DejMar

Your rules indicate the only operations permitted are concatenation ||, subtraction -, and division /. Is that correct? [spoiler=Or are these special operations and mathematical notations also permitted?]exponetiation (visually, does not require any special symbols when using simple superscription. The character symbols ^ and ** are only used when superscripting is not an option), tetration (the same is true for this operation, when superscripting the operation is alternately represented by ^^ or Knuth's double arrows. The other notation I inquire is whether one can use base notation. I am assuming you are using a modified interpretation of the concatenation operator, allowing it to concatenate the result of an operation and not simply two numerals. The operator precedence of the concatenation operator, in regards to the other operators, is last. Does this still hold true for this puzzle?

Is the task then to provide pseudo-code for an algorithm that correctly "maps" the maze using a virtual 'robot' as an exploration tool? And, though it must be preferred, is a most efficient algorithm - with the restraints given - required for the task?

Within the spoiler of my prior post, when I give the counts for the maximum length of a subsequence they are for that length and do not include the counts for lengths smaller. To be correct, each count should include the counts of the lesser length, but as indicated many, as yet, are unknown.

Here it is: https://en.m.wikipedia.org/wiki/Erdős–Szekeres_theorem Surprisingly, I think this might hold the key to this question. You could be right in that the theorem holds a key, but I am unsure on how to apply such for this problem. As to symmetry, it may apply to the number of ascending versus the number of descending subsequences, but not to the length of subsequences alone.

Sorry about that gavinksong. I didn't mean to attribute the quote incorrectly. The quoting structure is more complex than it should be, yet I will be more careful, as you request. I didn't remove any brackets. I used the quote tool on the toolbar, but, apparantly, it has changed with the upgrade(s) to the site. Unlike the Spoiler tag, it didn't provide anywhere for a reference to the quote -- except within the quote. i just picked the wrong name. And though the difference between 0 and 1 may seem to be trivial, it is not so trivial.

I apologize to gavinksong, he is correct. I guess lack of sleep had my mind thinking a bit off.

Why? The least number of apples Alice can place in a room is 0, not 1. 0 is a non-negative integer, and thus a valid option for Alice.

This is incorrect because the minimum requirement only advances three months at a time. It does not suddenly jump forward by a year.