Adrian Blau, "The Effective Number of Parties at Four Scales: Votes, Seats, Legislative Power and Cabinet Power," Party Politics, 14 (March, 2008), 167-187.

First paragraph:
Political scientists often disagree about how to count parties. In this article, I argue that how we count depends on where we look. The now-dominant 'effective number of parties' index is based on votes and seats, but says little directly about power; Duverger, Sartori and others look more at competition for government but make dubious assumptions about counting parties. I seek the best of both worlds by combining the technical advantages of the effective number of parties index with a focus on power. I therefore estimate two new measures: the effective number of parties in terms of legislative power, based on modifying Powell's (2000) method for estimating government and opposition influence, and the effective number of parties in terms of cabinet power, based on party shares of cabinet portfolios.

Figures and Tables:

Figure 1. Effective numbers of parties in votes, seats, legislative power and cabinet power

Table 1. Mean effective numbers of parties, UK and Germany

Table 2. Mean absolute and relative reductions in parties, UK and Germany

Table 3. Mean longer-term effective numbers of parties, UK and Germany

Table 4. Estimated legislative power in the UK

Table 5. Estimated legislative power in Germany

Table 6. Party shares of cabinet portfolios in UK coalition governments, 1931-1945

Table 7. Party shares of cabinet portfolios in German governments, 1949-2005

Last paragraph:
Overall, I hope I have advanced the debate about the 'right' method of counting parties: there is no single answer. And I hope I have advanced the debate about the 'right' description of the number of parties in a given system: we can count parties not just at one point only, but at four, or more when we include different regions and timescales. We can talk not of the number (singular) but of the numbers (plural) of parties. Given the importance of comparison in political science, we can compare in five ways: (a) we can compare numbers of parties at different stages of the political process, to see the reduction of parties over the political process; (b) we can compare numbers of parties to normative requirements, to see how well proportional or pluralitarian norms are met; (c) we can compare numbers of parties over time, to see how the system changes; (d) we can compare numbers of parties over space, to get cross-national or sub-national insights; and (e) we can compare numbers of parties to other important phenomena, like electoral systems, cabinet stability, policy output and so on. Of course, we must also address how we measure legislative and executive power. While my estimates of NL and NC are robust enough for the comparisons in this article, NL in particular must be refined in the future.