Idioms

Anyone may add idioms to the list. See also K by example.

Basic

Swap Variable Values	`x[2 5]:x[5 2]` `p:2 5; x[p]:x@\|p`
Coerce Array	`(),x`
Fill (right) to length	`fr: {@[y#z;!#x;:;x]}`
Fill (left) to length	`fl: {(#x)!@[y#z;!#x;:;x]}`

Numerical

Factorial */1+!:

Fibonacci {(x(|+\)\1 1)[;1]}

Fibonacci {x{x,+/-2#x}/!2}

Recursive Fibonacci {:[x<2;1;_f[x-1]+_f[x-2]]}

Recursive Fibonacci (memoized) {c::.();{v:c[a:`$$x];:[x<3;1;:[_n~v;c[a]:_f[x-1]+_f[x-2];v]]}x}

Maximum subsequence sum |/0(0|+)\

Positive divisors of n, inclusive {&~x!'!1+x}

Positive divisors of n, inclusive (faster) {d:&~x!'!1+_sqrt x;d,_ x%|d}

Reduced digit sum +/0$'$:

Collatz sequence (1<){:[x!2;1+3*x;_ x%2]}\

Collatz sequence (shorter) (1<){(_ x%2;1+3*x)x!2}\

Identity matrix

`{a=/:a:!x}`
`{(x,x)#1,x#0}`
`{(-!x)!\:1,(x-1)#0}`

GCD gcd:{:[~x;y;_f[y;x!y]]} gcd:{*{0<x 1}{{(y;x!y)}.(x)}/(x;y)}

LCM {_ x*y%gcd[x;y]}

Magnitude {_sqrt+/_sqr x}

Log base n logn: {(_log x)%_log y}

Vector Dot Product

`_dot`
`dot: {+/x*y}`

3×3 matrix determinant dt33:{-/{+/(*)/(!).'+(y*!3;x)}[x]'1 -1}

3-Vector Cross Product cross: {{-/{(*)/(!).'+(y*1+!2;x)}[x]'1 -1}2 3#x,y}

Unit Vector unit: {x%\:_sqrt x _dot x}

Primality

Primes to n, exclusive	`2_&{&/x!'2_!x}'!:`
Primes to n, sieve	`{2_&{:[x@y;x&@[1,-1_ z#(1_ y#1),0;y;:;1];x]}/[x#1;2_!__ceil_sqrt x;x]}`
Primes to n, sieve (Arthur 1)	`{:[x<4;,2;r,1_&~\|/x#'~!:'r: _f[_ _ceil _sqrt x]]}`
Is n a prime?	`isp: {(x>1)&&/x!'2_!1+_sqrt x} isp2: {2~#&~x!'!1+x}`
Next prime	`{~isp x}{x+1}/`
First n primes	`prsn: {(x-1){:[isp n:x+1;n;_f n]}\2}`
N’th prime	`prn: {*\|prsn x}`
Prime factors	pfac:{ n:2,3+2!.5_sqrt x d:n@&~x!'n m:d@&~1<+/~d!\:/:d p:,/{n:+/0=x!'y^1+!_ logn[x;y];n#y}[x;]'m :[_n~p;x;_dv[p,_ x%*/p;1]]}

Combinatorics

Pascal’s triangle with n+1 rows pasc: {x{+':0,x,0}\1}

Binomial {[n;k]pasc[n][n;k]}
{[n;k]i:!k-1;_*/(n-i)%i+1}

Is x a permutation vector? x~<<x

Random permutation of 0,…,n-1 n _draw -n

Nth permutation nperm: {y@{{<:<:y,x}/|x}'+({a-!a:#x}y)_vs(@x),:/x}

Permutation index {(a-!a:#x)_sv{+/x<*x}'(!#x)_\:x}

Permutations on n elements

`perm: {:[1<x;,/(>:'(x,x)#1,x#0)[;0,'1+_f x-1];,!x]}`
`perm2: {{x@<x}x{,/x(1!)\'x,'#*x}/,!0}`
`perm3: {nperm[!*/1+!x;!x]}`

Permutation cycles {{={x[y]@<x[y]}[x]'!#x}(x\'!#x)}

All permutations of x with the length y permlen:{x@v@&((y;1)~^=:)'v:!y##x}

Combinations, k out of n elements {{x@<x}@&:'k(?,/(1!)\'r,')/,(r:k=y)=&x-k:y&x-y}

Combinations, k out of n elements, unsorted, unoptimized {&:'y(?,/(1!)\'1,')/,&x-y}

Enumerate all values of length x in base y {(x#y)_vs!_ y^x}
{!x#y}

Bitstrings of length x {(x#2)_vs!_ 2^x}
{!x#2}
!2+&:

Power set {x@/:&:'2_vs!_2^#x}
{x[&:'!2+&#x]}

Are all values in the array different? {(#x)=#?x}

Sliding window {(!1+y-x)+\:!x}

Statistics

Arithmetic mean	`am: {(+/x)%#x}`
Harmonic mean	`%am@%:`
Geometric mean	`{(*/x)^%#x}`
Median	`{(x@<x)@_(#x)%2}`
Sample variance	`sv: {+/(_sqr x-am x)%#x}`
Standard deviation	`{_sqrt sv x}`
Histogram	`{c:w@*:'=w:x;@[(1+\|/c)#0;c;:;(#:'=w)]}`

Dates

Is x a leap year?	`{(+/~x!'4 100 400)!2}`
Day of the Week	`Mon`Tue`Wed`Thu`Fri`Sat`Sun@(_ _t%86400)!7

Sorting

Sort list	`x@<x`
Sort list	`x[<x]`
Sort list function	`{x@<x}`
Sort list descending	`x@>x`
Unique	`{x[*:'=x]}`
Non-unique elements	`{x@&1<#:'=x}`
Quicksort	`{f:*x@1?#x;:[0=#x;x;,/(_f x@&x<f;x@&x=f;_f x@&x>f)]}`

Selections

Overlapping infixes	`{:[y>#x;,x;x@(!y)+/:!(1-y)+#x]}`
Non-overlapping infixes	`{(y*!_ceil(#x)%y)_ x}`
Main diagonal	`{x ./:n,'n:!#x}`
Secondary diagonals	`{x ./:/:f@-1_1_=+/'f:,/n,/:\:n:!#x}`

Strings

Delete leading blanks	`dlb: {x@&\|\~x=" "}`
Delete trailing blanks	`{\|dlb@\|x}`
Delete multiple blanks	`{x@&a\|1_1!1,a:~x=" "}`
Left justify	`{(+/&\x=" ")!x}`
Right justify	`{(1-(x=" ")_sv 1)!x}`
Lower Case Letters	`lcase: _ci 97+!26`
Upper Case Letters	`ucase: _ci 65+!26`
To Lower Case	`{@[x;p;:;lcase@n@p:&26>n:ucase?/:x]}`
To Upper Case	`{@[x;p;:;ucase@n@p:&26>n:lcase?/:x]}`

Unsorted

Frequency table (sorted) {b@<b:(x@*:'a),'#:'a:=x}

Anagrams {x g@&1<#:'g:={x@<x}'x}

Rot 13 {a:+65 97+\:2 13#!26;_ci@[!256;a;:;|a]_ic x}

8-queens p@&{a:{(#x)=#?x};i:!#x;a[x[i]+i]&a[x[i]-i]}'(p:perm 8)
Variants:
p@&{&/{(#x)=#?x}'(x[i]+i;x[i]-i:!#x)}'(p:perm 8)
p@&{&/{(#x)=#?x}'(+;-).\:(x[i];i:!#x)}'(p:perm 8)

Perfect shuffle {x@<>(#x)#1 0}

Perfect shuffle (for 2*n)

`{{m:_((#x)%2);,/+(x[!m];x[m+!m])}\(!x*2)}`
`{x@<>(#x)#1 0}\!:2*`

Life (Knuth’s approach)

`{m:x;m:m+(1!'m)+-1!'m;m:m+(1!m)+-1!m;m:m+m-x;m _lin\:5 6 7}`
`{((2*+/,/2{-1 0 1!'\:x}/x)-x)_lin\:5 6 7}`
`{\|/(1;x)&3 4=\:+/,/2{-1 0 1!'\:x}/x}`

One dimensional cellular automata (rule 104) "_X"@{2=+/(0,x,0)@(!#x)+/:!3}\0 1 1 1 0 1 1 0 1 0 1 0 1 0 1 0 0 1 0 0