An excellent question certainly worthy of its own thread. I've been working on an answer to this question for about six months and I'll post the results in a future Met Blog, but some of the initial results are interesting. High EASTPAC totals equate to low Atlantic totals exactly half of the time. Lower seasonal EASTPAC totals seldom equate to higher Atlantic seasonal totals (the exception was 2010). Lower Atlantic totals correlated well with higher EASTPAC numbers although sometimes the Pacific activity was still below normal (examples include 1965 and 1977). The busiest Atlantic seasons always had lower EASTPAC totals - although sometimes those totals were representative of a normal EASTPAC season rather than a 'quiet' one.

With regard to ENSO conditions, here is my response on June 12, 2007, to a similar question:
"About a month ago I did a little research from a slightly different angle. Since ENSO conditions are only available from 1950, I decided to check El Nino Region 3.4 anomalies against all of the very active seasons since 1950. I defined an active season as 14 named storms or more (there have been 10 of these) and ENSO neutral conditions as anything from +0.5C to -0.5C. I used the average sea surface temperature anomaly for the three month period of May, June and July (figuring that a three month lag was reasonable between Pacific ENSO conditions and Atlantic activity). The results coincide nicely with the findings of Ostro & Lyons.

1953...14 storms...+0.4 anomaly
1969...17 storms...+0.4 anomaly
1990...14 storms...+0.3 anomaly
1995...19 storms...+0.1 anomaly
1998...14 storms...+0.1 anomaly
2000...14 storms...-0.6 anomaly (weakening La Nina)
2001...15 storms...+0.1 anomaly
2003...15 storms.....0.0 anomaly
2004...15 storms...+0.4 anomaly
2005...28 storms...+0.3 anomaly

Conclusion: Although the dataset is limited, it appears that very active seasons generally occur under initially 'neutral' ENSO conditions. A strong La Nina doesn't really enhance activity, but a strong El Nino (1992, 1997, 2006) will limit activity."

Of course since that was posted in 2007 there have been many more active Atlantic seasons that may ultimately limit the usefulness of the above conclusion.
Cheers,
ED