Re: Inequality with max I want to understand

Hi,

Thanks to the people who responded to this. I think I understand
what’s going on now.
A general question for anyone still reading. When I meet something
like this
inequality that I don't understand in a paper should I try to prove it
myself
or try to find some reference material that might have it described?
A few times I have spent many days on single lines like this trying to
understand
where what the author did came from.

For example in Knuth volume 2 3rd edition 19th printing I was looking
at the
answer to question 27 in section 4.6.3. The first line makes an
assumption
that a particular function (c(r)) is monotonically increasing. That’s
not proved
or even stated in the text. I spent quite a bit of time trying to
understand why
this was true. It also makes an assumption that another sequence is
monotonically
increasing. I asked one mathematician about this and he said he
thought it was
obvious. Not to me though!

For those interested c(r) is built on a sequence of values from a
function l(n)
with the following properties:

l(1) = 0, l(n+1) <= l(n) + 1, l(n) >= 0, n >= 1.

c(r) is defined as the first n with l(n) = r. So l(c(r)) = r.

So with this I was able to convince myself (prove even) that c(r) < c(r
+1)
and the same property for the other sequence. It became obvious to me
then by
thinking of l(n) as walking up stairs were you can only go up one
stair at a
time but you can slip down multiple stairs.

So I started thinking that there are likely books that contain things
like
this and maybe I should be reading them rather than trying to fill in
the
gaps myself. This particular case payed off as there are two problems
in the answer to this question the Knuth says he will write checks for
(the brackets are wrong and the text in the question is now out of
date
with the tables in the book).
Neill.
.

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