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?
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> 
> 


By condensation arguments, it can be seen that every representable sequence is already representable with a countable ordinal $\alpha$.


References would also be welcome.



Sequences of projecta in the constructible hierarchy
The short answer is that all finite sequences are representable.


Suppose you want the general constellation you gave. We work inside $L$. For $X=\langle X,\in \rangle$ a model of V=L (not necessarily transitive) define $S^X\_n(Y)$ to be the $\Sigma\_n$-Skolem Hull in $X$ of the set $Y$. Let $\tau$ be the least ordinal $> \omega\_{m-1}$ with $L\_\tau$ a model of $\Sigma\_{x\_1}$-Separation scheme. (This ensures that $\rho(x\_1,\tau)=\tau$.)
Set $H\_0= L\_\tau$. Now define by recursion on $j$ for $1\leq j\leq m-1$:
$$H\_j = S^{H\_{j-1}}\_{x\_j +1}(\omega\_{m-1-j}). $$


Let $L\_\alpha \simeq H\_m$.


This is admittedly easier to see with an example: say $(x\_1,x\_2,x\_3) = (2,4,6)$ with $m=3$ "drops". Let $\tau$ be the least above $\omega\_{m-1} =\omega\_2$ with $L\_\tau$ a model of $\Sigma\_2$-Separation. Then $\tau = \rho(\tau,2)> \omega\_2 =\rho(\tau,3) \ldots$. Then $$H\_1 = S^{L\_\tau}\_5(\omega\_1).$$ (If we were to take the Mostowski-Shepherdson collapse of this hull it would be an $L\_{\alpha'}$ with $\rho(\alpha',k)=\omega\_1$
for $k\geq 5$. One should check, e.g., that $\rho(\alpha',4)\neq \omega\_1$: but if this failed then there would be a $\Sigma\_4(L\_{\alpha'})$ map of $\omega\_1$ cofinally into $\alpha'$; but this is a $\Pi\_5$ statement, and would then be true in $L\_\tau$ which is absurd.) Then take $$H\_2 =S^{H\_1}\_7(\omega).$$


Let $L\_\alpha \simeq H\_2$. Similar checks reveal that $\rho(\alpha,6)=\omega\_1> \rho(\alpha,7) \ldots$ etc. and we have the correct pattern.


I do not know of any references for this.


