# Shifted Young tableaux & Hook numbers & Bulgarian Solitaire

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!

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Question not clear. Documentation of what? For Bulgarian solitaire, start with en.wikipedia.org/wiki/Bulgarian_solitaire and the link there to the Monthly article, or just type "Bulgarian solitaire" into the web to see what turns up. –  Gerry Myerson Aug 10 '12 at 0:07
I want to find articles or documentations about this full process and no Young Tableaux, Hook numbers or Bulgarian Solitaire alone, a mix of all. If I ask this, it's because I have reviewed a lot's of white papers & I do not found such mix. Sure that the state of the art of stackexchange users is better that mine, this is why I put that question on this forum!! –  Yvan Aug 10 '12 at 8:35
Then please edit your question so it's clear that this is what you are asking for. –  Gerry Myerson Aug 10 '12 at 9:51
There may be something in this paper: Kimmo Eriksson, Jonas Sjöstrand, Limiting shapes of birth-and-death processes on Young diagrams, Advances in Applied Mathematics, Volume 48, Issue 4, April 2012, Pages 575-602. –  Gerry Myerson Aug 10 '12 at 9:58
Gerry: Thank you for the pointed article. –  Yvan Aug 10 '12 at 10:40