Jeu de taquin

In the mathematical field of combinatorics, jeu de taquin is a construction due to Marcel-Paul Schützenberger (1977) which defines an equivalence relation on the set of skew standard Young tableaux. A jeu de taquin slide is a transformation where the numbers in a tableau are moved around in a way similar to how the pieces in the fifteen puzzle move. Two tableaux are jeu de taquin equivalent if one can be transformed into the other via a sequence of such slides.

"Jeu de taquin" (literally "teasing game") is the .

Given a skew standard Young tableau T of skew shape ${\displaystyle \lambda /\mu }$, pick an adjacent empty cell c that can be added to the skew diagram ${\displaystyle \lambda \setminus \mu }$; what this means is that c must share at least one edge with some cell in T, and ${\displaystyle \{c\}\cup \lambda \setminus \mu }$ must also be a skew diagram. There are two kinds of slide, depending on whether c lies to the upper left of T or to the lower right. Suppose to begin with that c lies to the upper left. Slide the number from its neighbouring cell into c; if c has neighbours both to its right and below, then pick the smallest of these two numbers. (This rule is designed so that the tableau property of having increasing rows and columns will be preserved.) If the cell that just has been emptied has no neighbour to its right or below, then the slide is completed. Otherwise, slide a number into that cell according to the same rule as before, and continue in this way until the slide is completed. After this transformation, the resulting tableau (with the now-empty cell removed) is still a skew (or possibly straight) standard Young tableau.

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