# Studio/Rank

Rank specifies the behaviour of a verb on certain subarrays of its arguments, which for a rank-k verb are referred to as the k-cells. Each verb is assigned a rank, and a rank may be otherwise specified with the rank conjunction.

The verb `<` (box) is useful for illustrating rank. `<` boxes its argument, and has unbounded rank, which means it boxes its entire argument:

]n=. i.2 3 4 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 <n NB. box n (entire array) +-----------+ | 0 1 2 3| | 4 5 6 7| | 8 9 10 11| | | |12 13 14 15| |16 17 18 19| |20 21 22 23| +-----------+ <"2 n NB. box the 2-cells (tables) +---------+-----------+ |0 1 2 3|12 13 14 15| |4 5 6 7|16 17 18 19| |8 9 10 11|20 21 22 23| +---------+-----------+ <"1 n NB. box the 1-cells (rows) +-----------+-----------+-----------+ |0 1 2 3 |4 5 6 7 |8 9 10 11 | +-----------+-----------+-----------+ |12 13 14 15|16 17 18 19|20 21 22 23| +-----------+-----------+-----------+ <"0 n NB. box the 0-cells (atoms) +--+--+--+--+ |0 |1 |2 |3 | +--+--+--+--+ |4 |5 |6 |7 | +--+--+--+--+ |8 |9 |10|11| +--+--+--+--+ +--+--+--+--+ |12|13|14|15| +--+--+--+--+ |16|17|18|19| +--+--+--+--+ |20|21|22|23| +--+--+--+--+

Here is `+/` (sum), also of unbounded rank, applied to the same argument:

sum=. +/ sum n NB. sum n 12 14 16 18 20 22 24 26 28 30 32 34 sum"2 n NB. sum the 2-cells 12 15 18 21 48 51 54 57 sum"1 n NB. sum the 1-cells 6 22 38 54 70 86

Unbounded rank means maximum rank applicable, which in this case is 3, since n has rank 3. Thus (`sum n`), (`sum"3 n`) and (`sum"_ n`) are all the same. Negative rank `r` means positive rank `a+r` where `a` is the rank of the argument. Thus

sum"_1 n NB. sum 2-cells (3+_1) 12 15 18 21 48 51 54 57 sum"_2 n NB. sum 1-cells (3+_2) 6 22 38 54 70 86

The composition conjunctions `@` (atop) and `@:` (at) differ in the ranks of the result. The rank of `u@v` is the rank of `v`; the rank of `u@:v` is unbounded. A common mistake is to use the wrong one:

]n=. 1 2;10 20 +---+-----+ |1 2|10 20| +---+-----+ +/@> 1 2;10 20 NB. add up each item of n 3 30 NB. rank 0, since > has rank 0 +/@:> 1 2;10 20 NB. add up items of n (rank _) 11 22

Here rank is used to specify how two arrays are joined together:

]a=.'XYZ' XYZ ]b=. >;:'bat cat dat' bat cat dat a,b NB. append a to b XYZ bat cat dat a,"1 1 b NB. append rows of a to rows of b XYZbat XYZcat XYZdat a,"0 1 b NB. append atoms of a to rows of b Xbat Ycat Zdat a,"1 0 b NB. append rows of a to atoms of b XYZb XYZa XYZt XYZc XYZa XYZt XYZd XYZa XYZt a,"0 0 b NB. append atoms of a to atoms of b Xb Xa Xt Yc Ya Yt Zd Za Zt