Cyclic sieving

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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.[1]

Definition[edit]

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 (XX(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[edit]

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 , 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

(of size 2)

and

(of size 4).

One can show[2] 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).

List of cyclic sieving phenomena[edit]

In the Reiner-Stanon-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 . Define the major index on words as the sum of all descents.

The triple exhibit a cyclic sieving phenomena, where the cyclic group acts via shift. Recall that , the q-multinomial coefficient. This formula is due to MacMahon.


The triple exhibit a cyclic sieving phenomena, where is the set of non-crossing (1,2)-configurations of [n-1]. [3]


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 exhibit the cyclic sieving phenomena. 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 exhibit the cyclic sieving phenomena. Here, and sλ is the Schur polynomial. [4]



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 for some . Let denote the set of increasing tableau with two rows of length n, and maximal entry . Then exhibit the cyclic sieving phenomena, where act via K-promotion.[5]


Let be the set of permutations of cycle type λ and exactly j excedances. Let , and let act on by conjugation.

Then exhibit the cyclic sieving phenomena. [6]

Notes and references[edit]

  1. ^ Reiner, Victor; Stanton, Dennis; White, Dennis (February 2014), "What is... Cyclic Sieving?" (PDF), Notices of the American Mathematical Society, 61 (2): 169–171, doi:10.1090/noti1084 
  2. ^ V. Reiner, D. Stanton and D. White, The cyclic sieving phenomenon, Journal of Combinatorial Theory, Series A, Volume 108 Issue 1, October 2004, Pages 17–50
  3. ^ Thiel, Marko (March 2017). "A new cyclic sieving phenomenon for Catalan objects". Discrete Mathematics. 340 (3): 426–429. arXiv:1601.03999Freely accessible. doi:10.1016/j.disc.2016.09.006. 
  4. ^ Rhoades, Brendon (January 2010). "Cyclic sieving, promotion, and representation theory". Journal of Combinatorial Theory, Series A. 117 (1): 38–76. doi:10.1016/j.jcta.2009.03.017. 
  5. ^ Pechenik, Oliver (July 2014). "Cyclic sieving of increasing tableaux and small Schröder paths". Journal of Combinatorial Theory, Series A. 125: 357–378. arXiv:1209.1355Freely accessible. doi:10.1016/j.jcta.2014.04.002. 
  6. ^ Sagan, Bruce; Shareshian, John; Wachs, Michelle L. (January 2011). "Eulerian quasisymmetric functions and cyclic sieving". Advances in Applied Mathematics. 46 (1-4): 536–562. doi:10.1016/j.aam.2010.01.013. 
  • Sagan, Bruce. The cyclic sieving phenomenon: a survey. Surveys in combinatorics 2011, 183–233, London Math. Soc. Lecture Note Ser., 392, Cambridge Univ. Press, Cambridge, 2011.