Given the setting of the birthday paradox ($n$ people with birthdays randomly drawn from the uniform distribution of 365 days), let us call the days on which at least one person was born birthdays and all other days non-birthdays. What can be said about the distribution of
a) The length in days of the longest streak of birthdays;
b) The length in days of the longest streak of non-birthdays?
Is there an analytical solution to, for example, the expected value of either of these?
(You can choose whether to break all streaks when the year changes; I think insightful analysis is more interesting than the precise formulation of the problem.)