This is the solution to the first problem in http://studentcompetitions-general.s3.amazonaws.com/dyalog-apl/phase_2/2013%20APL%20Problem%20Solving%20Competition%20Phase%20II%20Problems.pdf

Polyominoes.apl

⍝ The canonical representation.
⍝ The matrix with width <= height and has
⍝ the smalelst value if expressed as a binary number. 
CanonPoly←{
     ⍝ Remove zero rows
     rm←{(⍵∨.≠0)⌿⍵}
     allMatrix←{
     ⍝ Every mutation of the matrix(reflection and rotation)
         allHorMat←{(⍵)(⍉⊖⍉⊖⍵)(⊖⍵)(⍉⊖⍉⍵)}
         allVerMat←{(⍉⊖⍵)(⊖⍉⍵)(⍉⍵)(⊖⍉⊖⍵)}
     ⍝ We don't have to compute half of the matrices if
     ⍝ height doesn't equal to width.
         >/⍴⍵:allHorMat ⍵
         </⍴⍵:allVerMat ⍵
         (allHorMat ⍵),allVerMat ⍵
     }
     m←allMatrix rm⍉rm ⍵
     ⍝ Represent the matrices as a binary number, and pick the smallest.
     ⊃m[⊃⍒{2⊥,⍵}¨m]
 }
 
⍝ match a matrix in every direction. 
⍝ Implemented by rotate the matrix in each direction, do the match, and rotate back.
Match ← {(⍺⍷⍵)∨(⊖⍉⍺⍷⍉⊖⍵)∨(⊖⍉⊖⍉⍺⍷⍉⊖⍉⊖⍵)∨(⍉⊖⍺⍷⊖⍉⍵)} 

⍝ Two polyominoes are smiliar if and only if
⍝ they have the same canonical representation.
SimilarPoly ← {(CanonPoly ⍺) ≡ CanonPoly ⍵}
 
ValidPoly ← {
⍝ Expected input MUST BE a boolean matrix, otherwise undefined behavior.
⍝ First set a 1 to 2, then expand the area of 2 by BFS. 
bfs ← {({(1 2 Match ⍵)+⍵}⍣(+/+/⍵))⍵}
⍝ The matrix represents a polyominoes if there is no 1 in the resulting matrix.
~(1 ∊ bfs ⍵+((⍴⍵)⍴<\,⍵))
}
 
PolyGen←{
⍝ Add a border of 0's.
     br←{¯1⊖(⍵⍪0)⍪0}
⍝ remove duplicates.
     nub←{((⍵⍳⍵)≡¨⍳⍴⍵)/⍵}
⍝ matrix with exactly one 1.
     basis←{((×/⍵),⍵)⍴(1,(×/⍵)⍴0)}
⍝ Possible positions to add a new cell.
     deltas←{(,(0 1 Match ⍵))⌿basis(⍴⍵)}
⍝ all possible polyominoes within one move from input polyomino
     NextPoly←{nub CanonPoly¨⊂[2,3](deltas ⍵)+[2,3]⍵}
⍝ The set of polyominoes generated by adding one more cell to them
     NextPolys←{nub⊃,/NextPoly¨br¨⍉¨br¨⍵}
⍝ Iterate NextPolys, starting from a set of monomino.
     (NextPolys⍣(⍵-1)),⊂(1 1⍴1)
}