Let us substitute: ( - x, ) - y
Thus ()() becomes xyxy
())( becomes xyyx
Now clearly it can be seen, even while high, that the second one is and the first isn’t
for those too lazy to google,
palindrome /păl′ĭn-drōm″/
noun
A word, verse, or sentence, that is the same when read backward or forward.
"madam; Hannah; or Lewd did I live, & evil I did dwel."
Calm down, everyone. Brackets form a tree structure, and can be represented by a free magma, while strings with concatenation are equivalent to a free monoid. You're essentially asking for the two respective common involutory operations to be connected by this map, just because they're involutory, which put that way is a wild guess at best. In fact, reversing this string produces something outside the range of the map entirely, which is injective and so can't be surjective for combinatorical reasons.
... Yeah I might be the only person that finds that useful.
the part about the "wild" guess is, but this is a counterexample, and something like the reciprocal vs the negative of reals or rationals when moved across the log map would be an example. So, either you're a galaxybrain that just instantly knows if the transformation is structure-preserving in that way, or you're guessing to some degree as well.
The symbols and abstractions have touched me in no-no ways. I miss okaybuddyphd on r*ddit, they knew the pain.
I suppose I could also just say that characters which aren't just drawn asymmetrical, but actually point in a direction as part of their function, look wrong when reversed like this. So, (e) -> )e( is no good, but bed -> deb is fine.
I was saying unipotent at first instead of involutory, which was actually the wrong jargon because of the context, but I've fixed that now. Yes, they're all real.
Map, although in this context I could probably have just said function. I go with map by default when thinking bidirectionally.
I think most people here will know combinatorics, the study of the different possible configurations of something. The number of n-length strings with two possible characters is 2n, as coders should all know, and the number of trees turns out to be Catalan numbers, many of which have prime factors other than 2. This is an injective map from n node trees to 2n character strings, so it's possible, but you'll (almost?) never get a perfect match, so by the pigeonhole principle it can't be surjective.
I'm wondering now if Catalan numbers are O(n!). The equation has a lot of n! but it also has a certain smell like it might depend on big or little o.
Edit: D'oh, they must grow no faster than 22n; I just wrote that. So, exponential.
Yes it does bother me a little that the letters in the latter half of my username can't be written backwards. (Well, some can, and the p can become a q, but then it's not a p any more.)
This is especially terrible when lying in bed. With a keyboard or pen and paper you can make sense of it, it hurts my brain a bit to visualize it though.
It's not complicated at all: https://en.m.wikipedia.org/wiki/Palindrome. Not really something that's education-specific, in this instance (though I suppose it's commonly used in entry-level programming classes since it's a simple concept).
Yea but I'm just generally dumb in a pool of smart people. Not like I'm using palindromes in everyday conversation so when I see it I gotta look it up. Like when I saw a Fibonacci sequence and mentioned that it looks like something I've seen before but couldn't remember where. This doesn't even touch on why the syntax mentioned is a palindrome 😆
It's curious, but in my mind these types of mathematical or logical visualizations are the sheep I count, trying to sense the deeper flow and patterns where the emergent oddities even out.
That's when I know I will soon fall fast asleep, when my mind starts getting abstract.