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#The cyclic sieving phenomenon

See below for the definition of the cyclic sieving phenomenon (CSP). There are many instances of the CSP, which we have put into different categories based roughly on the type of combinatorial object it concerns.

#Definition

The cyclic sieving phenomenon (CSP) was introduced by V. Reiner, D. Stanton and D. White in [RSW04]. A nice survey by B. Sagan is given in [Sag11].

Let \(X\) be a set of combinatorial objects, \(\grpc_n = \langle g \rangle\) be a finite cyclic group of size \(n,\) and \(f(q) \in \setN[q].\) Then the triple \((X,\grpc_n,f(q))\) is said to exhibit the cyclic sieving phenomenon if for all \(d \in \setN,\) we have \[|\{x \in X : g^d \circ x =x \}| = f\left(\exp\left(2\pi i \frac{d}{n}\right)\right).\] In words, \(f(q)\) evaluated at certain roots of unity gives the number of elements in \(X\) fixed by powers of \(g.\) Note that \(f(1) = |X|,\) so in many instances, \(f(q)\) is a \(q\)-analog (or \(q\)-enumeration) of the set \(X.\) The cyclic action need not be faithful. If \(\grpc_h\) acts with a kernel of order \(d,\) then the CSP is equivalent to the faithful quotient action of \(\grpc_{h/d}\) together with a polynomial \(f'(q)\) satisfying \(f(q)=f'(q^d)\); see [Sec. 2.5, BR11].

Geng-Rui Zhang studies realizable and almost realizable integer sequences, where \(a_n\) counts periodic points of period dividing \(n\) for a self-map [Zha24]. This gives a useful neighboring dynamical viewpoint on root-of-unity and fixed-point phenomena: many classical combinatorial sequences are realizable, while Catalan, Motzkin, and Schröder-type sequences are not almost realizable.

Proposition

Let \(\xi\) be a primitive \(n^\thsup\) root of unity, and suppose that \(f \in \setN[q]\) satisfies \(f(\xi^j) \in \setZ\) for all \(j \in \setZ.\) Then for all \(j \in \setZ,\) \(f(\xi^j) = f( \xi^{\gcd(j,n)}).\)

Proof

In [De89] and [Lem. 2.2, AA19], it is proved that \(f\) (up to mod \(q^n-1\)) is a linear combination of \[h_d(q) \coloneqq \sum_{j=0}^{n/d-1} q^{dj} = \frac{[n]_q}{[d]_q} \qquad \text{ where } d \mid n.\] It then suffices to verify that \[h_d(\xi^j) = h_d( \xi^{\gcd(j,n)}) = \begin{cases} \frac{n}{d} &\text{ if } n \mid j d \\ 0 &\text{ otherwise} \end{cases}\] for all \(d \mid n,\) \(j\in \setZ,\) which is straightforward.

This proposition is very handy: Suppose we know that \(f\) is an integer at \(n^\thsup\) roots of unity. Then it suffices to verify that, for all \(d \mid n,\) \[|\{x \in X : g^d \circ x =x \}| = f\left(\exp\left(2\pi i \frac{d}{n}\right)\right)\] in order to prove CSP.

Theorem (From [RSW04]).

Given \(X\) and \(\grpc_n\) acting on \(X\) and \(f(q) \in \setN[q],\) then \((X,\grpc_n,f(q))\) exhibits the CSP if and only if \[f(q) \equiv \sum_{O \in Orb_{\grpc_n}(X)} \frac{q^n-1}{ q^{n/|O|} - 1} \mod (q^n-1),\] where \(Orb_\grpc(X)\) is the set of orbits of \(X\) under \(\grpc.\)

In other words, in a CSP triple \((X,\grpc_n,f(q))\) the polynomial \(f(q)\) is essentially uniquely determined by \(\grpc_n\) acting on \(X.\)

There is a converse to the above theorem.

Theorem (From [AA19]).

Let \(\xi\) be a primitive \(n\)th root of unity, and suppose \(f(q) \in \setN[q]\) has the property that \(f(\xi^j) \in \setN\) for all \(j\in \setN.\) Let \(S_k\) be defined as \[S_k \coloneqq \sum_{j|k} \mu(k/j) f(\xi^j).\] Then there is a \(\grpc_n\)-action on some \(X\) with cardinality \(f(1)\) and \((X,\grpc_n,f(q))\) exhibits the CSP if and only if all \(S_k\) are non-negative.

In other words, these are necessary and sufficient conditions on \(f(q) \in \setN[q]\) for there to be a CSP involving \(f(q).\)

Research situation: You have your favorite set \(X\) and a \(q\)-analog \(f(q).\) The previous theorem allows you to check (by computer, preferably) if there is some CSP phenomenon on \(X,\) involving \(f(q).\) Note that it is not sufficient for \(f(q)\) to evaluate to non-negative integers at roots of unity!

The number of orbits of size \(k\) is determined by \(\frac{1}{k}\sum_{j|k} \mu(k/j) f(\xi^j),\) and it follows that the total number of orbits is \[\sum_{k|n} \frac{1}{k}\sum_{j|k} \mu(k/j) f(\xi^j).\]

Example (Example 2.10 from [AA19]).

Take \(f(q)=q^5+3q^3+q+10.\) At any \(6^\thsup\) roots of unity, \(f(\xi^j)\) is a positive integer. However, for \(k=3,\) the sum \(\sum_{d|k} \mu(k/d) f(\xi^d)\) is \(-3.\) This sum represents the number of elements in an orbit of size \(3\) under the cyclic group, and this number is not allowed to be negative. Hence, there is no set \(X\) of size \(15\) with a cyclic group action \(C_6\) of order \(6,\) such that \((X,C_6,f(q))\) is a CSP-triple.

#Connection with representation theory

We can prove cyclic sieving using representation theory.

Suppose \(G\) (in this case the cyclic group \(C_n\)) acts on the set \(X.\) We can form the \(|X|\)-dimensional vector space \[V = \setC X = \{a_1 x_1 + a_2x_2+\dotsb + a_M x_M : a_1,\dotsc,a_m \in \setC \},\] and note that \(G\) acts on \(V.\) We can compute the character \(\chi(g)\) for \(g\in G,\) as a certain type of trace. Since \(G\) in our case acts via permutations, \(\chi(g)\) is simply the number of elements in \(X\) fixed by \(g.\)

One then wishes to find a different basis \(B\) for \(V\) (perhaps starting with the diagonal matrix with entries \((1,\zeta,\zeta^2,\dotsc,\zeta^{n-1})\) representing a generator of \(C_n\)) such that the action of \(G\) on \(B\) is diagonal. It is then easy to compute the trace, expressed in terms of roots of unity. If this trace of \(g^d\) is exactly \(f(\zeta^d)\) for our polynomial \(f(q),\) we have proved an instance of CSP.

Several examples are given in [Sag11]. One can also construct new CSP instances from existing ones by using representation theory, see [BER11, RSW04]. D. B. Rush gives a crystal/global-basis proof of B. Rhoades’s rectangular tableaux CSP theorem [Rus21]. Y.-T. Oh and E. Park use \(q\)-dimensions of highest weight crystals to obtain cyclic sieving phenomena [OP20].

#Universal CSP statistics

The notion of universal statistics was introduced in [AS18].

Suppose \((X,\grpc_n,f(q))\) exhibits the CSP, where \(f(q) = \sum_{x\in X} q^{\tau(x)}\) for some combinatorial statistic \(\tau:X \to \setN.\) Then \(\tau\) is said to be universal if for every \(\grpc_n\)-orbit \(O \subseteq X,\) the triple \((O,\grpc_n,\sum_{x\in O} q^{\tau(x)})\) exhibits the CSP.

The usual major index on words is not universal. The authors of [AS18] introduce a new (universal) statistic on words equidistributed with \(\maj\) on words of length \(n\) with fixed content \(\alpha.\)

#Proper statistic

The following definition has not been defined in any publication, to our knowledge. This is possibly related or equivalent to the notion of a universal CSP statistic. — Per

Suppose \(X\) is a combinatorial family with a statistic \(\sigma:X \to \setN,\) that defines the \(q\)-analog \(f(q)\) of \(X,\) and that \((X, \grpc_n, f(q))\) is a CSP-triple.

Then the statistic is called proper if, whenever \(d|n,\) \[g^{d}(x) = x \implies \sigma(x) \equiv_{n/d} 0 \text{ for all } x \in X.\]

Example

Let \(n=2\) and suppose \(g\) generates a \(\grpc_2\)-action on \(X.\) If \(\sigma\) is a proper statistic on \(X,\) and \(X^g \subseteq X\) is the set of elements fixed under \(g,\) then there should (in principle) exist an involution \(\psi : X\setminus X^g \to X\setminus X^g,\) such that \((-1)^{\sigma(x)}+(-1)^{\sigma(\psi(x))} = 0,\) which would provide a proof of CSP.

#Subset cyclic sieving

There is a natural generalization of the cyclic sieving phenomenon described in [ALP19]. Let \(X\) be a set of combinatorial objects and \(Y \subseteq X.\) Let \(\grpc_n = \langle g \rangle\) be a finite cyclic group of size \(n,\) and \(f(q) \in \setN[q].\) Then the triple \((Y\subseteq X,\grpc,f(q))\) is said to exhibit the subset cyclic sieving phenomenon if for all \(d \in \setN,\) we have \[|\{y \in Y : g^d \circ y = y \}| = f\left(\exp\left(2\pi i \frac{d}{n}\right)\right).\] Note that in particular, \(f(1)=|Y|.\)

The subset CSP can be thought of as a CSP on \(Y,\) but where the group action is allowed to leave \(Y.\) By using the main theorem in [AA19], one can show that if \(Y\subset X\) admits a subset CSP, then \(Y\) also admits a proper CSP, with some cyclic group action that does not leave \(Y.\)

Example (Taken from [ALP19]).

Let \(X_n\) be the set of binary words, and let \(Y_n \subseteq X_n\) be the set of words ending with a \(1.\) Furthermore, let \(\eta\) act on \(X_n\) by a twisted two-step cyclic shift \[\eta(b_1,b_2,\dotsc,b_{n-1},b_n) = (1-b_{n-1},1-b_n,b_1,b_2,\dotsc,b_{n-2}).\] It is easy to show that \(\langle \eta \rangle\) is a cyclic group of order \(n.\) The polynomial we use is \(f_n(q) = \prod_{j=1}^{n-1}(1+q^j).\) Then \[(Y_n \subseteq X_n, \langle \eta \rangle, f_n(q))\] exhibits the subset cyclic sieving phenomenon. Note that \((X_n, \langle \eta \rangle, 2f_n(q))\) is an instance of the classical CSP.

#Lyndon-like cyclic sieving

In [ALP19], we consider a special type of cyclic sieving phenomenon, on a family \(\{X_n\}_{n=1}^\infty\) of combinatorial objects. This idea captures the case where fixed points in \(X_n\) under the group action are in bijection with smaller members of the family.

Definition

Let \(\{(X_n, C_n, f_n(q))\}_{n=1}^\infty\) be a family of instances of CSP. The family is Lyndon-like if for all \(n\geq 1,\) \[f_{n/m}(1) = f_n\left( e^{\tfrac{2 \pi i}{m}} \right), \text{ whenever } m|n.\] By the definition of CSP, \[f_n\left( e^{\tfrac{2 \pi i}{m}} \right) = |\{ x \in X_n : g^{n/m}(x) = x \}|,\] where \(\langle g \rangle = C_n.\) The family is therefore Lyndon-like if and only if for every \(d|n\) the number of elements in \(X_n\) fixed under \(g^{d}\) is equal to \(|X_{d}|.\)

The connection with counting fixed-points under iterations of a map is also discussed in the earlier work [Zar08], where many related notions are discussed.

There are several examples of Lyndon-like families of CSP. For example, Cyclic sieving on words, CSP on circular Dyck paths and (conjecturally) CSP on non-attacking filling.

Notice that it is easy to gather computer evidence for a sequence of \(q\)-analogs \(\{f_n(q)\}_{n=1}^\infty\) to be Lyndon-like. Such evidence should give strong hints about possible corresponding cyclic group actions, as it should in principle behave as a type of cyclic shift on words.

#q-Gauß congruences

In [Gor19], O. Gorodetsky introduces the notion of \(q\)-Gauß congruences. A sequence of polynomials \(\{ a_n(q) \}_{n=1}^\infty\) is said to satisfy \(q\)-Gauß congruences if for every \(n \in \setP,\) we have \[\sum_{d|n} \mu(d) a_{n/d}(q^d) \equiv 0 \mod [n]_q.\] It can be verified that this condition on the family is equivalent to being Lyndon-like, that is \(a_{n/m}(1) = a_n(\xi)\) where \(\xi\) is an \(n\)th root of unity with order \(m,\) see [Cor. 2.5, Gor19]. O. Gorodetsky also uses this criterion to prove cyclic sieving phenomena for rotation on two word families [Thm. 1.2, Gor19]. The first family consists of words on \(\{0,1,2\}\) with prescribed difference between the numbers of \(0\)’s and \(2\)’s. The second consists of binary words with a prescribed number of \(1\)’s and no adjacent \(1\)’s cyclically; the sieving polynomial is a refinement of the \(q\)-Lucas numbers.

F. Gossow strengthens the link between \(q\)-Gauß congruences and cyclic sieving [Gos24]. The paper gives explicit formulae for \(q\)-Gauß congruences and develops a universal necklace model in which beads may vary in both colour and length, producing further CSP examples from necklaces, path walks, tubings, and related objects.

S. Narayan and N. U. Kiran study Euler–Gauß sequences and related \(q\)-analogues [NK25]. Their work includes CSP conditions for \(q\)-Euler–Gauß sequences and for sequences built from smallest and greatest prime factors.

#Orbital harmonics

In [OR21], J. Oh and B. Rhoades gives an introduction to a representation-theoretical approach to proving cyclic sieving results.

#Dihedral Sieving and \(G\)-sieving

It is possible to extend the cyclic sieving phenomenon to other groups, see [RS20]. The next natural example is the dihedral group, \(D_n.\) This group is presented as \(\langle r,s : r^n=s^2=e, rs=sr^{-1} \rangle.\) We can interpret this as \(r\) being a one-step rotation of an \(n\)-gon and \(s\) being a reflection. Several examples of dihedral sieving are given in [RS20]. Matthieu Josuat-Verges formulates a \(q,t\)-dihedral sieving phenomenon for cluster parking functions, relating their reduced homology to diagonal coinvariants up to a dihedral sign character [Jos26].

S. Rao and J. Suk have introduced the notion of \(G\)-sieving for arbitrary groups.

#Toric promotion with refraction

A. Adams, C. Defant, and J. Striker introduce toric promotion with reflections and refractions [ADS24]. This is a graph-labeling dynamical system motivated by combinatorial billiards in toric arrangements. For forests they give a formula for the orbit structure, and for even cycles with no reflection edges they obtain an instance of the cyclic sieving phenomenon.

#Permutation CSP from FindStat

A. Adams, J. Elder, N. Lafrenière, E. McNicholas, J. Striker, and A. Welch give a computer-assisted analysis of cyclic sieving on permutation maps and statistics from FindStat [AELM+25]. They prove several new CSP instances and record further conjectures, including examples involving long-cycle conjugation, reverse, complement, Lehmer-code rotation, and toric promotion. B. Deb gives cyclic sieving phenomena using continued-fraction identities for permutations, set partitions, perfect matchings, and \(D\)-permutations [Deb25]. The proofs pass through weighted lattice paths and include involutions exchanging crossings and nestings, including a Genocchi–Corteel involution for \(D\)-permutations. M. Bousquet-M{\'e}lou and C. Krattenthaler prove cyclic sieving phenomena for corner-rooted plane trees and tree-rooted planar maps [BK25]. They treat several tree classes and root-moving cyclic actions, including actions restricted to leaves, non-leaf corners, or corners at nodes of prescribed degree.

#Homomesy

The term homomesy was introduced by J. Propp and T. Roby in [PR13] and is defined as follows. Let \(X\) be some combinatorial set and let \(\langle g \rangle\) act on \(X\) such that every orbit is finite, and let \(\sigma : X \to \setZ\) be a statistic on \(X.\) Then the triple \((X,\langle g \rangle,\sigma)\) is said to exhibit homomesy if for every orbit \(O \subseteq X,\) we have \[\frac{1}{|O|} \sum_{x \in O} \sigma(x) = \frac{1}{|X|} \sum_{x \in X} \sigma(x).\] In other words, the average value of \(\sigma\) is the same on every orbit.

The first instance of the homomesy phenomenon was observed by D. Panyushev in [Pan09], when studying a rowmotion-type action on antichains of positive roots. Since then, many more instances have been found.

For homomesy on minuscule posets, see [Oka21], and for the Foata map, see [LR20]. For a spectral-theory approach to homomesy, see [Pro21].

W. Dowling and N. Lafrenière carry out a systematic search for permutation statistics that are homomesic under rotation [DL25]. They prove several instances found through FindStat, and show that the same homomesies hold for a broader class of toggling actions related to rowmotion-style toggle factorizations.

J. Elder, N. Lafrenière, E. McNicholas, J. Striker, and A. Welch systematically analyze permutation maps and statistics in FindStat and prove \(122\) homomesy instances [ELMS+24]. Their maps include Lehmer-code rotation, reverse, complement, the Foata bijection, and Kreweras complement.

An action is homometric if the total sum of the statistic on orbits of the same size is the same. For example, for all orbits of size 4, the sum of the statistic on each orbit must be the same. This notion was introduced by S. Elizalde, M. Plante, T. Roby and B. Sagan.

Homomesy and homometry for promotion on rooted trees is considered in [PDS23].

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