is a complex matter, which attracted much attention in the past [8, 9].
Cerf, "The three dimensional polyominoes
of minimal area," Electronic Journal of Combinatorics, vol.
A comparison study of polyominoes
explorations in a physical and virtual manipulative environment.
Just like polyominoes
, congruent situations are considered to be the same.
, which are the basis for thousands of mathematical puzzles, are shapes that cover connected squares on a checkerboard.
are classified according to how many cells they have: monomino, domino, tromino, etc.
We might think laterally and link the word 'tile' with Polyominoes
, or Polyiamonds, as in Solomon Golomb's game of Pentominoes, or variants of Blokus (Figure 3).
He fully described polyominoes
and pentominoes while a graduate student at Harvard where he obtained his Ph.D.
Major index for 01-fillings of moon polyominoes
. Journal of Combinatorial Theory, 117(8):1058-1081, 2010.
The last equality follows from the fact that for polyominoes
the difference between the number of salient and reflex angle is exactly 4.
* The Geometry Junkyard, Polyominoes
and Other Animals: www.ics.uci.edu/~eppstein/junkyard/ polyomino.html
It is well known that Catalan numbers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] enumerate many combinatorial objects, such as binary trees and parallelogram polyominoes
. Several bijective proofs in the literature show that parallelogram polyominoes
are enumerated by Catalan numbers, the two most classical being Delest-Viennot's bijection with Dyck paths [DV84] and Viennot's bijection with bicolored Motzkin paths [DV84].