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 French name for the fifteen puzzle.
Definition of a jeu de taquin slide
Given a skew standard Young tableau T of skew shape λ / μ {\displaystyle \lambda /\mu } , pick an adjacent empty cell c that can be added to the skew diagram λ ∖ μ {\displaystyle \lambda \setminus \mu } ; what this means is that c must share at least one edge with some cell in T, and { c } ∪ λ ∖ μ {\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, favoring the one below. (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. The other kind of slide, when c lies to the lower right of T, just goes in the opposite direction. In this case, one slides numbers into an empty cell from the neighbour to its left or above, picking the larger number whenever there is a choice. The two types of slides are mutual inverses – a slide of one kind can be undone using a slide of the other kind. The two slides described above are referred to as slides into the cell c. The first kind of slide (when c lies to the upper left of T) is said to be an inward slide; the second kind is referred to as an outward slide. The word "slide" is synonymous to the French word "glissement", which is occasionally also used in English literature.
Subtleties Jeu-de-taquin slides change not only the relative order of the entries of a tableau, but also its shape. In the definition given above, the result of a jeu-de-taquin slide is given as a skew diagram along with a skew standard tableau having it as shape. Often, it is better to work with skew shapes rather than skew diagrams. (Recall that every skew shape λ / μ {\displaystyle \lambda /\mu } gives rise to a skew diagram λ ∖ μ {\displaystyle \lambda \setminus \mu } , but this is not an injective correspondence because, e. g., the distinct skew shapes ( 2 , 1 ) / ( 2 ) {\displaystyle (2,1)/(2)} and ( 1 , 1 ) / ( 1 ) {\displaystyle (1,1)/(1)} yield the same skew diagram.) For this reason, it is useful to modify the above definition of a jeu-de-taquin slide in such a way that, when given a skew shape along with a skew standard tableau and an addable cell as an input, it yields a well-defined skew shape along with a skew standard tableau at its output. This is done as follows: An inward slide of a skew tableau T of skew shape λ / μ {\displaystyle \lambda /\mu } into a cell c is defined as above when c is a corner of μ {\displaystyle \mu } (that is, when μ ∖ { c } {\displaystyle \mu \setminus \left\{c\right\}} is a Young diagram), and the resulting skew shape is set to be ( λ ∖ { d } ) / ( μ ∖ { c } ) {\displaystyle (\lambda \setminus \left\{d\right\})/(\mu \setminus \left\{c\right\})} where d is the empty cell at the end of the sliding procedure. An outward slide of a skew tableau T of skew shape λ / μ {\displaystyle \lambda /\mu } into a cell c is defined as above when c is a cocorner of λ {\displaystyle \lambda } (that is, when λ ∪ { c } {\displaystyle \lambda \cup \left\{c\right\}} is a Young diagram), and the resulting skew shape is set to be ( λ ∪ { c } ) / ( μ ∪ { d } ) {\displaystyle (\lambda \cup \left\{c\right\})/(\mu \cup \left\{d\right\})} where d is the empty cell at the end of the sliding procedure.
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