This is a good question which may be even more well received on Mathematics StackExchange. After a couple of days, if you don't get satisfaction here, I recommend posting it at math.stackexchange.com with a link to this question. Also, you might find it useful to fix n to 1, 2 , 3 successively and doing computations using those constraints. Of course, an upper bound will be suggested by letting x be an extreme element and A an extreme antichain, and the literature will support your work above. Gerhard "Post Your Computational Result Summary" Paseman, 2015.09.02

For finite lattices, one could consider a ratio such as length of a maximal antichain to the number of all lattice members, or a similar ratio involving a disjoint union of all disjoint antichains of maximal length. The latter means that a chain is also very flat. Gerhard "Lay It On Its Side" Paseman, 2015.08.31

23252a232b2_23272_232_25232a2723252b232 . Length 39. a and b represent the primes 11 and 13. Fill in the three blanks with 17, 19, and 23. Use Chinese Remainder Theorem to calculate the numbers corresponding to the digits, or lookup Westzynthius. Gerhard "And There Are Larger Examples" Paseman, 2015.08.30

There is likely an example computed already (I know Westzynthius has one in his paper for primes up to 23). The best link I have right now is mathoverflow.net/questions/49400/question-in-prime-numbers . Gerhard "Yes, It's Been Done Before" Paseman, 2015.08.29

Thank you for the clarifying edit. Gerhard "Needs A Universal Algebra Brushup" Paseman, 2015.08.28

Others have thought that was the upper bound, and were wrong. It may not be the best way, but it is (according to Terry Tao and others in their December ArXiv preprint) the way used by most researchers on lower bounds of prime gaps. If you edit the question to reflect the viewpoint I suggested above, it might be closed as a duplicate on this forum. If you edit it to show clearly a key idea or insight using your definition, I am willing to consider it. Gerhard "Always Looking For New Ideas" Paseman, 2015.08.29

Oh, and I suspect the maximum length is bounded from above (for all but finitely many $a$) by $p_a(\log p_a)^2$, and more likely by something not much larger than $p_a\log p_a$. Gerhard "Speaking From Somewhat Limited Experience" Paseman, 2015.08.29

I am still having trouble understanding your definition of denizen. To me it looks like you are looking at sequences of the form L(a+1),L(a+2),...,L(a+m) where a and m are positive integers and L() is the least prime factor. If so, there are better ways of characterising denizens, and there is literature and some computation involving longer denizens. Search on this forum for "Westzynthius" for some detail. Gerhard "Good To Be Back Home" Paseman, 2015.08.29

If one were able to nearly cover a large S_m by embeddings of a smaller S_{2^n}, that would help give you some idea as to how distributive S_m can be. Even knowing how S_2 embeds in S_5 might help in figuring out f(n) in general. Gerhard "Also Look At Their Clones" Paseman, 2015.08.09

You might be able to show by induction that the permutation (1,n/2)(2,n/2+1)...(n/2-1,n) or something similar is at near maximal distance in the graph from the identity. Even if not, constructing parts of the graph for n up to 5 should give some more clues. Gerhard "Examine The Output More Closely" Paseman, 2015.08.08

Are these algebras rigid? Primal? Can you "nearly" embed S_n into a larger S_m ? What literature on Laver tables have you not read yet? Gerhard "Always More Questions Than Answers" Paseman, 2015.08.08

Actually, we need to make exceptions in the assumption for p=2 (so k=1) and p=3 (so k=2). However, for larger primes p, small even values of k seem to work. Gerhard "Except For Finitely Many Exceptions" Paseman, 2015.08.03

I don't know that you have anything more special than two bilinear operators (convolutions) over (the countable power of) a group. If you have some equational relations involving the two convolutions, you might be looking at something special which has been studied before. There are forms (cf. Movsisyan) of "hyperdistributivity", where certain distributive relations are presumed to hold, but your convolutions may not fit that. Gerhard "Consider The Free Such Algebra?" Paseman, 2015.08.03

Just to be clear, you should affirm or deny something like the following: Each coin in the bag is two-sided (one side heads, the other not heads) and each coin d is weighted with probability p_d landing heads up after a flip. Then you can later tie p_d to D(p) as needed. Gerhard "Unless There's Something More Clear" Paseman, 2015.08.02

The question as it stands implicitly asks for references on a topic in computability theory which could be started in an undergraduate class, but not finished there. It also asks (in an interesting way) for that part of the state of the art where contributions can be made. If amateurs found a way to, say, classify state machines so that some classes are recognizably non halting, that would help in mathematical logic and elsewhere. I am struggling to understand why this question is "off-topic". Gerhard "Please Explain Or Please Reopen" Paseman, 2015.07.28

Right now there is no polynomial guarantee that such a bound exists. The Hardy-Littlewood conjectures concerning prime k-tuples and associated calculations can tell you how many such are expected to appear (something like $Cx/(\log x)^{2n}$ for an effectively computable C ). If you are willing to take those conjectures into account, you may get what you want. Gerhard "And Hopefully What You Need" Paseman, 2015.07.27