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.
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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(e2πid/n) is the number of elements fixed by cd. 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
and
One can show that evaluating the q-binomial coefficient when q is an nth 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).