Size of partition:

Shape of partition:

Number of permutations with this shape:

Number of standard tableaux with this shape:

Number of semistandard tableaux with this shape and maximum entry   : 1

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About Young Diagrams

Young diagrams are a way of encoding integer partitions. A partition of an integer \(N\) is a tuple of positive integers \((p_1, p_2, \ldots, p_\ell)\), written in weakly decreasing order, that add up to \(N\). The numbers in this list are called parts of the partition. The Young diagram corresponding to such a partition will have \(p_1\) boxes in the first row, \(p_2\) boxes in the second, etc. A diagram constructed in this way will have rows that weakly decrease from top to bottom, and columns that weakly decrease from left to right.

The size of a partition is the integer \(N\). The shape of the partition is the tuple \((p_1, p_2, \ldots, p_\ell)\). The hook length of a box \(b\) in a Young diagram, denoted hook(\(b\)), is the total number of boxes to the right of and below that box, plus one for the box itself.

If a permutation is written in cycle notation, its shape is the tuple composed of the lengths of each cycle, written in decreasing order. For instance, the shape of the permutation \[ (1 \ 4 \ 7)(2 \ 5 \ 6)(3 \ 8)(9) \] is \((3,3,2,1)\), since the permutation has two cycles of length 3, one cycle of length 2, and one cycle of length 1. More than one permutation may have the same shape. We can also write the shape of a permutation can also be written \((1^{a_1}, 2^{a_2}, 3^{a_3}, \ldots )\), where the permutation has \(a_1\) cycles of length 1, \(a_2\) cycles of length 2, etc. With this notation, we can calculate the number of permutations on \(n\) letters with shape \((1^{a_1}, 2^{a_2}, 3^{a_3}, \ldots )\) using the formula \[ \frac{n!}{1^{a_1}a_1!2^{a_2}a_2!3^{a_3}a_3!\ldots } \]
A standard tableau is a Young diagram with \(N\) boxes, where each box is filled with a distinct number from the set \( \{1,2,3,\ldots,N\} \) such that the numbers strictly increase from left to right across each row, and strictly increase from top to bottom along each column. The number of standard tableaux with a given shape \((p_1, p_2, \ldots, p_\ell)\) with \(N\) boxes can be computed by the hook length formula \[ \frac{N!}{\prod_{b \in T} \mbox{hook}(b)} \]
A semistandard tableau is a Young diagram where each box is filled with a positive integer, but not necessarily distinct as in a standard tableau. The numbers are required to strictly increase from top to bottom along each column, but may only weakly increase from left to right across each row. The number of semistandard tableau with shape \((p_1, p_2, \ldots, p_\ell)\) and maximum entry \(m\) can be computed by the formula \[ \prod_{b \in T} \frac{m + b_c - b_r}{\mbox{hook}(b)} \] where \(b_r\) is the row of the tableau in which box \(b\) appears, and \(b_c\) is the column. Note that \(m\) must be greater than or equal to the number of rows of the tableau to have a possible semistandard tableau.

Using the Applet

Tap on the white region to change the tableau. Information about the tableau will be updated as the shape is changed. The hook length display and shading can be turned off by unchecking the boxes on the right.

About this Applet

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