# Ken Iverson’s Favourite APL Expression?

What was Ken Iverson’s favourite APL expression? I don’t know that he had one and if he had I don’t know what it was, but if I have to guess …

The expression `(0,x)+(x,0)` or its commute, which generates the next set of binomial coefficients, is present in the document that introduced APL\360 in 1967 [20, Fig.1] and the one that introduced J in 1990 [21, Gc&Gd]; in Elementary Functions: An Algorithmic Treatment in 1966 [22, p.69], in APL\360 User’s Manual in 1968 [23, A.5], in Algebra: An Algorithmic Treatment in 1972 [24, p.141], in Introducing APL to Teachers in 1972 [25, p.22], in An Introduction to APL for Scientists and Engineers in 1973 [26, p.19], in Elementary Analysis in 1976 [27, ex.1.68], in Programming Style in APL in 1978 [28, §6], in Notation as a Tool of Thought in 1980 [29, A.3], in A Dictionary of APL in 1987 [30, `m∇n`], and probably others.

The expression in action:

``````   ⎕←x←,1
1
⎕←x←(0,x)+(x,0)
1 1
⎕←x←(0,x)+(x,0)
1 2 1
⎕←x←(0,x)+(x,0)
1 3 3 1
⎕←x←(0,x)+(x,0)
1 4 6 4 1
⎕←x←(0,x)+(x,0)
1 5 10 10 5 1
⎕←x←(0,x)+(x,0)
1 6 15 20 15 6 1
``````

It is easily seen from the expression that the `n`-th vector of binomial coefficients is a palindrome and that its sum is `2*n`.