Editing Young tableau (section)

=== Diagrams <!-- [[Young diagram]] currently redirects to this section]]--> ===
[[Image:Young diagram for 541 partition.svg|thumb|right|150px|Young diagram of shape (5, 4, 1), English notation]]
[[Image:Young diagram for 541 partition-French.svg|thumb|right|150px|Young diagram of shape (5, 4, 1), French notation]]

A '''Young diagram''' (also called a [[Ferrers diagram]], particularly when represented using dots) is a finite collection of boxes, or cells, arranged in left-justified rows, with the row lengths in non-increasing order. Listing the number of boxes in each row gives a [[integer partition|partition]] {{mvar|''λ''}} of a non-negative integer {{mvar|''n''}}, the total number of boxes of the diagram. The Young diagram is said to be of shape {{mvar|''λ''}}, and it carries the same information as that partition. Containment of one Young diagram in another defines a [[partial ordering]] on the set of all partitions, which is in fact a [[lattice (order)|lattice]] structure, known as [[Young's lattice]]. Listing the number of boxes of a Young diagram in each column gives another partition, the '''conjugate''' or ''transpose'' partition of {{mvar|''λ''}}; one obtains a Young diagram of that shape by reflecting the original diagram along its main diagonal.

There is almost universal agreement that in labeling boxes of Young diagrams by pairs of integers, the first index selects the row of the diagram, and the second index selects the box within the row. Nevertheless, two distinct conventions exist to display these diagrams, and consequently tableaux: the first places each row below the previous one, the second stacks each row on top of the previous one. Since the former convention is mainly used by [[English-speaking world|Anglophones]] while the latter is often preferred by [[Francophone]]s, it is customary to refer to these conventions respectively as the ''English notation'' and the ''French notation''; for instance, in his book on [[symmetric function]]s, [[Ian G. Macdonald|Macdonald]] advises readers preferring the French convention to "read this book upside down in a mirror" (Macdonald 1979, p.&nbsp;2). This nomenclature probably started out as jocular. The English notation corresponds to the one universally used for matrices, while the French notation is closer to the convention of [[Cartesian coordinates]]; however, French notation differs from that convention by placing the vertical coordinate first. The figure on the right shows, using the English notation, the Young diagram corresponding to the partition (5, 4, 1) of the number 10. The conjugate partition, measuring the column lengths, is (3, 2, 2, 2, 1).

==== Arm and leg length ====
In many applications, for example when defining [[Jack function]]s, it is convenient to define the '''arm length''' ''a''<sub>λ</sub>(''s'') of a box ''s'' as the number of boxes to the right of ''s'' in the diagram λ in English notation. Similarly, the '''leg length''' ''l''<sub>λ</sub>(''s'') is the number of boxes below ''s''. The '''hook length''' of a box ''s'' is the number of boxes to the right of ''s'' or below ''s'' in English notation, including the box ''s'' itself; in other words, the hook length is ''a''<sub>λ</sub>(''s'') + ''l''<sub>λ</sub>(''s'') + 1.