Editing Young tableau (section)

=== Skew tableaux ===
[[Image:Skew tableau 5422-21.svg|thumb|right|150px|Skew tableau of shape (5, 4, 2, 2) / (2, 1), English notation]]

A '''skew shape''' is a pair of partitions ({{math|''λ''}}, {{math|''μ''}}) such that the Young diagram of {{math|''λ''}} contains the Young diagram of {{math|''μ''}}; it is denoted by {{math|''λ''/''μ''}}.  If {{math|''λ'' {{=}} (''λ''<sub>1</sub>, ''λ''<sub>2</sub>, ...)}} and {{math|''μ'' {{=}} (''μ''<sub>1</sub>, ''μ''<sub>2</sub>, ...)}}, then the containment of diagrams means that {{math|''μ''<sub>''i''</sub>&nbsp;≤&nbsp;''λ''<sub>''i''</sub>}} for all {{mvar|i}}. The '''skew diagram''' of a skew shape {{math|''λ''/''μ''}} is the set-theoretic difference of the Young diagrams of {{mvar|''λ''}} and {{mvar|''μ''}}: the set of squares that belong to the diagram of {{mvar|''λ''}} but not to that of {{mvar|''μ''}}. A '''skew tableau''' of shape {{math|''λ''/''μ''}} is obtained by filling the squares of the corresponding skew diagram; such a tableau is semistandard if entries increase weakly along each row, and increase strictly down each column, and it is standard if moreover all numbers from 1 to the number of squares of the skew diagram occur exactly once. While the map from partitions to their Young diagrams is injective, this is not the case for the map from skew shapes to skew diagrams;<ref>For instance the skew diagram consisting of a single square at position (2,4) can be obtained by removing the diagram of {{math|''μ''&nbsp;{{=}}&nbsp;(5,3,2,1)}} from the one of {{math|''λ'' {{=}} (5,4,2,1)}}, but also in (infinitely) many other ways. In general any skew diagram whose set of non-empty rows (or of non-empty columns) is not contiguous or does not contain the first row (respectively column) will be associated to more than one skew shape.</ref> therefore the shape of a skew diagram cannot always be determined from the set of filled squares only. Although many properties of skew tableaux only depend on the filled squares, some operations defined on them do require explicit knowledge of {{mvar|''λ''}} and {{mvar|''μ''}}, so it is important that skew tableaux do record this information: two distinct skew tableaux may differ only in their shape, while they occupy the same set of squares, each filled with the same entries.<ref>A somewhat similar situation arises for matrices: the 3-by-0 matrix {{mvar|''A''}} must be distinguished from the 0-by-3 matrix {{mvar|''B''}}, since {{math|''AB''}} is a 3-by-3 (zero) matrix while {{math|''BA''}} is the 0-by-0 matrix, but both {{mvar|''A''}} and {{mvar|''B''}} have the same (empty) set of entries; for skew tableaux however such distinction is necessary even in cases where the set of entries is not empty.</ref> Young tableaux can be identified with skew tableaux in which {{mvar|''μ''}} is the empty partition (0) (the unique partition of 0).

Any skew semistandard tableau {{mvar|''T''}} of shape {{math|''λ''/''μ''}} with positive integer entries gives rise to a sequence of partitions (or Young diagrams), by starting with {{mvar|''μ''}}, and taking for the partition {{mvar|''i''}} places further in the sequence the one whose diagram is obtained from that of {{mvar|''μ''}} by adding all the boxes that contain a value&nbsp; ≤&nbsp;{{mvar|''i''}} in {{mvar|''T''}}; this partition eventually becomes equal to&nbsp;{{mvar|''λ''}}. Any pair of successive shapes in such a sequence is a skew shape whose diagram contains at most one box in each column; such shapes are called '''horizontal strips'''. This sequence of partitions completely determines {{mvar|''T''}}, and it is in fact possible to define (skew) semistandard tableaux as such sequences, as is done by Macdonald (Macdonald 1979, p.&nbsp;4). This definition incorporates the partitions {{mvar|''λ''}} and {{mvar|''μ''}} in the data comprising the skew tableau.