A Young diagram , also called Ferrers diagram, is a graphical representation of an unordered integer partition ). If is a partition of then the Young diagram has boxes. A partition can be addressed as a multiset over .
There are two widely used such representations. The English one uses matrix-like indices, and the French one uses Cartesian coordinate-like indices for the boxes in the diagram .
In the English representation the boxes are adjusted to the north-west in the 4th quadrant of a 2-dimensional Cartesian coordinate system, with the ‘y’-axis being downward oriented. For instance the diagram representing the partition of is given in the English representation as:
Let be the set of Young diagrams. Important functions on Young diagrams include:
conjugation: denoted by a prime reflects the Young diagram along its main diagonal (north-west to south-east). In the above example the conjugated partition would be .
weight: provides the number of boxes.
length: provides the number of rows or equivalently the number of positive parts of the partition . The length of the conjugated diagram gives the number of columns.
times: the unordered union of the multisets. It follows that .
A filling of a Young diagram with elements from a set is called a Young tableau.
Skew Young diagram
A generalization of a Young diagram is a skew Young diagram. Let be two partitions, and let be defined as (possibly adding trailing zeros). The skew Young diagram is given by the Young diagram with all boxes belonging to when superimposed removed. If and then looks like:
A skew diagram is called connected if all boxes share an edge.
A skew diagram is called a horizontal strip if every column contains at most one box.
A skew diagram is called a vertical strip if every row contains at most one box.
conjugation, weight, length extend to skew diagrams accordingly.
For a quick online introduction to Young diagrams, try:
Alexander Yong, What is … a Young Tableau, Notices of the American Mathematical Society54 (February 2007), 240–241. (pdf)
A nice introduction to Young diagrams can be found here:
William Fulton and Joe Harris, Representation Theory: a First Course, Springer, Berlin, 1991.
A more detailed reference is:
William Fulton, Young Tableaux, with Applications to Representation Theory and Geometry, Cambridge U. Press, 1997.
A connection to algebraic geometry:
C. de Concini,D. Eisenbud, C. Procesi, Young diagrams and determinantal varieties, Invent. Math. 56 (1980), 129-165.