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Let  be a Young diagram. A filling of  is a labelling of the boxes in  by positive integers. For example, consider the Young diagram with shape
 .
One filling of this Young diagram is
A filling is a Young tableau if it includes each label from  to  exactly once. One Young tableau with shape  is
Each Young tableau with shape
 corresponds to a set partition of
![$ [n]=\{1,\dots,n\}$](http://images.planetmath.org:8080/cache/objects/9039/l2h/img12.png "$ [n]=\{1,\dots,n\}$") .
A filling is a semi-standard tableau if the labels monotonically increase in each row and strictly increase in each column. One semi-standard tableau with shape  is
Finally, a semi-standard tableau is a standard Young tableau if it includes each label from  to  exactly once. Hence a standard Young tableau is both a semi-standard tableau and a Young tableau. One standard Young tableau with shape  is
There is some variation in this terminology. For example, Fulton uses the terms tableau and Young tableau interchangeably for what we call a semi-standard Young tableau.
- 1
- William Fulton. Young tableaux: with applications to representation theory and geometry. Cambridge University Press, 1997.
- 2
- Bruce E. Sagan. The symmetric group: representations, combinatorial algorithms, and symmetric functions, 2nd ed. Springer, 2001.
- 3
- Richard P. Stanley. Enumerative combinatorics, volume 2. Cambridge University Press, 1999.
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