  The problem of entanglement detection for arbitrary spin systems is analyzed.
We demonstrate how a single measurement of the squared total spin can
probabilistically discern separable from entangled many-particle states. For
achieving this goal, we construct a tripartite analogy between the degeneracy
of entanglement witness eigenstates, tensor products of $SO(3)$ representations
and classical lattice walks with special constraints. Within this framework,
degeneracies are naturally given by generalized Catalan numbers and determine
the fraction of states that are decidedly entangled and also known to be
somewhat protected against decoherence. In addition, we introduce the concept
of a "sterile entanglement witness", which for large enough systems detects
entanglement without affecting much the system's state. We discuss when our
proposed entanglement witness can be regarded as a sterile one.
