Party system polarization and the effective number of parties

Polarization is a key characteristic of party systems, but scholars disagree about how polarization relates to the number of parties in a system. Different authors find positive, negative, or null relationships. I claim that when polarization is measured using the weighted standard deviation of standardized party positions, seat-level polarization is equal to (N_S - 1) / (1/√2 + N_S - 1), where N_S is the
effective number of seat-winning parties. This relationship is what one would expect if parties were drawn randomly from a super-population with an effective sample size somewhere between the effective and raw number of parties. I test this claim using multiple datasets which report party positions and seat shares, before extending my analysis to consider vote-level polarization, the range of positions, and
polarization in presidential and parliamentary regimes. My work extends the Taageperaan research agenda of building interlocking networks of equations relating key quantities of electoral and party systems.