# Grade Down of Grade Up

## March 21, 2023

One of the amazing things about old APL books is that no matter how many years have passed, no matter how many new language features have been added, there are invariably many useful nuggets to be gleaned from them. A lot of very smart people used APL back in the day.

Consider the case of `⍒⍋`

and its twin `⍒⍒`

. A while back on the APL Farm there was a discussion of the interpretation and usefulness of these combinations, which culminated with:

So the conclusion is that if we are only dealing in permutations

`⍒⍋`

is worthless since it's simply a slower`⌽`

, and if we are not dealing in permutations even Adám doesn't know what`⍒⍋`

does so it is probably equally worthless.

I read this at the time with an uneasy feeling that somewhere, sometime, I had a use for `⍒⍋`

, but promptly forgot about the whole thing. And then just the other day I was refactoring a test for a function that computes the average rank. And lo and behold:

```
averageRankUp←{0.5×(⍋⍋⍵)+⍒⍋⌽⍵}
```

and its sibling:

```
averageRankDown←{0.5×(⍋⍒⍵)+⍒⍒⌽⍵}
```

I wonder if and how average rank could be computed as efficiently or succinctly without `⍒⍋`

. I know it can be computed using two outer products, but that is very expensive in time, space, and tokens. I also wonder if there are any other documented uses of these pairings. Given the discussion in the APL Farm where it was observed that `⍒⍋p`

is `⌽p`

for permutation `p`

, it is probably no accident that this application of `⍒⍋`

involves `⌽`

.

I was pretty sure I did not come up with these expressions on my own, and a short search of my library revealed that I had most likely lifted it from APL2 in Depth , a 1995 text by legends Norman Thompson and Ray Polivka. Don't get rid of those old APL books!