced by a spherical overdensity (SO) finder reproduce both
the measured CiC multiplicity function and the projected correlation function,
while mocks based on a Friends-of-Friends (FoF) halo catalog has a deficit of
close pairs at ~1 Mpc/h separations. Because the CiC method relies on higher
order statistics of close pairs, it is robust to the choice of halo finder. In
a companion paper we will apply this technique to optimize Finger-of-God (FOG)
compression to eliminate the 1-halo contribution to the LRG power spectrum.
