I just invented 16 new pieces (observation)

3 * 3 * 3 * 3 = 3 ^ 4 = 81 possible pieces

I mean like hg muller Said, you could combine the pieces

Oh i forgot the - 1

Then pls also read his comment in #8 of this thread

I think my initial calculation of 2^16-1 was correct after all. I just got confused by the idea that squares with a don't-care occupancy would play a role. But that was a mistake; a piece that has a don't-care amongst the 4 squares can be considered a compound of two pieces that do care about the square occupancy: one that requires it to be occupied, the other that requires it to be empty.

So all potential intermediate squares must always be taken into account, and with 4 such squares there are 2^4 = 16 occupancy patterns. A piece can selectively allow the move for any sub-set of those. And there are 2^16 such sub-sets. But the empty sub-set does not have the possibility to move to the destination under any circumstance, and thus should be excluded. The piece that moves to the destination in a single unblockable leap (i.e. the Zebra) would be the compound of all 16 elementary lame and hopping pieces.

Of course it could still be a matter of taste which squares to consider as potential intermediates. E.g. for the (2,1) Knight move you could either only consider the (1,0) and (1,1) squares intermediates, but you could also include the (2,0) and (0,1) squares, allowing only orthogonal steps in the shortest path. And for a (3,1) Camel move you might want to include (2,2) and (1,-1), as it would still be possible to reach the destination over these with three diagonal steps.

And then there is the possibility to have moves that 'overshoot' the target. E.g. a (1,0) step that would go over (2,0), and would only be allowed if the latter is empty. In my variants Makromachy and Megalomachy I introduced 'airlift moves', which where slides that would overshoot their destination by 1 square, and would require that this square was occupied. So the pieces having this move would only be able to go to the furthest empty square on the path if that path was blocked by a piece rather than an edge. (A sort of 'Bouncing Grasshopper' move.) This to make it possible to transport slow pieces quickly over large distances on a huge board without increasing their value very much by adding a lot of mobility.

You forgot that a zebra can not be blocked in any case, even though a compound of the other pieces which could be blocked can.

The unblockable Zebra is not one of the 16 elementary pieces the move of which is dependent on the occupation of all 4 intermediate squares. Because it has a don't care for all these squares, not a "must be occupied" or "must be vacant". So it is one of the 81 possible elementary pieces that also include those dependent on the occupancy of only 3, 2, 1 or 0 intermediate squares.

But those 81 would indeed partly 'eclipse' each other when combined. Like you mention, the unblockable Zebra would eclipse anything you combine it with. But it is the elementary piece that depends on 0 intermediate squares, and is not in the set of 16 that depend on all 4 squares.

For the ease of understanding let me propose a name of a class of pieces, which I originally had in mind.
Let's call it lame leapers type I - just for now.

Let's simplify the scenario and invent the class of pieces that leap from d4 to e6, and let's only consider d5 and e5 as intermediate squares.

All pieces that have this leap and are potentially blocked on either d5 and e5 - but without interdependency between these squares - are in this class.
(I'm not even yet talking about the other directions e.g. d4 to f5 and so on, I just assume that lame leapers type I are defined symmetrical.)

Then this class has 4 pieces in it, because there are two options for each of the two intermediate squares:
1. Can always move
2. Can only move if d5 is vacant
3. Can only move if e5 is vacant
4. Can only move if both d5 and e5 are vacant

The first piece is the FIDE Knight, the second piece is the Xiangqi Knight and the 4rth piece is the Superchess Archer.

@aserew12 made the valuable remark that we can consider a third option, which is a mandatory occupied square.
That would give us a more extended class which we can call lame leapers type II, with 9 pieces:
1. Can always move
2. Can only move if e5 is vacant
3. Can only move if e5 is occupied
4. Can only move if d5 is vacant
5. Can only move if both d5 and e5 are vacant
6. Can only move if d5 is vacant and e5 is occupied
7. Can only move if d5 is occupied
8. Can only move if d5 is occupied and e5 is vacant
9. Can only move if both d5 and e5 are occupied

Now we could also invent another piece that can move from d4 to e6 only if:
(d5 is occupied and e5 is vacant) OR (d5 is vacant and e5 is occupied)This piece would not belong to the class of lame leapers type I or type II, because this is what I would call an interdependency between the intermediate squares.

Question is now:
How would an even more extended class of lame leapers be defined? How many pieces are in this class? And how is this calculated?

A piece that is common enough that it even has an established name is the Moo. Which can move if d5 OR e5 is vacant.

Your first list is one that classifies the intermediate squares either as "must be empty" or "don't care". There are indeed 4 of those in this case. But these are not the 'elementary moves' I was talking about. These are:

1. Can only move if e5 and d5 are both empty
2. Can only move if d5 is empty and e5 is occupied
3. Can only move if e5 is empty and d5 is occupied
4. Can only move if d5 and e5 are both occupied

The Knight is the compound of all four of those. The Mao is the compound of 1 and 2, The Moa of 1 and 3, and the Moo of 1, 2 and 3. But there is also the Mao-hopper, the compound of 3 and 4, etcetera. In total 16 possibilities, one of which cannot move at all.

The list of 9 is overdoing it; these cases cannot independently be enabled or disabled, as enabling some could eclipse many others. Any don't cares can be simulated as a compound of the 'elementaries', by taking both the case that has the square empty, and the one that only differs from it by requiring the square is occupied.

But the possibilities in this case are not limited to describing various degrees of lameness; it also covers hoppers and multi-hoppers. And if the hopping is not 'color blind' there would be even more possibilities. (Namely 3^2 = 9 elementaries, and 2^9 - 1 compounds.)

I get it now...

511 compounds

OK, so what you're doing is:
Each subset (S) of all intermediate squares (I), defines one elementary move from d4 to e6 while having exactly all squares in S occupied and all squares in I - S vacant.
If I has n squares then there are 2 ^ n such elementary moves.
So we can combine these elementary moves into variant pieces in 2 ^ (2 ^ n) ways.

In terms of Power sets each element of P(I) defines an elementary move, and each element of P(P(I)) defines a piece definition.
The piece that has no move at all is included in these piece definitions, and it's a matter of taste whether we want to allow that.

Indeed. I excluded the 'null piece' (Flag?) because initially we were talking about pieces that could reach the Zebra destinations, and a piece that never has any moves obviously cannot.

It is of course also a matter of taste whether one would consider d5 and e6 the only relevant squares for the d4-e6 Knight move. There are people that look upon a Knight move as going along an L-shaped orthogonal-steps-only trajectory, and to those people also putting requirements on the occupancy of d6 and e4 would seem natural.