I would like to find articles or documentation regarding this process:
Starting from what ever integer partition, e.g. 5,2 for the number 7. Construct his Young tableaux and then fill it with Hook numbers, like that:
$\square\square$ $\space$ 6 2
$\square\square$ $\space$ 5 1
$\square$ $\space\space\space\space$ 3
$\square$ $\space\space\space\space$ 2
$\square$ $\space\space\space\space$ 1
Take the 1st column (you can reconstruct the Young tableaux only with the 1st column), apply the Bulgarian Solitaire to this set of values & for each Bulgarian loop, reconstruct the corresponding Young tableaux.
6 $\space$ 5 $\space$ 5 $\space$ 5 $\space$ 5
5 $\space$ 5 $\space$ 4 $\space$ 4 $\space$ 4
3 $\space$ 4 $\space$ 4 $\space$ 3 $\space$ 3
2 $\space$ 2 $\space$ 3 $\space$ 3 $\space$ 2
1 $\space$ 1 $\space\space$ 1 $\space\space$ 2 $\space$ 2
$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space$1
Shifted Young Tableaux of 5, 2
$\square\square$ $\space$ $\square$ $\space\space\space\space\space$ $\square$ $\space\space\space\space$ $\square$
$\square\square$ $\space$ $\square\square$ $\space\space$ $\square$ $\space\space\space\space$ $\square$
$\square$ $\space\space\space\space$ $\square\square$ $\space\space$ $\square\square$ $\space$ $\square$
$\square$ $\space\space\space\space$ $\square$ $\space\space\space\space\space$ $\square\square$ $\space$ $\square\square$
$\square$ $\space\space\space\space$ $\square$ $\space\space\space\space\space$ $\square$ $\space\space\space\space$ $\square\square$
For the last set 5 4 3 2 2 1, we can't reconstruct a Young tableaux. For a defined partition, enumeration of shifted Young tableaux end when the Bulgarian Solitaire number of deck increases or deacreses in size. If you follow the Bulgarian Soliatire loop, you end up with a cycle, e.g.
6 $\space$ 5 $\space$ 5 $\space$ 5 $\space$ 5 $\space$ 6 $\space$ 6
5 $\space$ 5 $\space$ 4 $\space$ 4 $\space$ 4 $\space$ 4 $\space$ 5
3 $\space$ 4 $\space$ 4 $\space$ 3 $\space$ 3 $\space$ 3 $\space$ 3
2 $\space$ 2 $\space$ 3 $\space$ 3 $\space$ 2 $\space$ 2 $\space$ 2
1 $\space$ 1 $\space\space$ 1 $\space\space$ 2 $\space$ 2 $\space$ 1 $\space$ 1
$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space$1 $\space\space$ 1
Some articles or documentations? Thanks a lot!