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Spending 1-2 years on a research direction that in the end does not work for one or another reason is not at all uncommon in our profession. It is perhaps better if it first happens during doctoral studies, since at this point your supervisor and other more experienced colleagues can actually help you through it - at later stages in your life you may be more on your own when it happens. View it as a useful life experience; I am assuming from the fact that you are doing a doctorate in mathematics that your passion for mathematical research is strong enough to not let this discourage you.
What exactly worries you though? Let's think about the case $A_0$ being the ground field for a second. Under various finite dimensionality assumptions necessary for all this to make sense, the graded dual of an algebra is a coalgebra (and that is what is implicit in the first recipe when you split out the first tensor factor). All the information about duality can be 'packaged' by writing $Id_{A_1}\in End(A_1)\cong A_1^*\otimes A_1$ via a basis and a dual basis, which is precisely what is being used in the second recipe.
The MO question linked by Darij has a hidden/deleted answer by Gjergji Zaimi which does not solve this problem but indicates that the same answer appears in a similar problem about sequences of points in general position on the plane containing either an "$l$-cap" or a "$k$-cup" (natural notions expressed using slopes computed via differences of coordinates), and that problem is solved in the original paper of Erdős–Szekeres (numdam.org/item/?id=CM_1935__2__463_0). Probably you just have to do a careful search through lemmas and propositions of papers that cite Erdős–Szekeres...
@LSpice you honestly assumed that the authors had a say about anything relating to the English translation of their work back in the USSR? :) In any case, a quick look at Zelmanov's homepage should be convincing enough.
@LSpice : thank you for adding names of articles to the references I posted. Please note that the spelling of Zelmanov's name with an apostrophe only arises in translations (and with the extra apostrophe in "'s" looks truly absurd). I changed this back to the way it was; please do not edit it further.
What an interesting observation! I came across some sort of similar situations where this sort of behaviour ($n$ replaced by $1-n$, and at the same time, $x$ replaced by $-x$ in the generating function; THAT is a key hint) was a manifestation of Koszul duality for some algebraic objects. I'll definitely try to think about your observation with this in mind...
I see. Thank you for the explanation. Altogether implementing this from scratch is probably beyond my personal programming skills. Do you happen to know of any people who did implement similar things in the past?
Indeed, you are choosing to fill the matrix with elements of $\mathbb{Q}S_8$, so my second concern does not literally apply. But just because you decided to "hide" the large number in the elements of the matrix, does not change the fact that the presentation matrix is an element of a vector space whose dimension is in hundreds of thousands or rather millions, as far as the first step is concerned.
Also, and probably much more importantly, the binary trees have labelled leaves, so the number of binary trees with 8 leaves is (oeis.org/A001147) equal to 13!!=135135 - much more than 429. So I fear that the approach you suggest is highly unlikely to help.
Dear John, alas the presentation you write is not enough, these are not all consequences. For example, there are consequences of the identity $F(a_1,a_2,a_3,a_4)$ of the form $a_1F(a_2,T_3,T_4,T_5)$, where $T_3$, $T_4$, $T_5$ are some trees, etc. In other words, you can precompose and postcompose with something. So the computation is much bigger.
