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Exercise N

Posted 02/25/2015 and updated 04/08/2015
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A partition `\lambda` of integer `n` can be represented by a unique Young diagram. The parition `lambda` is called the shape of the Young diagram.

For example, `\lambda=(4,2,1)` is a partition of `7` then the Young diagram of shape `(4,2,1)` has `7` boxes and the bottom level has `4` boxes, the second level has `2` boxes and the third level has `1` box. In the picture, Young diagrams of shape `(4,2,1)`, `(5,3^2,2)` and `(3,1^3)` are drawn respectively.


A standard Young tableau of shape `\lambda` (`\lambda` is a partition of `n`) is obtained by filling `\{1,2,\cdots,n\}` in `n` boxes such that each column is increasing from bottom to top and each row is increasing from left to right.

For example, two standard Young tableaux of shape `(4,2,1)` are as follows,


The number of standard Young tableaux can be calculated by hook-length fomula.

`a(n)` is the number of standard Young tableaux of shape `(n,n-1,n-2,\cdots,2,1)` with the largest number not in the top box.

Find `a(n)`.



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Updated 04/08/2015

Typos are fixed.