Tableau promotion is a bijection on the standard Young tableaux of a fixed shape. To promote a tableau
containing the entries 1, ..., 
, erase the 1 in its upper-left box. Repeatedly slide the smaller
of the entries immediately to the right of and below the empty box into that box,
until the empty box reaches an outer corner. Then decrease every remaining entry
by 1 and place 
in the empty box (Schützenberger 1972).
Repeated promotion partitions the standard Young tableaux of a fixed shape into promotion orbits. The orbit length of a tableau is the number
of distinct tableaux in its group orbit, equivalently
the least positive integer 
for which the 
th promotion returns to the original tableau. The promotion
order of a shape is the least positive integer 
for which the 
th iterate fixes every tableau
of that shape. Haiman (1992, p. 96) proved that the 
th iterate of promotion fixes all rectangular 
standard Young tableaux,
where 
and 
are positive integers. Promotion on rectangular
tableaux is also a principal example of the cyclic
sieving phenomenon.
Catania et al. (2026) used graphical constructions called 
-diagrams to compute the orbit length of an individual rectangular
standard Young tableau. They also obtained a formula
for promotion orbit lengths of rectangular column semistandard Young
tableaux.
