Cyclic sieving

In combinatorial mathematics, cyclic sieving is a phenomenon by which evaluating a generating function for a finite set at roots of unity counts symmetry classes of objects acted on by a cyclic group.
Definition.
Let "C" be a cyclic group generated by an element "c" of order "n". Suppose "C" acts on a set "X". Let "X"("q") be a polynomial with integer coefficients. Then the triple ("X", "X"("q"), "C") is said to exhibit the cyclic sieving phenomenon (CSP) if for all integers "d", the value "X"("e"2"id"/"n") is the number of elements fixed by "c""d". In particular "X"(1) is the cardinality of the set "X", and for that reason "X"("q") is regarded as a generating function for "X".
Examples.
The "q"-binomial coefficient
is the polynomial in "q" defined by
It is easily seen that its value at "q" = 1 is the usual binomial coefficient formula_3, so it is a generating function for the subsets of {1, 2, ..., "n"} of size "k". These subsets carry a natural action of the cyclic group "C" of order "n" which acts by adding 1 to each element of the set, modulo "n". For example, when "n" = 4 and "k" = 2, the group orbits are
and
One can show that evaluating the "q"-binomial coefficient when "q" is an "n"th root of unity gives the number of subsets fixed by the corresponding group element.
In the example "n" = 4 and "k" = 2, the "q"-binomial coefficient is
evaluating this polynomial at "q" = 1 gives 6 (as all six subsets are fixed by the identity element of the group); evaluating it at "q" = −1 gives 2 (the subsets {1, 3} and {2, 4} are fixed by two applications of the group generator); and evaluating it at "q" = ±"i" gives 0 (no subsets are fixed by one or three applications of the group generator).
List of cyclic sieving phenomena.
In the Reiner–Stanton–White paper, the following example is given:
Let α be a composition of "n", and let "W"("α") be the set of all words of length "n" with αi
letters equal to "i". A "descent" of a word "w" is any index "j" such that formula_7.
Define the major index formula_8 on words as the sum of all descents.
The triple formula_9 exhibit a cyclic sieving phenomenon, where formula_10 is the set of non-crossing (1,2)-configurations of ["n" − 1].
Let "λ" be a rectangular partition of size "n", and let "X" be the set of standard Young tableaux 
of shape "λ". Let "C" = "Z"/"nZ" act on "X" via promotion. Then formula_11 exhibit the cyclic sieving phenomenon. Note that the polynomial is a "q"-analogue of the hook length formula.
Furthermore, let "λ" be a rectangular partition of size "n", and let "X" be the set of semi-standard Young tableaux of shape "λ". Let "C" = "Z"/"kZ" act on "X" via "k"-promotion.
Then formula_12
exhibit the cyclic sieving phenomenon. Here, formula_13
and "s"λ is the Schur polynomial.
An increasing tableau is a semi-standard Young tableau, where both rows and columns are strictly increasing,
and the set of entries is of the form formula_14 for some formula_15.
Let formula_16 denote the set of increasing tableau with two rows of length "n",
and maximal entry formula_17. Then
formula_18
exhibit the cyclic sieving phenomenon, where formula_19 act via "K"-promotion.
Let formula_20 be the set of permutations of cycle type "λ" and exactly "j" excedances.
Let formula_21,
and let formula_22 act on formula_20 by conjugation.
Then formula_24 exhibit the cyclic sieving phenomenon.