Genady

1. You need to derive the equations for generic variables r and n. Working with specific values, such as r=4 and n=2, will not give you a general derivation. 2. The upper limit of summation is still wrong. 3. I don't know where the second part of these two simple sums came from. From nowhere? 4. You still did not derive equation (1). I've given you a fat hint for that. If you derive it, you will see where your summation limit is wrong.

Here you go:

It is not the order of summation you need to change. It is the upper limit of summation.

I think, the difference appears because of the problem which I've pointed to earlier:

Yes, both equations (1) and (2) relate to distinguishable objects and distinguishable cells. Eq (1) is recursive. That is, assume that there are A(x, n) ways to distribute x distinguishable objects between n distinguishable cells, with no empty cells. Knowing this for any number of distinguishable objects x > n, we want to find A(r, n+1), number of ways to distribute r distinguishable objects between n+1 distinguishable cells, with no empty cells. Start thinking about it in the following way. Take one of the n+1 cells aside. Let's call it a "new" cell. You have one new cell and n old cells. You have to put some number of objects, k>0, in the new cell. The rest r-k objects will go into the n old cells. This can be done in A(r-k, n) ways, the number that we assume we know. Can you continue from here so we get the expression for A(r, n+1), which is the eq (1)?

In the eq (1) here your upper limit of summation is k=r. Are you sure about it? I don't think it is correct. The case (2) doesn't make sense. You cannot distribute 2 objects in 3 cells such that there are no empty cells. I think, you have to have at least as many objects as there are cells for these formulas to make sense. That's why I think that the upper limit of summation in eq (1) is incorrect. Did you derive the eq (1) by a combinatorial argument, as requested?

You can substitute A(r-k, n) in eq.(1) using the right side of eq.(2) while instead of (n-v)r you should put (n-v)r-k. Let's see what you get.

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Yes, this formula is correct. However, the equation (2) is not. Take for example r=2 objects and n=2 cells. There is only 1 way to distribute them with no empty cells, i.e. 1+1. But the equation (2) gives 2. This would be so if, for example, the objects were distinguishable.

OK. Now, the first formula in the OP says, A(r,n) = C(r-1,n-1) Where did it come from?

Here is your other post: Of course v or k doesn't make a difference. There is an actual difference between (2) above and the second formula in the OP: Do you see it? Plus, I don't see anything there about the equivalence you're talking about here.

I just did. The equation (2) there is very similar although not identical to your formula 2 here. And, nothing there says that the two formulas in the OP are equivalent, does it?

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I don't know about this, but they are different for r=n=2. What is the source of the statement above?

They are not equivalent.

Yes, you did say splicing. It is my fault that I've missed it. I'm sorry. In my misunderstanding, I thought that X,Y,Z in your strings stand for arbitrary nucleotide bases. Now, after you've mentioned "hypothetical protein", I think they stand for arbitrary amino acids. Is this correct? Are we talking about mRNA splicing that occur after transcription and before translation? When introns get excised and exons get connected? I don't know about any redundancy associated with this splicing. I do know that alternative splicing is used to make different polypeptides from the same DNA sequence. This is somewhat opposite to redundancy.

They are not the same. There is no cyclic symmetry. Your example looks like a frameshift mutation, which would lead to a very different peptide chain. Frameshift mutation - Wikipedia

Looks good!

In fact, there is a lot of redundancy, i.e. the same amino acid being coded by several mRNA codons. There are 64 codons coding for only 20 amino acids: There seems to be quite a few modified bases occurring naturally, defined chemically. Here is a recent review: Natural, modified DNA bases - ScienceDirect

I don't see how fermentation could be "the oldest", because it uses organic compounds, which had to be produced from an inorganic matter first. Chemolithotrophs could've done that, for example.

Generally, yes. I don't think it is so for all people, all blanks.

Yes, maybe. Anyway, I hope that if they err, then only by undercounting.

I suspect that the corals were spawning and the fish were feasting on the eggs which were slowly floating upward.

Sometimes it seems to be a self-inflicted impairment. Just yesterday I had a phone call from a Russian acquaintance living long time in the US. Evidently, she didn't see the program that gave the nuclear bomb reasoning for shelling Kharkov, and started telling me about the Ukrainian ultra-nationalists shelling their own people to create the anti-Russian public opinion. On my questions, she said that Russian TV is the only truthful source of information about what is going on there, and all the rest is just fake news. (I've told her to never call me again and hang up.)

Right, no more doubts and confusion, everything is crystal clear now. Жить стало лучше, жить стало веселей.