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@Teo: I have a feel for how standard "cd" is/is not, and point out that, two sentences later, mentioned performing searches with no notations. These are examples, not an exhaustive list of what all I did.
@Geoff Robinson: While I've used variants of that first paragraph, I've never seen one this succinct nor this broad before. Thanks especially for that.
To record for posterity, the result can be made stronger with a little less work. We can prove @Martin Rubey's observation directly (alternating (k), (n-k), (k+1), (n-k-1), etc. yields a strictly increasing sequence) using only the second and third computation.
