For $n$ a natural number, $\alpha$ an ordinal, let $\rho(n,\alpha)$ be the $n$-th projectum of $J_\alpha$, where $J$ is the Jensen hierarchy for $L$.
Call a finite sequence $s:=(x_1,\dots,x_m)$ of integers projectum-representable iff there is an ordinal $\alpha$ such that $$\rho(1,\alpha)=\rho(2,\alpha)=\dots=\rho(x_1,\alpha)\gt$$ $$\rho(x_1+1,\alpha)=\dots=\rho(x_2,\alpha)\gt$$ $$\rho(x_2+1,\alpha)=\dots=\rho(x_3,\alpha)\gt$$ $$\dots=\rho(x_m,\alpha),$$ i.e. if the sequence of projecta of $J_\alpha$ consists of a sequence of $x_1$ identical terms, then drops, then has $x_2$ identical terms, then drops again etc.
Which finite sequences are projectum-representable?
By condensation arguments, it can be seen that every representable sequence is already representable with a countable ordinal $\alpha$.
References would also be welcome.