# Backward records

Given a sequence $A=(a_1,\ a_2,\ \ldots)$ one can define the records of $A$ as numbers $a_n:n\in\mathbb{Z}^+$ such that $a_n>a_m$ whenever $n>m.$ So you start at 1 and write down every number larger than all preceding numbers. (Of course you could also look for record-small numbers in just the same way.)

Is there a standard term for the terms $a_n:n\in\mathbb{Z}^+$ such that $a_n<a_m$ for all $n<m$? Essentially, you start at $+\infty$ and walk down the positive integers, writing down every number which is smaller than all numbers you encountered first. (Alternately you could write down the ones which are larger, but either way the important point is that you walk down from infinity rather than up from 1.)

These are somewhat more tricky to work with since you can't prove membership by checking finitely many values. But they are often defined and useful. Surely there is a standard term for this somewhere; I'm loathe to invent terminology except when absolutely necessary.

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How about high-water and low-water marks? – Théophile Jul 31 '12 at 20:27
@Théophile: To me these suggest the two kinds of records in the first paragraph, where you're walking from 1 upward and looking for the smallest yet or the largest yet. – Charles Jul 31 '12 at 20:42
What you call records are sometimes called champions in the Number Theory literature. But I can't say I've ever come across a term for the other concept. – Gerry Myerson Aug 1 '12 at 2:09
@Charles Ah, right. I spent a few minutes just now thinking about a good prefix for high/low-water mark, but arrived at nothing but barbarisms. – Théophile Aug 1 '12 at 21:25
@Théophile: I appreciate that, thanks for your time. – Charles Aug 1 '12 at 21:52

There is precedent for the term anti-champion. Bukor, Filakovszky, and Toth, On the diophantine equation $x_1x_2\cdots x_n=h(n)(x_1+x_2+\cdots+x_n)$, Ann Math Silesiannae 12 (1998) 123-130, available here, write on page 128,
Denote by $f(n)$ the number of solution [sic] of $$x_1x_2\cdots x_n=n(x_1+x_2+\cdots+x_n),\qquad x_1\le x_2\le\cdots\le x_n.$$ A number $n$ is called a champion if $f(n)\gt f(m)$ for every $m\lt n$, it is called [an] anti-champion if for every $m\gt n\quad f(m)\gt f(n)$. It is true that the anti-champions are always primes?