Re: Dick Lipton and Ken Regan • (1) • (2)
I need to think a little about the context Lipton and Regan have wrapped around the Frankl Conjecture, if not exactly about the problem itself. This will be a scratch-worky post-in-progress (PIP❢), or a series of posts, and I may even delete it down the road if it runs out of gas or turns into too much nonsense.
Just to get started, let me post the passage from the 2nd of the above articles where I thought I spied something familiar, though it was late, past my bedtime, and I was pondering weak and weary, etc.
Let us use 
Obviously the following is true:
There must be a better notation than 
In order to understand the above formulas, we have to back up a little and pick up a couple of earlier definitions, rephrasing them slightly in the style of notation to which I would like you to become accustomed.
Let 
![\displaystyle I_{i}(f) = \mathrm{Prob}_{x}[ f(x) \neq f(x^{i}) ],](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+I_%7Bi%7D%28f%29+%3D+%5Cmathrm%7BProb%7D_%7Bx%7D%5B+f%28x%29+%5Cneq+f%28x%5E%7Bi%7D%29+%5D%2C&bg=ffffff&fg=000000&s=0&c=20201002)
where 
There is another way of expressing the bit flip operation 
Definition. Let the function 
![{[1, k]}](https://s0.wp.com/latex.php?latex=%7B%5B1%2C+k%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
It is conventional to write logical conjunction as multiplication, with or without a dot between conjuncts. It is also convenient in many contexts to write logical negation 
![\begin{matrix} \lnot_j (x_1, \ldots, x_j, \ldots, x_k) & = & x_1 x_2 \ldots x_{j-1} \tilde{x}_j x_{j+1} \ldots x_{k-1} x_k \\[4pt] & = & x_1 x_2 \ldots x_{j-1} \texttt{(} x_j \texttt{)} x_{j+1} \ldots x_{k-1} x_k \end{matrix}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Bmatrix%7D++%5Clnot_j+%28x_1%2C+%5Cldots%2C+x_j%2C+%5Cldots%2C+x_k%29++%26+%3D+%26++x_1+x_2+%5Cldots+x_%7Bj-1%7D+%5Ctilde%7Bx%7D_j+x_%7Bj%2B1%7D+%5Cldots+x_%7Bk-1%7D+x_k++%5C%5C%5B4pt%5D++%26+%3D+%26++x_1+x_2+%5Cldots+x_%7Bj-1%7D+%5Ctexttt%7B%28%7D+x_j+%5Ctexttt%7B%29%7D+x_%7Bj%2B1%7D+%5Cldots+x_%7Bk-1%7D+x_k++%5Cend%7Bmatrix%7D&bg=ffffff&fg=000000&s=0&c=20201002)
Visually speaking, 
This looks like a good place to break and start a second post in the series.
To be continued …
to integers, either non-negatives 
or positives 
I was especially interested in codings that were also surjective, thereby revealing something about the target domain of arithmetic.
and finite partial functions from 
All of this comes straight out of the primes factorizations. That type of bijection may remind some people of Dana Scott’s 
Corresponding to the positive integers there arose two species of graphical structures, which I dubbed “riffs” and “rotes”. See these links for more info:
Inquiry Into Inquiry 