6077:
3392:
5638:
4452:
284:. Little is lost by working with permutation groups in such a setting, so in these applications, when a group is considered, it is a permutation representation of the group which will be worked with, and thus, a group action must be specified. Algebraists, on the other hand, are more interested in the groups themselves and would be more concerned with the
6072:{\displaystyle \sum \limits _{n\geq 1}y^{n}Z(S_{n})=\sum \limits _{n\geq 1}\sum _{j_{1}+2j_{2}+3j_{3}+\cdots +nj_{n}=n}\prod _{k=1}^{n}{\frac {a_{k}^{j_{k}}y^{kj_{k}}}{k^{j_{k}}j_{k}!}}=\prod \limits _{k\geq 1}\sum \limits _{j_{k}\geq 0}{\frac {(a_{k}y^{k})^{j_{k}}}{k^{j_{k}}j_{k}!}}=\prod \limits _{k\geq 1}\exp \left({\frac {a_{k}y^{k}}{k}}\right),}
6326:
5587:
6851:
4745:
3631:
These edge rotations rotate about the axis that passes through the midpoints of opposite edges not incident on the same face and parallel to each other and exchanges the two faces that are incident on the first edge, the two faces incident on the second edge, and the two faces that share two vertices
415:
is to be read as: each element is sent to the element on its right, but the last element is sent to the first one (it "cycles" to the beginning). With cycle notation, it does not matter where a cycle starts, so (1 2 3 4 5) and (3 4 5 1 2) and (5 1 2 3 4) all represent the same permutation. The
4186:
5070:
3045:
Eight permutations that fix one vertex and produce a three-cycle for the three vertices not fixed: These permutations create two three-cycles of edges, one containing those not incident on the vertex, and another one containing those incident on the vertex; the contribution is
558:
6115:
5342:
3857:
580:) variables in the following way: a variable is needed for each distinct cycle length of the cycles that appear in the cycle decomposition of the permutation. In the previous example there were three different cycle lengths, so we will use three variables,
390:
7569:
The cycle index is a polynomial in several variables and the above results show that certain evaluations of this polynomial give combinatorially significant results. As polynomials they may also be formally added, subtracted, differentiated and
3381:
6575:
2989:
Six permutations that exchange two vertices: These permutations preserve the edge that connects the two vertices as well as the edge that connects the two vertices not exchanged. The remaining edges form two two-cycles and the contribution is
4478:
1008:
1824:
2528:
2019:
As the above example shows, the cycle index depends on the group action and not on the abstract group. Since there are many permutation representations of an abstract group, it is useful to have some terminology to distinguish them.
5227:
2007:
1295:
2268:
3583:
This time we rotate about the axis passing through two opposite vertices (the endpoints of a main diagonal). This creates two three-cycles of faces (the faces incident on the same vertex form a cycle). The contribution is
1505:
2829:
7317:
4447:{\displaystyle Z(D_{n})={\frac {1}{2}}Z(C_{n})+{\begin{cases}{\frac {1}{2}}a_{1}a_{2}^{(n-1)/2},&n{\mbox{ odd, }}\\{\frac {1}{4}}\left(a_{1}^{2}a_{2}^{(n-2)/2}+a_{2}^{n/2}\right),&n{\mbox{ even.}}\end{cases}}}
4154:
6966:
1126:
4856:
7464:
6564:
2533:
Often, when an author does not wish to use the group action terminology, the permutation group involved is given a name which implies what the action is. The following three examples illustrate this point.
437:
3086:
Three permutations that exchange two vertex pairs at the same time: These permutations preserve the two edges that connect the two pairs. The remaining edges form two two-cycles and the contribution is
6422:
1321:. Its elements are completely determined by the images of just the corners of the square. By labeling these corners 1, 2, 3 and 4 (consecutively going clockwise, say) we can represent the elements of
6321:{\displaystyle {\begin{aligned}Z(S_{n})&=\prod \limits _{k\geq 1}\exp \left({\frac {a_{k}y^{k}}{k}}\right)\\&=\exp \left(\sum \limits _{k\geq 1}{\frac {a_{k}y^{k}}{k}}\right).\end{aligned}}}
2141:
7075:
4820:
6120:
3142:
Six permutations that cycle the vertices in a four-cycle: These permutations create a four-cycle of edges (those that lie on the cycle) and exchange the remaining two edges; the contribution is
5582:{\displaystyle {\frac {n!}{\prod _{k=1}^{n}(k!)^{j_{k}}}}\prod _{k=1}^{n}\left({\frac {k!}{k}}\right)^{j_{k}}\prod _{k=1}^{n}{\frac {1}{j_{k}!}}={\frac {n!}{\prod _{k=1}^{n}k^{j_{k}}j_{k}!}}.}
423:
Not all permutations are cyclic permutations, but every permutation can be written as a product of disjoint (having no common element) cycles in essentially one way. As a permutation may have
732:
3683:
2624:
in an axis passing through a vertex and the midpoint of the opposite edge: These fix one edge (the one not incident on the vertex) and exchange the remaining two; the contribution is
3572:
3138:
3041:
312:
3938:
3460:
We rotate about the axis passing through the centers of the face and the face opposing it. This will fix the face and the face opposing it and create a four-cycle of the faces
401:= {1, 2, 3, 4, 5} which sends 1 ↦ 2, 2 ↦ 3, 3 ↦ 4, 4 ↦ 5 and 5 ↦ 1. This can be read off from the columns of the notation. When the top row is understood to be the elements of
3510:
2904:
6846:{\displaystyle Z(S_{n})={\frac {1}{n!}}\sum _{g\in S_{n}}\prod _{k=1}^{n}a_{k}^{j_{k}(g)}={\frac {1}{n!}}\sum _{l=1}^{n}{n-1 \choose l-1}\;(l-1)!\;a_{l}\;(n-l)!\;Z(S_{n-l})}
6477:
3233:
3183:
2665:
4740:{\displaystyle Z(A_{n})=\sum _{j_{1}+2j_{2}+3j_{3}+\cdots +nj_{n}=n}{\frac {1+(-1)^{j_{2}+j_{4}+\cdots }}{\prod _{k=1}^{n}k^{j_{k}}j_{k}!}}\prod _{k=1}^{n}a_{k}^{j_{k}}.}
3668:
3620:
3082:
6371:
3449:
2985:
2616:
5334:
3521:
We rotate about the same axis as in the previous case, but now there is no four cycle of the faces parallel to the axis, but rather two two-cycles. The contribution is
2354:
states that every abstract group has a regular permutation representation given by the group acting on itself (as a set) by (right) multiplication. This is called the
891:
2700:
6107:
5261:
6976:
Throughout this section we will modify the notation for cycle indices slightly by explicitly including the names of the variables. Thus, for the permutation group
5296:
1693:
6503:
565:
This permutation is the product of three cycles, one of length two, one of length three, and a fixed point. The elements in these cycles are disjoint subsets of
5630:
5610:
2924:
2401:
5085:
1896:
1175:
2154:
1399:
2849:(vertex-edge dual) and hence the edge permutation group induced by the vertex permutation group is the same as the vertex permutation group, namely
2722:
7190:
4054:
7718:
There is a convention to not write the fixed points in the cycle notation for a permutation, but these must be represented in the cycle index.
6862:
5065:{\displaystyle Z(S_{n})=\sum _{j_{1}+2j_{2}+3j_{3}+\cdots +nj_{n}=n}{\frac {1}{\prod _{k=1}^{n}k^{j_{k}}j_{k}!}}\prod _{k=1}^{n}a_{k}^{j_{k}}}
3407:. It permutes the six faces of the cube. (We could also consider edge permutations or vertex permutations.) There are twenty-four symmetries.
1607:= {2,4}. These elements can be thought of as the sides and diagonals of the square or, in a completely different setting, as the edges of the
1023:
5232:
This formula is obtained by counting how many times a given permutation shape can occur. There are three steps: first partition the set of
405:
in an appropriate order, only the second row need be written. In this one-line notation, our example would be . This example is known as a
7346:
6508:
553:{\displaystyle \left({\begin{matrix}1&2&3&4&5&6\\2&1&3&5&6&4\end{matrix}}\right)=(12)(3)(456).}
7709:
This notation is common amongst geometers and combinatorialists. It is used instead of the more common g(x) for traditional reasons.
6384:
7562:(also known as the Not Burnside's lemma, but traditionally called Burnside's lemma) and the weighted version of the result is
1333:
consists of the four permutations (1 4 3 2), (1 3)(2 4), (1 2 3 4) and e = (1)(2)(3)(4) which represent the counter-clockwise
7847:
7789:
7667:
Up to the different ways one can write a cycle and the fact that disjoint cycles commute so they can be written in any order.
2047:
6988:
4761:
7976:
577:
668:
7886:
7868:
7829:
7807:
3852:{\displaystyle Z(C)={\frac {1}{24}}\left(a_{1}^{6}+6a_{1}^{2}a_{4}+3a_{1}^{2}a_{2}^{2}+8a_{3}^{2}+6a_{2}^{3}\right).}
7563:
2669:
Two rotations, one clockwise, the other counterclockwise: These create a cycle of three edges; the contribution is
2023:
When an abstract group is defined in terms of permutations, it is a permutation group and the group action is the
429:(elements that are unchanged by the permutation), these will be represented by cycles of length one. For example:
288:
of the group actions, which measure how much is lost in passing from the group to its permutation representation.
7326:
2867:). This is not the case for complete graphs on more than three vertices, since these have strictly more edges (
2368:
in its regular representation contains the six permutations (one-line form of the permutation is given first):
7911:
7586:
385:{\displaystyle \left({\begin{matrix}1&2&3&4&5\\2&3&4&5&1\end{matrix}}\right)}
7581:
The question of what the cycle structure of a random permutation looks like is an important question in the
7170:
2557:
in the
Euclidean plane. This permits us to use geometric language to describe the permutations involved as
3524:
3090:
2993:
2952:
The identity: This permutation maps all vertices (and hence, edges) to themselves and the contribution is
136:
7903:
4008:
3886:
261:. A given group can have many different permutation representations, corresponding to different actions.
2537:
235:), is a permutation group. The group homomorphism can be thought of as a means for permitting the group
2929:
253:
3467:
3376:{\displaystyle Z(G)={\frac {1}{24}}\left(a_{1}^{6}+9a_{1}^{2}a_{2}^{2}+8a_{3}^{2}+6a_{2}a_{4}\right).}
3202:
Rotation by 120 degrees about the axis passing through a vertex and the midpoint of the opposite face.
143:
operations on these polynomials and then interpreting the results combinatorially lies at the core of
2870:
425:
51:
4249:
6427:
2621:
3145:
2627:
3635:
3587:
3049:
1003:{\displaystyle 0\leq j_{k}(g)\leq \lfloor n/k\rfloor {\mbox{ and }}\sum _{k=1}^{n}k\,j_{k}(g)=n.}
762:, while the cycle index monomial of the permutation (1 2)(3 4)(5)(6 7 8 9)(10 11 12 13) would be
6334:
3419:
2955:
2586:
7971:
7582:
7575:
5305:
411:
because it "cycles" the numbers around, and a third notation for it would be (1 2 3 4 5). This
112:
of a permutation group is the average of the cycle index monomials of its elements. The phrase
3880:
This group contains one permutation that fixes every element (this must be a natural action).
2672:
2579:
in its natural action, given above) induces an edge permutation. These are the permutations:
1819:{\displaystyle Z(C_{4})={\frac {1}{4}}\left(a_{1}^{6}+a_{1}^{2}a_{2}^{2}+2a_{2}a_{4}\right).}
1310:
144:
7946:
7145:
6085:
5239:
3986:
3189:
2554:
272:(that is, a group action exists). In combinatorial applications the interest is in the set
172:
5270:
2523:{\displaystyle Z(C_{6})={\frac {1}{6}}\left(a_{1}^{6}+a_{2}^{3}+2a_{3}^{2}+2a_{6}\right).}
2323:) if the only permutation in the group that has fixed points is the identity permutation.
8:
7559:
6482:
3461:
2351:
1885:
228:
128:
3632:
but no edge with the two edges, i.e. there are three two-cycles and the contribution is
3199:
Reflection in the plane that contains one edge and the midpoint of the edge opposing it.
2934:
This is entirely analogous to the three-vertex case. These are the vertex permutations (
7818:
5615:
5595:
5302:. But we do not distinguish between cycles of the same size, i.e. they are permuted by
5222:{\displaystyle Z(S_{n})={\frac {B_{n}(0!\,a_{1},1!\,a_{2},\dots ,(n-1)!\,a_{n})}{n!}}.}
2909:
576:
The cycle structure of a permutation can be coded as an algebraic monomial in several (
407:
216:
66:
62:
7936:
7919:
3205:
Rotation by 180 degrees about the axis connecting the midpoints of two opposite edges.
7882:
7864:
7843:
7825:
7803:
7785:
4469:
794:
196:
124:
39:
32:
6331:
There is a useful recursive formula for the cycle index of the symmetric group. Set
2930:
The cycle index of the edge permutation group of the complete graph on four vertices
2002:{\displaystyle Z(C_{4})={\frac {1}{4}}\left(a_{1}^{16}+a_{2}^{8}+2a_{4}^{4}\right).}
1290:{\displaystyle Z(G)={\frac {1}{|G|}}\sum _{g\in G}\prod _{k=1}^{n}a_{k}^{j_{k}(g)}.}
92:
31:
in several variables which is structured in such a way that information about how a
7931:
4751:
4016:
1875:
816:
285:
7578:
provides combinatorial interpretations of the results of these formal operations.
1341:
permutation e is the only permutation with fixed points in this representation of
304:}. A permutation in this setting can be represented by a two-line notation. Thus,
5076:
4839:
4755:
3970:
2263:{\displaystyle Z(S_{3})={\frac {1}{6}}\left(a_{1}^{3}+3a_{1}a_{2}+2a_{3}\right).}
1318:
176:
7688:
Technically we are using the notion of equivalence of group actions, replacing
4173:
3209:
2339:
1608:
1500:{\displaystyle Z(C_{4})={\frac {1}{4}}\left(a_{1}^{4}+a_{2}^{2}+2a_{4}\right).}
212:
7965:
7641:
This notational style is frequently found in the computer science literature.
2824:{\displaystyle Z(G)={\frac {1}{6}}\left(a_{1}^{3}+3a_{1}a_{2}+2a_{3}\right).}
123:
Knowing the cycle index polynomial of a permutation group, one can enumerate
17:
5592:
The formula may be further simplified if we sum up cycle indices over every
7700:. For the purposes of exposition, it is better to slide over these details.
7312:{\displaystyle \sum _{k=0}^{n}f_{k}t^{k}=Z(G;1+t,1+t^{2},\ldots ,1+t^{n}),}
4177:
3957:
2327:
2024:
1350:
1338:
296:
Finite permutations are most often represented as group actions on the set
248:
132:
35:
2583:
The identity: No vertices are permuted, and no edges; the contribution is
1617:. Acting on this new set, the four group elements are now represented by (
7856:
7784:(5th ed.), Upper Saddle River, NJ: Prentice Hall, pp. 541–575,
7692:
acting on the corners of the square by the permutation representation of
4149:{\displaystyle Z(C_{n})={\frac {1}{n}}\sum _{d|n}\varphi (d)a_{d}^{n/d}.}
43:
20:
7912:
Cycle indices of the set / multiset operator and the exponential formula
6961:{\displaystyle Z(S_{n})={\frac {1}{n}}\sum _{l=1}^{n}a_{l}\;Z(S_{n-l}).}
2846:
2842:
1121:{\displaystyle \prod _{c\in g}a_{|c|}=\prod _{k=1}^{n}a_{k}^{j_{k}(g)}}
140:
58:
28:
3386:
156:
7877:
van Lint, J.H.; Wilson, R.M. (1992), "35.Pólya theory of counting",
7459:{\displaystyle \sum _{k=0}^{n}F_{k}t^{k}/k!=Z(G;1+t,1,1,\ldots ,1).}
3391:
800:
is the average of the cycle index monomials of all the permutations
7571:
2558:
1334:
188:
74:
6559:{\displaystyle {\begin{matrix}{\frac {l!}{l}}=(l-1)!\end{matrix}}}
3192:. This yields the following description of the permutation types.
1391:
respectively. Thus, the cycle index of this permutation group is:
3993:
47:
7798:
Cameron, Peter J. (1994), "15. Enumeration under group action",
291:
7838:
Roberts, Fred S.; Tesman, Barry (2009), "8.5 The Cycle Index",
1314:
1862:) (in this case we would also have ordered pairs of the form (
46:
and exponents. This compact way of storing information in an
7585:. An overview of the most important results may be found at
3188:
We may visualize the types of permutations geometrically as
7881:, Cambridge: Cambridge University Press, pp. 461–474,
7802:, Cambridge: Cambridge University Press, pp. 245–256,
4440:
3862:
3400:
6417:{\displaystyle {\begin{matrix}1\leq l\leq n.\end{matrix}}}
3981:
elements equally spaced around a circle. This group has φ(
1337:
by 90°, 180°, 270° and 360° respectively. Notice that the
7842:(2nd ed.), Boca Raton: CRC Press, pp. 472–479,
2709:
of edge permutations induced by vertex permutations from
4180:, but also includes reflections. In its natural action,
1888:
at each vertex). The cycle index in this case would be:
243:(using the permutations associated with the elements of
7920:"Cycle indices of linear, affine and projective groups"
280:
and knowing what structures might be left invariant by
150:
6513:
6505:
elements of the cycle and every such choice generates
6430:
6389:
4431:
4316:
3416:
There is one such permutation and its contribution is
2136:{\displaystyle S_{3}=\{e,(23),(12),(123),(132),(13)\}}
941:
446:
321:
7349:
7193:
6991:
6865:
6578:
6511:
6485:
6387:
6337:
6118:
6088:
6082:
thus giving a simplified form for the cycle index of
5641:
5618:
5598:
5345:
5308:
5273:
5242:
5088:
4859:
4764:
4481:
4189:
4057:
3889:
3686:
3638:
3590:
3527:
3470:
3422:
3236:
3148:
3093:
3052:
2996:
2958:
2912:
2873:
2725:
2675:
2630:
2589:
2404:
2157:
2050:
1899:
1696:
1402:
1178:
1026:
894:
834:
has a unique decomposition into disjoint cycles, say
671:
440:
315:
7152:
in these actions respectively. By convention we set
7070:{\displaystyle Z(G)=Z(G;a_{1},a_{2},\ldots ,a_{n}).}
4815:{\displaystyle {\frac {1}{|A_{n}|}}={\frac {2}{n!}}}
3673:
The conclusion is that the cycle index of the group
3403:
in three-space and its group of symmetries, call it
7944:
7917:
7904:
Pólya's enumeration theorem and the symbolic method
7863:(3rd ed.), New York: Wiley, pp. 365–371,
1359:, and this permutation representation of it is its
743:. The cycle index monomial of our example would be
7817:
7780:Brualdi, Richard A. (2010), "14. Pólya Counting",
7458:
7311:
7069:
6960:
6845:
6558:
6497:
6471:
6416:
6365:
6320:
6101:
6071:
5624:
5604:
5581:
5328:
5290:
5255:
5221:
5064:
4814:
4739:
4446:
4148:
3932:
3851:
3662:
3614:
3566:
3504:
3443:
3387:The cycle index of the face permutations of a cube
3375:
3177:
3132:
3076:
3035:
2979:
2941:in its natural action) and the edge permutations (
2918:
2898:
2823:
2694:
2659:
2610:
2522:
2262:
2135:
2001:
1818:
1499:
1289:
1120:
1002:
726:
552:
384:
247:). Such a group homomorphism is formally called a
6762:
6733:
4011:, giving the number of natural numbers less than
2890:
2877:
1838:can also act on the ordered pairs of elements of
7963:
4472:in its natural action as a permutation group is
2561:of the triangle. Every permutation in the group
1519:also acts on the unordered pairs of elements of
727:{\displaystyle \prod _{k=1}^{n}a_{k}^{j_{k}(g)}}
7650:Cyclic permutations are functions and the term
5632:to keep track of the total size of the cycles:
4850:in its natural action is given by the formula:
1329:= {1,2,3,4}. The permutation representation of
7876:
7763:
7739:
3216:The cycle index of the edge permutation group
7837:
7815:
7727:
7676:
7605:
6463:
6434:
5075:that can be also stated in terms of complete
3464:to the axis of rotation. The contribution is
2948:acting on unordered pairs) that they induce:
292:Disjoint cycle representation of permutations
7800:Combinatorics:Topics, Techniques, Algorithms
2130:
2064:
937:
923:
7531:. The number of orbits of this action is Z(
6929:
6817:
6798:
6787:
6768:
1685:), and the cycle index of this action is:
653:in the cycle decomposition of permutation
104:is the number of cycles of π of size
7935:
5191:
5153:
5133:
2027:homomorphism. This is referred to as the
1842:in the same natural way. Any permutation
971:
7816:Dixon, John D.; Mortimer, Brian (1996),
3863:Cycle indices of some permutation groups
3390:
420:is the number of elements in the cycle.
7797:
7779:
7751:
7629:
7617:
4457:
2542:of the complete graph on three vertices
2041:in its natural action has the elements
1874:could be thought of as the arcs of the
251:and the image of the homomorphism is a
163:onto itself is called a permutation of
7964:
7855:
7608:, pg. 2, section 1.2 Symmetric groups
5236:labels into subsets, where there are
4825:
167:, and the set of all permutations of
135:. This is the main ingredient in the
4159:
3867:
3567:{\displaystyle 3a_{1}^{2}a_{2}^{2}.}
3133:{\displaystyle 3a_{1}^{2}a_{2}^{2}.}
3036:{\displaystyle 6a_{1}^{2}a_{2}^{2}.}
2546:We will identify the complete graph
2315:. A transitive permutation group is
649:) is the number of cycles of length
151:Permutation groups and group actions
7924:Linear Algebra and Its Applications
6262:
6166:
6008:
5904:
5888:
5691:
5643:
4023:. In the regular representation of
3933:{\displaystyle Z(E_{n})=a_{1}^{n}.}
3190:symmetries of a regular tetrahedron
2834:It happens that the complete graph
2014:
276:; for instance, counting things in
116:is also sometimes used in place of
95:of this partition: the exponent of
13:
6737:
6438:
3943:
3578:Eight 120-degree vertex rotations:
2881:
1523:in a natural way. Any permutation
139:. Performing formal algebraic and
14:
7988:
7951:Beiträge zur Elektronischen Musik
7895:
7106:-tuples of distinct elements of
5612:, while using an extra variable
3516:Three 180-degree face rotations:
3505:{\displaystyle 6a_{1}^{2}a_{4}.}
1363:. The cycle index monomials are
853:... . Let the length of a cycle
42:can be simply read off from the
7947:"Enumeration in Musical Theory"
7859:(1995), "9.3 The Cycle Index",
7757:
7745:
7733:
7721:
7712:
7327:exponential generating function
6971:
3969:is the group of rotations of a
2899:{\displaystyle {\binom {n}{2}}}
7703:
7682:
7670:
7661:
7644:
7635:
7623:
7611:
7599:
7450:
7408:
7303:
7241:
7094:also induces an action on the
7061:
7010:
7001:
6995:
6952:
6933:
6882:
6869:
6840:
6821:
6811:
6799:
6781:
6769:
6686:
6680:
6595:
6582:
6546:
6534:
6472:{\textstyle {n-1 \choose l-1}}
6354:
6341:
6247:
6234:
6162:
6149:
6139:
6126:
5953:
5929:
5684:
5671:
5388:
5378:
5267:. Every such subset generates
5202:
5185:
5173:
5124:
5105:
5092:
4876:
4863:
4787:
4772:
4600:
4590:
4498:
4485:
4378:
4366:
4294:
4282:
4238:
4225:
4206:
4193:
4117:
4111:
4099:
4074:
4061:
3906:
3893:
3696:
3690:
3626:Six 180-degree edge rotations:
3246:
3240:
2735:
2729:
2421:
2408:
2287:if for every pair of elements
2174:
2161:
2127:
2121:
2115:
2109:
2103:
2097:
2091:
2085:
2079:
2073:
1916:
1903:
1713:
1700:
1419:
1406:
1279:
1273:
1209:
1201:
1188:
1182:
1113:
1107:
1057:
1049:
988:
982:
917:
911:
719:
713:
601:(in general, use the variable
544:
538:
535:
529:
526:
520:
397:corresponds to a bijection on
1:
7937:10.1016/S0024-3795(96)00530-7
7773:
7754:, pg. 248, Proposition 15.3.1
7587:random permutation statistics
7558:This result follows from the
7519:induces an action on the set
7475:be a group acting on the set
7086:be a group acting on the set
6479:ways to choose the remaining
3455:Six 90-degree face rotations:
2705:The cycle index of the group
2330:transitive permutation group
2319:(or sometimes referred to as
877:) be the number of cycles of
784:
7945:Harald Fripertinger (1992).
7918:Harald Fripertinger (1997).
7730:, pg. 9, Corollary 1.4A(iii)
7171:ordinary generating function
7111:
3178:{\displaystyle 6a_{2}a_{4}.}
2660:{\displaystyle 3a_{1}a_{2}.}
2146:and so, its cycle index is:
1547:is the image of the element
657:. We can then associate the
7:
7564:Pólya's enumeration theorem
6569:This yields the recurrence
6377:of the cycle that contains
4750:The numerator is 2 for the
3663:{\displaystyle 6a_{2}^{3}.}
3615:{\displaystyle 8a_{3}^{2}.}
3077:{\displaystyle 8a_{3}^{2}.}
1325:as permutations of the set
50:form is frequently used in
10:
7993:
7782:Introductory Combinatorics
7764:van Lint & Wilson 1992
7740:van Lint & Wilson 1992
7507:) is also a function from
6366:{\displaystyle Z(S_{0})=1}
4758:. The 2 is needed because
3444:{\displaystyle a_{1}^{6}.}
2980:{\displaystyle a_{1}^{6}.}
2611:{\displaystyle a_{1}^{3}.}
1300:
815:be a permutation group of
254:permutation representation
7977:Enumerative combinatorics
7879:A Course in Combinatorics
7728:Dixon & Mortimer 1996
7677:Roberts & Tesman 2009
7606:Dixon & Mortimer 1996
5329:{\displaystyle S_{j_{k}}}
4032:, a permutation of order
2395:Thus its cycle index is:
137:Pólya enumeration theorem
52:combinatorial enumeration
7592:
1345:. As an abstract group,
610:to correspond to length
569:and form a partition of
175:of mappings, called the
171:forms a group under the
57:Each permutation π of a
7766:, pg. 463, Theorem 35.1
7742:, pg. 464, Example 35.1
7632:, pg. 231, section 14.3
4838:The cycle index of the
4468:The cycle index of the
3395:Cube with colored faces
2856:and the cycle index is
2695:{\displaystyle 2a_{3}.}
2538:The cycle index of the
91:, … that describes the
7824:, New York: Springer,
7583:analysis of algorithms
7576:symbolic combinatorics
7523:of all functions from
7460:
7370:
7313:
7214:
7071:
6962:
6918:
6847:
6729:
6659:
6560:
6499:
6473:
6418:
6373:and consider the size
6367:
6322:
6103:
6073:
5803:
5626:
5606:
5583:
5542:
5485:
5427:
5377:
5330:
5292:
5257:
5223:
5066:
5039:
4985:
4816:
4741:
4711:
4657:
4448:
4150:
3934:
3853:
3664:
3616:
3568:
3506:
3445:
3396:
3377:
3179:
3134:
3078:
3037:
2981:
2920:
2900:
2825:
2696:
2661:
2612:
2540:edge permutation group
2524:
2356:regular representation
2299:there is at least one
2264:
2137:
2003:
1820:
1551:under the permutation
1501:
1361:regular representation
1291:
1252:
1122:
1086:
1004:
967:
728:
692:
623:will be raised to the
614:cycles). The variable
554:
386:
110:cycle index polynomial
7861:Applied Combinatorics
7840:Applied Combinatorics
7461:
7350:
7314:
7194:
7144:denote the number of
7072:
6963:
6898:
6848:
6709:
6639:
6561:
6500:
6474:
6419:
6368:
6323:
6104:
6102:{\displaystyle S_{n}}
6074:
5783:
5627:
5607:
5584:
5522:
5465:
5407:
5357:
5331:
5293:
5258:
5256:{\displaystyle j_{k}}
5224:
5067:
5019:
4965:
4817:
4742:
4691:
4637:
4449:
4151:
3985: ) elements of
3935:
3854:
3665:
3617:
3569:
3507:
3446:
3399:Consider an ordinary
3394:
3378:
3180:
3135:
3079:
3038:
2982:
2921:
2901:
2826:
2697:
2662:
2613:
2525:
2265:
2138:
2004:
1821:
1502:
1311:rotational symmetries
1292:
1232:
1157:Then the cycle index
1123:
1066:
1005:
947:
729:
672:
555:
387:
195: ) is called a
33:group of permutations
7560:orbit counting lemma
7347:
7191:
6989:
6863:
6576:
6509:
6483:
6428:
6385:
6335:
6116:
6086:
5639:
5616:
5596:
5343:
5306:
5291:{\displaystyle k!/k}
5271:
5240:
5086:
4857:
4762:
4479:
4187:
4055:
3887:
3684:
3636:
3588:
3525:
3468:
3420:
3234:
3146:
3091:
3050:
2994:
2956:
2910:
2871:
2723:
2673:
2628:
2587:
2555:equilateral triangle
2402:
2374:= (1)(2)(3)(4)(5)(6)
2275:A permutation group
2155:
2048:
2034:The symmetric group
1897:
1870:)). The elements of
1694:
1400:
1176:
1024:
892:
826:. Every permutation
669:
659:cycle index monomial
438:
313:
239:to "act" on the set
71:cycle index monomial
7658:of these functions.
6980:we will now write:
6690:
6498:{\displaystyle l-1}
5828:
5061:
4733:
4416:
4390:
4355:
4306:
4142:
3926:
3840:
3819:
3798:
3783:
3752:
3731:
3656:
3608:
3560:
3545:
3488:
3437:
3338:
3317:
3302:
3281:
3126:
3111:
3070:
3029:
3014:
2973:
2770:
2604:
2570:vertex permutations
2495:
2474:
2456:
2209:
1990:
1969:
1951:
1781:
1766:
1748:
1472:
1454:
1305:Consider the group
1283:
1117:
811:More formally, let
739:to the permutation
723:
264:Suppose that group
125:equivalence classes
7820:Permutation Groups
7456:
7309:
7067:
6958:
6843:
6660:
6638:
6566:different cycles.
6556:
6554:
6495:
6469:
6414:
6412:
6363:
6318:
6316:
6276:
6180:
6099:
6069:
6022:
5925:
5902:
5807:
5782:
5705:
5657:
5622:
5602:
5579:
5326:
5288:
5253:
5219:
5062:
5040:
4958:
4812:
4737:
4712:
4580:
4458:Alternating group
4444:
4439:
4435:
4394:
4356:
4341:
4320:
4272:
4146:
4120:
4107:
3930:
3912:
3849:
3826:
3805:
3784:
3769:
3738:
3717:
3660:
3642:
3612:
3594:
3564:
3546:
3531:
3502:
3474:
3441:
3423:
3397:
3373:
3324:
3303:
3288:
3267:
3175:
3130:
3112:
3097:
3074:
3056:
3033:
3015:
3000:
2977:
2959:
2916:
2896:
2821:
2756:
2692:
2657:
2608:
2590:
2520:
2481:
2460:
2442:
2321:sharply transitive
2260:
2195:
2133:
1999:
1976:
1955:
1937:
1816:
1767:
1752:
1734:
1497:
1458:
1440:
1287:
1253:
1231:
1118:
1087:
1042:
1000:
945:
724:
693:
550:
511:
408:cyclic permutation
382:
376:
217:group homomorphism
183:, and denoted Sym(
7849:978-1-4200-9982-9
7791:978-0-13-602040-0
6896:
6760:
6707:
6616:
6614:
6529:
6461:
6304:
6261:
6218:
6165:
6060:
6007:
6002:
5903:
5887:
5882:
5706:
5690:
5642:
5625:{\displaystyle y}
5605:{\displaystyle n}
5574:
5506:
5446:
5405:
5298:cycles of length
5214:
5017:
4882:
4810:
4792:
4752:even permutations
4689:
4504:
4470:alternating group
4434:
4334:
4319:
4260:
4220:
4090:
4088:
4044:cycles of length
3710:
3260:
2919:{\displaystyle n}
2906:) than vertices (
2888:
2749:
2435:
2383:= (1 4)(2 5)(3 6)
2361:The cyclic group
2188:
1930:
1727:
1433:
1216:
1214:
1131:in the variables
1027:
944:
795:permutation group
418:length of a cycle
197:permutation group
7984:
7958:
7941:
7939:
7891:
7873:
7852:
7834:
7823:
7812:
7794:
7767:
7761:
7755:
7749:
7743:
7737:
7731:
7725:
7719:
7716:
7710:
7707:
7701:
7686:
7680:
7674:
7668:
7665:
7659:
7648:
7642:
7639:
7633:
7627:
7621:
7615:
7609:
7603:
7483:a function from
7465:
7463:
7462:
7457:
7395:
7390:
7389:
7380:
7379:
7369:
7364:
7318:
7316:
7315:
7310:
7302:
7301:
7277:
7276:
7234:
7233:
7224:
7223:
7213:
7208:
7076:
7074:
7073:
7068:
7060:
7059:
7041:
7040:
7028:
7027:
6967:
6965:
6964:
6959:
6951:
6950:
6928:
6927:
6917:
6912:
6897:
6889:
6881:
6880:
6852:
6850:
6849:
6844:
6839:
6838:
6797:
6796:
6767:
6766:
6765:
6759:
6748:
6736:
6728:
6723:
6708:
6706:
6695:
6689:
6679:
6678:
6668:
6658:
6653:
6637:
6636:
6635:
6615:
6613:
6602:
6594:
6593:
6565:
6563:
6562:
6557:
6555:
6530:
6525:
6517:
6504:
6502:
6501:
6496:
6478:
6476:
6475:
6470:
6468:
6467:
6466:
6460:
6449:
6437:
6423:
6421:
6420:
6415:
6413:
6372:
6370:
6369:
6364:
6353:
6352:
6327:
6325:
6324:
6319:
6317:
6310:
6306:
6305:
6300:
6299:
6298:
6289:
6288:
6278:
6275:
6246:
6245:
6227:
6223:
6219:
6214:
6213:
6212:
6203:
6202:
6192:
6179:
6161:
6160:
6138:
6137:
6108:
6106:
6105:
6100:
6098:
6097:
6078:
6076:
6075:
6070:
6065:
6061:
6056:
6055:
6054:
6045:
6044:
6034:
6021:
6003:
6001:
5997:
5996:
5987:
5986:
5985:
5984:
5969:
5968:
5967:
5966:
5965:
5951:
5950:
5941:
5940:
5927:
5924:
5917:
5916:
5901:
5883:
5881:
5877:
5876:
5867:
5866:
5865:
5864:
5849:
5848:
5847:
5846:
5845:
5827:
5826:
5825:
5815:
5805:
5802:
5797:
5781:
5774:
5773:
5752:
5751:
5736:
5735:
5720:
5719:
5704:
5683:
5682:
5667:
5666:
5656:
5631:
5629:
5628:
5623:
5611:
5609:
5608:
5603:
5588:
5586:
5585:
5580:
5575:
5573:
5569:
5568:
5559:
5558:
5557:
5556:
5541:
5536:
5520:
5512:
5507:
5505:
5501:
5500:
5487:
5484:
5479:
5464:
5463:
5462:
5461:
5451:
5447:
5442:
5434:
5426:
5421:
5406:
5404:
5403:
5402:
5401:
5400:
5376:
5371:
5355:
5347:
5335:
5333:
5332:
5327:
5325:
5324:
5323:
5322:
5297:
5295:
5294:
5289:
5284:
5263:subsets of size
5262:
5260:
5259:
5254:
5252:
5251:
5228:
5226:
5225:
5220:
5215:
5213:
5205:
5201:
5200:
5163:
5162:
5143:
5142:
5123:
5122:
5112:
5104:
5103:
5077:Bell polynomials
5071:
5069:
5068:
5063:
5060:
5059:
5058:
5048:
5038:
5033:
5018:
5016:
5012:
5011:
5002:
5001:
5000:
4999:
4984:
4979:
4960:
4957:
4950:
4949:
4928:
4927:
4912:
4911:
4896:
4895:
4875:
4874:
4826:Symmetric group
4821:
4819:
4818:
4813:
4811:
4809:
4798:
4793:
4791:
4790:
4785:
4784:
4775:
4766:
4756:odd permutations
4754:, and 0 for the
4746:
4744:
4743:
4738:
4732:
4731:
4730:
4720:
4710:
4705:
4690:
4688:
4684:
4683:
4674:
4673:
4672:
4671:
4656:
4651:
4635:
4634:
4633:
4626:
4625:
4613:
4612:
4582:
4579:
4572:
4571:
4550:
4549:
4534:
4533:
4518:
4517:
4497:
4496:
4453:
4451:
4450:
4445:
4443:
4442:
4436:
4432:
4421:
4417:
4415:
4411:
4402:
4389:
4385:
4364:
4354:
4349:
4335:
4327:
4321:
4318: odd,
4317:
4305:
4301:
4280:
4271:
4270:
4261:
4253:
4237:
4236:
4221:
4213:
4205:
4204:
4155:
4153:
4152:
4147:
4141:
4137:
4128:
4106:
4102:
4089:
4081:
4073:
4072:
4017:relatively prime
4009:Euler φ-function
4007: ) is the
3939:
3937:
3936:
3931:
3925:
3920:
3905:
3904:
3858:
3856:
3855:
3850:
3845:
3841:
3839:
3834:
3818:
3813:
3797:
3792:
3782:
3777:
3762:
3761:
3751:
3746:
3730:
3725:
3711:
3703:
3669:
3667:
3666:
3661:
3655:
3650:
3621:
3619:
3618:
3613:
3607:
3602:
3573:
3571:
3570:
3565:
3559:
3554:
3544:
3539:
3511:
3509:
3508:
3503:
3498:
3497:
3487:
3482:
3450:
3448:
3447:
3442:
3436:
3431:
3382:
3380:
3379:
3374:
3369:
3365:
3364:
3363:
3354:
3353:
3337:
3332:
3316:
3311:
3301:
3296:
3280:
3275:
3261:
3253:
3184:
3182:
3181:
3176:
3171:
3170:
3161:
3160:
3139:
3137:
3136:
3131:
3125:
3120:
3110:
3105:
3083:
3081:
3080:
3075:
3069:
3064:
3042:
3040:
3039:
3034:
3028:
3023:
3013:
3008:
2986:
2984:
2983:
2978:
2972:
2967:
2925:
2923:
2922:
2917:
2905:
2903:
2902:
2897:
2895:
2894:
2893:
2880:
2830:
2828:
2827:
2822:
2817:
2813:
2812:
2811:
2796:
2795:
2786:
2785:
2769:
2764:
2750:
2742:
2701:
2699:
2698:
2693:
2688:
2687:
2666:
2664:
2663:
2658:
2653:
2652:
2643:
2642:
2617:
2615:
2614:
2609:
2603:
2598:
2529:
2527:
2526:
2521:
2516:
2512:
2511:
2510:
2494:
2489:
2473:
2468:
2455:
2450:
2436:
2428:
2420:
2419:
2389:= (1 6 5 4 3 2).
2386:= (1 5 3)(2 6 4)
2380:= (1 3 5)(2 4 6)
2352:Cayley's theorem
2269:
2267:
2266:
2261:
2256:
2252:
2251:
2250:
2235:
2234:
2225:
2224:
2208:
2203:
2189:
2181:
2173:
2172:
2142:
2140:
2139:
2134:
2060:
2059:
2015:Types of actions
2008:
2006:
2005:
2000:
1995:
1991:
1989:
1984:
1968:
1963:
1950:
1945:
1931:
1923:
1915:
1914:
1876:complete digraph
1825:
1823:
1822:
1817:
1812:
1808:
1807:
1806:
1797:
1796:
1780:
1775:
1765:
1760:
1747:
1742:
1728:
1720:
1712:
1711:
1506:
1504:
1503:
1498:
1493:
1489:
1488:
1487:
1471:
1466:
1453:
1448:
1434:
1426:
1418:
1417:
1349:is known as the
1296:
1294:
1293:
1288:
1282:
1272:
1271:
1261:
1251:
1246:
1230:
1215:
1213:
1212:
1204:
1195:
1127:
1125:
1124:
1119:
1116:
1106:
1105:
1095:
1085:
1080:
1062:
1061:
1060:
1052:
1041:
1013:We associate to
1009:
1007:
1006:
1001:
981:
980:
966:
961:
946:
942:
933:
910:
909:
733:
731:
730:
725:
722:
712:
711:
701:
691:
686:
559:
557:
556:
551:
516:
512:
391:
389:
388:
383:
381:
377:
187: ). Every
7992:
7991:
7987:
7986:
7985:
7983:
7982:
7981:
7962:
7961:
7898:
7889:
7871:
7850:
7832:
7810:
7792:
7776:
7771:
7770:
7762:
7758:
7750:
7746:
7738:
7734:
7726:
7722:
7717:
7713:
7708:
7704:
7687:
7683:
7675:
7671:
7666:
7662:
7649:
7645:
7640:
7636:
7628:
7624:
7616:
7612:
7604:
7600:
7595:
7391:
7385:
7381:
7375:
7371:
7365:
7354:
7348:
7345:
7344:
7337:
7297:
7293:
7272:
7268:
7229:
7225:
7219:
7215:
7209:
7198:
7192:
7189:
7188:
7181:
7165:
7158:
7143:
7134:
7055:
7051:
7036:
7032:
7023:
7019:
6990:
6987:
6986:
6974:
6940:
6936:
6923:
6919:
6913:
6902:
6888:
6876:
6872:
6864:
6861:
6860:
6828:
6824:
6792:
6788:
6761:
6749:
6738:
6732:
6731:
6730:
6724:
6713:
6699:
6694:
6674:
6670:
6669:
6664:
6654:
6643:
6631:
6627:
6620:
6606:
6601:
6589:
6585:
6577:
6574:
6573:
6553:
6552:
6518:
6516:
6512:
6510:
6507:
6506:
6484:
6481:
6480:
6462:
6450:
6439:
6433:
6432:
6431:
6429:
6426:
6425:
6411:
6410:
6388:
6386:
6383:
6382:
6348:
6344:
6336:
6333:
6332:
6315:
6314:
6294:
6290:
6284:
6280:
6279:
6277:
6265:
6260:
6256:
6241:
6237:
6225:
6224:
6208:
6204:
6198:
6194:
6193:
6191:
6187:
6169:
6156:
6152:
6142:
6133:
6129:
6119:
6117:
6114:
6113:
6093:
6089:
6087:
6084:
6083:
6050:
6046:
6040:
6036:
6035:
6033:
6029:
6011:
5992:
5988:
5980:
5976:
5975:
5971:
5970:
5961:
5957:
5956:
5952:
5946:
5942:
5936:
5932:
5928:
5926:
5912:
5908:
5907:
5891:
5872:
5868:
5860:
5856:
5855:
5851:
5850:
5841:
5837:
5833:
5829:
5821:
5817:
5816:
5811:
5806:
5804:
5798:
5787:
5769:
5765:
5747:
5743:
5731:
5727:
5715:
5711:
5710:
5694:
5678:
5674:
5662:
5658:
5646:
5640:
5637:
5636:
5617:
5614:
5613:
5597:
5594:
5593:
5564:
5560:
5552:
5548:
5547:
5543:
5537:
5526:
5521:
5513:
5511:
5496:
5492:
5491:
5486:
5480:
5469:
5457:
5453:
5452:
5435:
5433:
5429:
5428:
5422:
5411:
5396:
5392:
5391:
5387:
5372:
5361:
5356:
5348:
5346:
5344:
5341:
5340:
5318:
5314:
5313:
5309:
5307:
5304:
5303:
5280:
5272:
5269:
5268:
5247:
5243:
5241:
5238:
5237:
5206:
5196:
5192:
5158:
5154:
5138:
5134:
5118:
5114:
5113:
5111:
5099:
5095:
5087:
5084:
5083:
5054:
5050:
5049:
5044:
5034:
5023:
5007:
5003:
4995:
4991:
4990:
4986:
4980:
4969:
4964:
4959:
4945:
4941:
4923:
4919:
4907:
4903:
4891:
4887:
4886:
4870:
4866:
4858:
4855:
4854:
4849:
4840:symmetric group
4836:
4834:
4802:
4797:
4786:
4780:
4776:
4771:
4770:
4765:
4763:
4760:
4759:
4726:
4722:
4721:
4716:
4706:
4695:
4679:
4675:
4667:
4663:
4662:
4658:
4652:
4641:
4636:
4621:
4617:
4608:
4604:
4603:
4599:
4583:
4581:
4567:
4563:
4545:
4541:
4529:
4525:
4513:
4509:
4508:
4492:
4488:
4480:
4477:
4476:
4466:
4464:
4438:
4437:
4430:
4425:
4407:
4403:
4398:
4381:
4365:
4360:
4350:
4345:
4340:
4336:
4326:
4323:
4322:
4315:
4310:
4297:
4281:
4276:
4266:
4262:
4252:
4245:
4244:
4232:
4228:
4212:
4200:
4196:
4188:
4185:
4184:
4170:
4168:
4160:Dihedral group
4133:
4129:
4124:
4098:
4094:
4080:
4068:
4064:
4056:
4053:
4052:
4031:
3968:
3954:
3952:
3921:
3916:
3900:
3896:
3888:
3885:
3884:
3878:
3876:
3868:Identity group
3865:
3835:
3830:
3814:
3809:
3793:
3788:
3778:
3773:
3757:
3753:
3747:
3742:
3726:
3721:
3716:
3712:
3702:
3685:
3682:
3681:
3651:
3646:
3637:
3634:
3633:
3603:
3598:
3589:
3586:
3585:
3555:
3550:
3540:
3535:
3526:
3523:
3522:
3493:
3489:
3483:
3478:
3469:
3466:
3465:
3432:
3427:
3421:
3418:
3417:
3389:
3359:
3355:
3349:
3345:
3333:
3328:
3312:
3307:
3297:
3292:
3276:
3271:
3266:
3262:
3252:
3235:
3232:
3231:
3226:
3210:rotoreflections
3166:
3162:
3156:
3152:
3147:
3144:
3143:
3121:
3116:
3106:
3101:
3092:
3089:
3088:
3065:
3060:
3051:
3048:
3047:
3024:
3019:
3009:
3004:
2995:
2992:
2991:
2968:
2963:
2957:
2954:
2953:
2947:
2940:
2932:
2911:
2908:
2907:
2889:
2876:
2875:
2874:
2872:
2869:
2868:
2866:
2855:
2840:
2807:
2803:
2791:
2787:
2781:
2777:
2765:
2760:
2755:
2751:
2741:
2724:
2721:
2720:
2715:
2683:
2679:
2674:
2671:
2670:
2648:
2644:
2638:
2634:
2629:
2626:
2625:
2599:
2594:
2588:
2585:
2584:
2578:
2567:
2552:
2544:
2506:
2502:
2490:
2485:
2469:
2464:
2451:
2446:
2441:
2437:
2427:
2415:
2411:
2403:
2400:
2399:
2377:= (1 2 3 4 5 6)
2367:
2246:
2242:
2230:
2226:
2220:
2216:
2204:
2199:
2194:
2190:
2180:
2168:
2164:
2156:
2153:
2152:
2055:
2051:
2049:
2046:
2045:
2040:
2017:
1985:
1980:
1964:
1959:
1946:
1941:
1936:
1932:
1922:
1910:
1906:
1898:
1895:
1894:
1883:
1837:
1802:
1798:
1792:
1788:
1776:
1771:
1761:
1756:
1743:
1738:
1733:
1729:
1719:
1707:
1703:
1695:
1692:
1691:
1616:
1518:
1483:
1479:
1467:
1462:
1449:
1444:
1439:
1435:
1425:
1413:
1409:
1401:
1398:
1397:
1390:
1383:
1376:
1369:
1358:
1319:Euclidean plane
1303:
1267:
1263:
1262:
1257:
1247:
1236:
1220:
1208:
1200:
1199:
1194:
1177:
1174:
1173:
1153:
1144:
1137:
1101:
1097:
1096:
1091:
1081:
1070:
1056:
1048:
1047:
1043:
1031:
1025:
1022:
1021:
976:
972:
962:
951:
943: and
940:
929:
905:
901:
893:
890:
889:
872:
857:be denoted by |
852:
846:
840:
787:
780:
774:
768:
761:
755:
749:
707:
703:
702:
697:
687:
676:
670:
667:
666:
644:
631:
622:
609:
600:
593:
586:
510:
509:
504:
499:
494:
489:
484:
478:
477:
472:
467:
462:
457:
452:
445:
441:
439:
436:
435:
375:
374:
369:
364:
359:
354:
348:
347:
342:
337:
332:
327:
320:
316:
314:
311:
310:
294:
177:symmetric group
159:map from a set
153:
114:cycle indicator
103:
90:
83:
12:
11:
5:
7990:
7980:
7979:
7974:
7960:
7959:
7942:
7915:
7909:Marko Riedel,
7907:
7901:Marko Riedel,
7897:
7896:External links
7894:
7893:
7892:
7887:
7874:
7869:
7853:
7848:
7835:
7830:
7813:
7808:
7795:
7790:
7775:
7772:
7769:
7768:
7756:
7744:
7732:
7720:
7711:
7702:
7681:
7669:
7660:
7643:
7634:
7622:
7610:
7597:
7596:
7594:
7591:
7574:. The area of
7469:
7468:
7467:
7466:
7455:
7452:
7449:
7446:
7443:
7440:
7437:
7434:
7431:
7428:
7425:
7422:
7419:
7416:
7413:
7410:
7407:
7404:
7401:
7398:
7394:
7388:
7384:
7378:
7374:
7368:
7363:
7360:
7357:
7353:
7333:
7323:
7322:
7321:
7320:
7308:
7305:
7300:
7296:
7292:
7289:
7286:
7283:
7280:
7275:
7271:
7267:
7264:
7261:
7258:
7255:
7252:
7249:
7246:
7243:
7240:
7237:
7232:
7228:
7222:
7218:
7212:
7207:
7204:
7201:
7197:
7177:
7166:= 1. We have:
7163:
7156:
7139:
7130:
7118:= 2), for 1 ≤
7080:
7079:
7078:
7077:
7066:
7063:
7058:
7054:
7050:
7047:
7044:
7039:
7035:
7031:
7026:
7022:
7018:
7015:
7012:
7009:
7006:
7003:
7000:
6997:
6994:
6973:
6970:
6969:
6968:
6957:
6954:
6949:
6946:
6943:
6939:
6935:
6932:
6926:
6922:
6916:
6911:
6908:
6905:
6901:
6895:
6892:
6887:
6884:
6879:
6875:
6871:
6868:
6854:
6853:
6842:
6837:
6834:
6831:
6827:
6823:
6820:
6816:
6813:
6810:
6807:
6804:
6801:
6795:
6791:
6786:
6783:
6780:
6777:
6774:
6771:
6764:
6758:
6755:
6752:
6747:
6744:
6741:
6735:
6727:
6722:
6719:
6716:
6712:
6705:
6702:
6698:
6693:
6688:
6685:
6682:
6677:
6673:
6667:
6663:
6657:
6652:
6649:
6646:
6642:
6634:
6630:
6626:
6623:
6619:
6612:
6609:
6605:
6600:
6597:
6592:
6588:
6584:
6581:
6551:
6548:
6545:
6542:
6539:
6536:
6533:
6528:
6524:
6521:
6515:
6514:
6494:
6491:
6488:
6465:
6459:
6456:
6453:
6448:
6445:
6442:
6436:
6409:
6406:
6403:
6400:
6397:
6394:
6391:
6390:
6362:
6359:
6356:
6351:
6347:
6343:
6340:
6329:
6328:
6313:
6309:
6303:
6297:
6293:
6287:
6283:
6274:
6271:
6268:
6264:
6259:
6255:
6252:
6249:
6244:
6240:
6236:
6233:
6230:
6228:
6226:
6222:
6217:
6211:
6207:
6201:
6197:
6190:
6186:
6183:
6178:
6175:
6172:
6168:
6164:
6159:
6155:
6151:
6148:
6145:
6143:
6141:
6136:
6132:
6128:
6125:
6122:
6121:
6096:
6092:
6080:
6079:
6068:
6064:
6059:
6053:
6049:
6043:
6039:
6032:
6028:
6025:
6020:
6017:
6014:
6010:
6006:
6000:
5995:
5991:
5983:
5979:
5974:
5964:
5960:
5955:
5949:
5945:
5939:
5935:
5931:
5923:
5920:
5915:
5911:
5906:
5900:
5897:
5894:
5890:
5886:
5880:
5875:
5871:
5863:
5859:
5854:
5844:
5840:
5836:
5832:
5824:
5820:
5814:
5810:
5801:
5796:
5793:
5790:
5786:
5780:
5777:
5772:
5768:
5764:
5761:
5758:
5755:
5750:
5746:
5742:
5739:
5734:
5730:
5726:
5723:
5718:
5714:
5709:
5703:
5700:
5697:
5693:
5689:
5686:
5681:
5677:
5673:
5670:
5665:
5661:
5655:
5652:
5649:
5645:
5621:
5601:
5590:
5589:
5578:
5572:
5567:
5563:
5555:
5551:
5546:
5540:
5535:
5532:
5529:
5525:
5519:
5516:
5510:
5504:
5499:
5495:
5490:
5483:
5478:
5475:
5472:
5468:
5460:
5456:
5450:
5445:
5441:
5438:
5432:
5425:
5420:
5417:
5414:
5410:
5399:
5395:
5390:
5386:
5383:
5380:
5375:
5370:
5367:
5364:
5360:
5354:
5351:
5336:. This yields
5321:
5317:
5312:
5287:
5283:
5279:
5276:
5250:
5246:
5230:
5229:
5218:
5212:
5209:
5204:
5199:
5195:
5190:
5187:
5184:
5181:
5178:
5175:
5172:
5169:
5166:
5161:
5157:
5152:
5149:
5146:
5141:
5137:
5132:
5129:
5126:
5121:
5117:
5110:
5107:
5102:
5098:
5094:
5091:
5073:
5072:
5057:
5053:
5047:
5043:
5037:
5032:
5029:
5026:
5022:
5015:
5010:
5006:
4998:
4994:
4989:
4983:
4978:
4975:
4972:
4968:
4963:
4956:
4953:
4948:
4944:
4940:
4937:
4934:
4931:
4926:
4922:
4918:
4915:
4910:
4906:
4902:
4899:
4894:
4890:
4885:
4881:
4878:
4873:
4869:
4865:
4862:
4845:
4835:
4830:
4824:
4808:
4805:
4801:
4796:
4789:
4783:
4779:
4774:
4769:
4748:
4747:
4736:
4729:
4725:
4719:
4715:
4709:
4704:
4701:
4698:
4694:
4687:
4682:
4678:
4670:
4666:
4661:
4655:
4650:
4647:
4644:
4640:
4632:
4629:
4624:
4620:
4616:
4611:
4607:
4602:
4598:
4595:
4592:
4589:
4586:
4578:
4575:
4570:
4566:
4562:
4559:
4556:
4553:
4548:
4544:
4540:
4537:
4532:
4528:
4524:
4521:
4516:
4512:
4507:
4503:
4500:
4495:
4491:
4487:
4484:
4465:
4462:
4456:
4455:
4454:
4441:
4429:
4426:
4424:
4420:
4414:
4410:
4406:
4401:
4397:
4393:
4388:
4384:
4380:
4377:
4374:
4371:
4368:
4363:
4359:
4353:
4348:
4344:
4339:
4333:
4330:
4325:
4324:
4314:
4311:
4309:
4304:
4300:
4296:
4293:
4290:
4287:
4284:
4279:
4275:
4269:
4265:
4259:
4256:
4251:
4250:
4248:
4243:
4240:
4235:
4231:
4227:
4224:
4219:
4216:
4211:
4208:
4203:
4199:
4195:
4192:
4174:dihedral group
4169:
4164:
4158:
4157:
4156:
4145:
4140:
4136:
4132:
4127:
4123:
4119:
4116:
4113:
4110:
4105:
4101:
4097:
4093:
4087:
4084:
4079:
4076:
4071:
4067:
4063:
4060:
4027:
3964:
3953:
3948:
3942:
3941:
3940:
3929:
3924:
3919:
3915:
3911:
3908:
3903:
3899:
3895:
3892:
3877:
3872:
3866:
3864:
3861:
3860:
3859:
3848:
3844:
3838:
3833:
3829:
3825:
3822:
3817:
3812:
3808:
3804:
3801:
3796:
3791:
3787:
3781:
3776:
3772:
3768:
3765:
3760:
3756:
3750:
3745:
3741:
3737:
3734:
3729:
3724:
3720:
3715:
3709:
3706:
3701:
3698:
3695:
3692:
3689:
3671:
3670:
3659:
3654:
3649:
3645:
3641:
3628:
3627:
3623:
3622:
3611:
3606:
3601:
3597:
3593:
3580:
3579:
3575:
3574:
3563:
3558:
3553:
3549:
3543:
3538:
3534:
3530:
3518:
3517:
3513:
3512:
3501:
3496:
3492:
3486:
3481:
3477:
3473:
3457:
3456:
3452:
3451:
3440:
3435:
3430:
3426:
3413:
3412:
3388:
3385:
3384:
3383:
3372:
3368:
3362:
3358:
3352:
3348:
3344:
3341:
3336:
3331:
3327:
3323:
3320:
3315:
3310:
3306:
3300:
3295:
3291:
3287:
3284:
3279:
3274:
3270:
3265:
3259:
3256:
3251:
3248:
3245:
3242:
3239:
3224:
3214:
3213:
3212:by 90 degrees.
3206:
3203:
3200:
3197:
3186:
3185:
3174:
3169:
3165:
3159:
3155:
3151:
3140:
3129:
3124:
3119:
3115:
3109:
3104:
3100:
3096:
3084:
3073:
3068:
3063:
3059:
3055:
3043:
3032:
3027:
3022:
3018:
3012:
3007:
3003:
2999:
2987:
2976:
2971:
2966:
2962:
2945:
2938:
2931:
2928:
2915:
2892:
2887:
2884:
2879:
2864:
2853:
2838:
2832:
2831:
2820:
2816:
2810:
2806:
2802:
2799:
2794:
2790:
2784:
2780:
2776:
2773:
2768:
2763:
2759:
2754:
2748:
2745:
2740:
2737:
2734:
2731:
2728:
2713:
2703:
2702:
2691:
2686:
2682:
2678:
2667:
2656:
2651:
2647:
2641:
2637:
2633:
2618:
2607:
2602:
2597:
2593:
2576:
2565:
2550:
2543:
2536:
2531:
2530:
2519:
2515:
2509:
2505:
2501:
2498:
2493:
2488:
2484:
2480:
2477:
2472:
2467:
2463:
2459:
2454:
2449:
2445:
2440:
2434:
2431:
2426:
2423:
2418:
2414:
2410:
2407:
2393:
2392:
2391:
2390:
2387:
2384:
2381:
2378:
2375:
2365:
2358:of the group.
2340:if and only if
2273:
2272:
2271:
2270:
2259:
2255:
2249:
2245:
2241:
2238:
2233:
2229:
2223:
2219:
2215:
2212:
2207:
2202:
2198:
2193:
2187:
2184:
2179:
2176:
2171:
2167:
2163:
2160:
2144:
2143:
2132:
2129:
2126:
2123:
2120:
2117:
2114:
2111:
2108:
2105:
2102:
2099:
2096:
2093:
2090:
2087:
2084:
2081:
2078:
2075:
2072:
2069:
2066:
2063:
2058:
2054:
2038:
2029:natural action
2016:
2013:
2012:
2011:
2010:
2009:
1998:
1994:
1988:
1983:
1979:
1975:
1972:
1967:
1962:
1958:
1954:
1949:
1944:
1940:
1935:
1929:
1926:
1921:
1918:
1913:
1909:
1905:
1902:
1881:
1835:
1829:
1828:
1827:
1826:
1815:
1811:
1805:
1801:
1795:
1791:
1787:
1784:
1779:
1774:
1770:
1764:
1759:
1755:
1751:
1746:
1741:
1737:
1732:
1726:
1723:
1718:
1715:
1710:
1706:
1702:
1699:
1614:
1609:complete graph
1516:
1510:
1509:
1508:
1507:
1496:
1492:
1486:
1482:
1478:
1475:
1470:
1465:
1461:
1457:
1452:
1447:
1443:
1438:
1432:
1429:
1424:
1421:
1416:
1412:
1408:
1405:
1388:
1381:
1374:
1367:
1356:
1302:
1299:
1298:
1297:
1286:
1281:
1278:
1275:
1270:
1266:
1260:
1256:
1250:
1245:
1242:
1239:
1235:
1229:
1226:
1223:
1219:
1211:
1207:
1203:
1198:
1193:
1190:
1187:
1184:
1181:
1149:
1142:
1135:
1129:
1128:
1115:
1112:
1109:
1104:
1100:
1094:
1090:
1084:
1079:
1076:
1073:
1069:
1065:
1059:
1055:
1051:
1046:
1040:
1037:
1034:
1030:
1011:
1010:
999:
996:
993:
990:
987:
984:
979:
975:
970:
965:
960:
957:
954:
950:
939:
936:
932:
928:
925:
922:
919:
916:
913:
908:
904:
900:
897:
868:
850:
844:
838:
786:
783:
778:
772:
766:
759:
753:
747:
737:
736:
735:
734:
721:
718:
715:
710:
706:
700:
696:
690:
685:
682:
679:
675:
640:
636:) power where
627:
618:
605:
598:
591:
584:
563:
562:
561:
560:
549:
546:
543:
540:
537:
534:
531:
528:
525:
522:
519:
515:
508:
505:
503:
500:
498:
495:
493:
490:
488:
485:
483:
480:
479:
476:
473:
471:
468:
466:
463:
461:
458:
456:
453:
451:
448:
447:
444:
413:cycle notation
395:
394:
393:
392:
380:
373:
370:
368:
365:
363:
360:
358:
355:
353:
350:
349:
346:
343:
341:
338:
336:
333:
331:
328:
326:
323:
322:
319:
293:
290:
227: ). The
213:abstract group
207: |. Let
152:
149:
145:species theory
99:
88:
81:
65:that set into
9:
6:
4:
3:
2:
7989:
7978:
7975:
7973:
7972:Combinatorics
7970:
7969:
7967:
7956:
7952:
7948:
7943:
7938:
7933:
7929:
7925:
7921:
7916:
7914:
7913:
7908:
7906:
7905:
7900:
7899:
7890:
7888:0-521-42260-4
7884:
7880:
7875:
7872:
7870:0-471-59504-7
7866:
7862:
7858:
7854:
7851:
7845:
7841:
7836:
7833:
7831:0-387-94599-7
7827:
7822:
7821:
7814:
7811:
7809:0-521-45761-0
7805:
7801:
7796:
7793:
7787:
7783:
7778:
7777:
7765:
7760:
7753:
7748:
7741:
7736:
7729:
7724:
7715:
7706:
7699:
7695:
7691:
7685:
7678:
7673:
7664:
7657:
7654:really means
7653:
7647:
7638:
7631:
7626:
7620:, pp. 227–228
7619:
7614:
7607:
7602:
7598:
7590:
7588:
7584:
7579:
7577:
7573:
7567:
7565:
7561:
7556:
7554:
7550:
7546:
7542:
7538:
7534:
7530:
7526:
7522:
7518:
7514:
7510:
7506:
7502:
7498:
7494:
7490:
7486:
7482:
7478:
7474:
7453:
7447:
7444:
7441:
7438:
7435:
7432:
7429:
7426:
7423:
7420:
7417:
7414:
7411:
7405:
7402:
7399:
7396:
7392:
7386:
7382:
7376:
7372:
7366:
7361:
7358:
7355:
7351:
7343:
7342:
7341:
7340:
7339:
7338:is given by:
7336:
7332:
7328:
7306:
7298:
7294:
7290:
7287:
7284:
7281:
7278:
7273:
7269:
7265:
7262:
7259:
7256:
7253:
7250:
7247:
7244:
7238:
7235:
7230:
7226:
7220:
7216:
7210:
7205:
7202:
7199:
7195:
7187:
7186:
7185:
7184:
7183:
7182:is given by:
7180:
7176:
7172:
7167:
7162:
7155:
7151:
7147:
7142:
7138:
7133:
7129:
7125:
7121:
7117:
7114:for the case
7113:
7109:
7105:
7101:
7097:
7093:
7089:
7085:
7064:
7056:
7052:
7048:
7045:
7042:
7037:
7033:
7029:
7024:
7020:
7016:
7013:
7007:
7004:
6998:
6992:
6985:
6984:
6983:
6982:
6981:
6979:
6955:
6947:
6944:
6941:
6937:
6930:
6924:
6920:
6914:
6909:
6906:
6903:
6899:
6893:
6890:
6885:
6877:
6873:
6866:
6859:
6858:
6857:
6835:
6832:
6829:
6825:
6818:
6814:
6808:
6805:
6802:
6793:
6789:
6784:
6778:
6775:
6772:
6756:
6753:
6750:
6745:
6742:
6739:
6725:
6720:
6717:
6714:
6710:
6703:
6700:
6696:
6691:
6683:
6675:
6671:
6665:
6661:
6655:
6650:
6647:
6644:
6640:
6632:
6628:
6624:
6621:
6617:
6610:
6607:
6603:
6598:
6590:
6586:
6579:
6572:
6571:
6570:
6567:
6549:
6543:
6540:
6537:
6531:
6526:
6522:
6519:
6492:
6489:
6486:
6457:
6454:
6451:
6446:
6443:
6440:
6407:
6404:
6401:
6398:
6395:
6392:
6380:
6376:
6360:
6357:
6349:
6345:
6338:
6311:
6307:
6301:
6295:
6291:
6285:
6281:
6272:
6269:
6266:
6257:
6253:
6250:
6242:
6238:
6231:
6229:
6220:
6215:
6209:
6205:
6199:
6195:
6188:
6184:
6181:
6176:
6173:
6170:
6157:
6153:
6146:
6144:
6134:
6130:
6123:
6112:
6111:
6110:
6094:
6090:
6066:
6062:
6057:
6051:
6047:
6041:
6037:
6030:
6026:
6023:
6018:
6015:
6012:
6004:
5998:
5993:
5989:
5981:
5977:
5972:
5962:
5958:
5947:
5943:
5937:
5933:
5921:
5918:
5913:
5909:
5898:
5895:
5892:
5884:
5878:
5873:
5869:
5861:
5857:
5852:
5842:
5838:
5834:
5830:
5822:
5818:
5812:
5808:
5799:
5794:
5791:
5788:
5784:
5778:
5775:
5770:
5766:
5762:
5759:
5756:
5753:
5748:
5744:
5740:
5737:
5732:
5728:
5724:
5721:
5716:
5712:
5707:
5701:
5698:
5695:
5687:
5679:
5675:
5668:
5663:
5659:
5653:
5650:
5647:
5635:
5634:
5633:
5619:
5599:
5576:
5570:
5565:
5561:
5553:
5549:
5544:
5538:
5533:
5530:
5527:
5523:
5517:
5514:
5508:
5502:
5497:
5493:
5488:
5481:
5476:
5473:
5470:
5466:
5458:
5454:
5448:
5443:
5439:
5436:
5430:
5423:
5418:
5415:
5412:
5408:
5397:
5393:
5384:
5381:
5373:
5368:
5365:
5362:
5358:
5352:
5349:
5339:
5338:
5337:
5319:
5315:
5310:
5301:
5285:
5281:
5277:
5274:
5266:
5248:
5244:
5235:
5216:
5210:
5207:
5197:
5193:
5188:
5182:
5179:
5176:
5170:
5167:
5164:
5159:
5155:
5150:
5147:
5144:
5139:
5135:
5130:
5127:
5119:
5115:
5108:
5100:
5096:
5089:
5082:
5081:
5080:
5078:
5055:
5051:
5045:
5041:
5035:
5030:
5027:
5024:
5020:
5013:
5008:
5004:
4996:
4992:
4987:
4981:
4976:
4973:
4970:
4966:
4961:
4954:
4951:
4946:
4942:
4938:
4935:
4932:
4929:
4924:
4920:
4916:
4913:
4908:
4904:
4900:
4897:
4892:
4888:
4883:
4879:
4871:
4867:
4860:
4853:
4852:
4851:
4848:
4844:
4841:
4833:
4829:
4823:
4806:
4803:
4799:
4794:
4781:
4777:
4767:
4757:
4753:
4734:
4727:
4723:
4717:
4713:
4707:
4702:
4699:
4696:
4692:
4685:
4680:
4676:
4668:
4664:
4659:
4653:
4648:
4645:
4642:
4638:
4630:
4627:
4622:
4618:
4614:
4609:
4605:
4596:
4593:
4587:
4584:
4576:
4573:
4568:
4564:
4560:
4557:
4554:
4551:
4546:
4542:
4538:
4535:
4530:
4526:
4522:
4519:
4514:
4510:
4505:
4501:
4493:
4489:
4482:
4475:
4474:
4473:
4471:
4461:
4427:
4422:
4418:
4412:
4408:
4404:
4399:
4395:
4391:
4386:
4382:
4375:
4372:
4369:
4361:
4357:
4351:
4346:
4342:
4337:
4331:
4328:
4312:
4307:
4302:
4298:
4291:
4288:
4285:
4277:
4273:
4267:
4263:
4257:
4254:
4246:
4241:
4233:
4229:
4222:
4217:
4214:
4209:
4201:
4197:
4190:
4183:
4182:
4181:
4179:
4175:
4167:
4163:
4143:
4138:
4134:
4130:
4125:
4121:
4114:
4108:
4103:
4095:
4091:
4085:
4082:
4077:
4069:
4065:
4058:
4051:
4050:
4049:
4047:
4043:
4039:
4035:
4030:
4026:
4022:
4018:
4014:
4010:
4006:
4002:
3998:
3995:
3991:
3988:
3984:
3980:
3976:
3974:
3967:
3963:
3959:
3951:
3947:
3944:Cyclic group
3927:
3922:
3917:
3913:
3909:
3901:
3897:
3890:
3883:
3882:
3881:
3875:
3871:
3846:
3842:
3836:
3831:
3827:
3823:
3820:
3815:
3810:
3806:
3802:
3799:
3794:
3789:
3785:
3779:
3774:
3770:
3766:
3763:
3758:
3754:
3748:
3743:
3739:
3735:
3732:
3727:
3722:
3718:
3713:
3707:
3704:
3699:
3693:
3687:
3680:
3679:
3678:
3676:
3657:
3652:
3647:
3643:
3639:
3630:
3629:
3625:
3624:
3609:
3604:
3599:
3595:
3591:
3582:
3581:
3577:
3576:
3561:
3556:
3551:
3547:
3541:
3536:
3532:
3528:
3520:
3519:
3515:
3514:
3499:
3494:
3490:
3484:
3479:
3475:
3471:
3463:
3459:
3458:
3454:
3453:
3438:
3433:
3428:
3424:
3415:
3414:
3411:The identity:
3410:
3409:
3408:
3406:
3402:
3393:
3370:
3366:
3360:
3356:
3350:
3346:
3342:
3339:
3334:
3329:
3325:
3321:
3318:
3313:
3308:
3304:
3298:
3293:
3289:
3285:
3282:
3277:
3272:
3268:
3263:
3257:
3254:
3249:
3243:
3237:
3230:
3229:
3228:
3223:
3219:
3211:
3207:
3204:
3201:
3198:
3196:The identity.
3195:
3194:
3193:
3191:
3172:
3167:
3163:
3157:
3153:
3149:
3141:
3127:
3122:
3117:
3113:
3107:
3102:
3098:
3094:
3085:
3071:
3066:
3061:
3057:
3053:
3044:
3030:
3025:
3020:
3016:
3010:
3005:
3001:
2997:
2988:
2974:
2969:
2964:
2960:
2951:
2950:
2949:
2944:
2937:
2927:
2913:
2885:
2882:
2863:
2859:
2852:
2848:
2844:
2837:
2818:
2814:
2808:
2804:
2800:
2797:
2792:
2788:
2782:
2778:
2774:
2771:
2766:
2761:
2757:
2752:
2746:
2743:
2738:
2732:
2726:
2719:
2718:
2717:
2712:
2708:
2689:
2684:
2680:
2676:
2668:
2654:
2649:
2645:
2639:
2635:
2631:
2623:
2619:
2605:
2600:
2595:
2591:
2582:
2581:
2580:
2575:
2571:
2564:
2560:
2556:
2549:
2541:
2535:
2517:
2513:
2507:
2503:
2499:
2496:
2491:
2486:
2482:
2478:
2475:
2470:
2465:
2461:
2457:
2452:
2447:
2443:
2438:
2432:
2429:
2424:
2416:
2412:
2405:
2398:
2397:
2396:
2388:
2385:
2382:
2379:
2376:
2373:
2372:
2371:
2370:
2369:
2364:
2359:
2357:
2353:
2349:
2345:
2341:
2337:
2333:
2329:
2324:
2322:
2318:
2314:
2310:
2306:
2302:
2298:
2294:
2290:
2286:
2282:
2278:
2257:
2253:
2247:
2243:
2239:
2236:
2231:
2227:
2221:
2217:
2213:
2210:
2205:
2200:
2196:
2191:
2185:
2182:
2177:
2169:
2165:
2158:
2151:
2150:
2149:
2148:
2147:
2124:
2118:
2112:
2106:
2100:
2094:
2088:
2082:
2076:
2070:
2067:
2061:
2056:
2052:
2044:
2043:
2042:
2037:
2032:
2030:
2026:
2021:
1996:
1992:
1986:
1981:
1977:
1973:
1970:
1965:
1960:
1956:
1952:
1947:
1942:
1938:
1933:
1927:
1924:
1919:
1911:
1907:
1900:
1893:
1892:
1891:
1890:
1889:
1887:
1880:
1877:
1873:
1869:
1865:
1861:
1857:
1853:
1849:
1845:
1841:
1834:
1813:
1809:
1803:
1799:
1793:
1789:
1785:
1782:
1777:
1772:
1768:
1762:
1757:
1753:
1749:
1744:
1739:
1735:
1730:
1724:
1721:
1716:
1708:
1704:
1697:
1690:
1689:
1688:
1687:
1686:
1684:
1680:
1676:
1672:
1668:
1664:
1660:
1656:
1652:
1648:
1644:
1640:
1636:
1633:
1629:
1626:
1623:
1620:
1613:
1610:
1606:
1602:
1598:
1594:
1590:
1586:
1582:
1578:
1574:
1570:
1566:
1562:
1558:
1554:
1550:
1546:
1542:
1538:
1534:
1530:
1526:
1522:
1515:
1494:
1490:
1484:
1480:
1476:
1473:
1468:
1463:
1459:
1455:
1450:
1445:
1441:
1436:
1430:
1427:
1422:
1414:
1410:
1403:
1396:
1395:
1394:
1393:
1392:
1387:
1380:
1373:
1366:
1362:
1355:
1352:
1348:
1344:
1340:
1336:
1332:
1328:
1324:
1320:
1316:
1312:
1308:
1284:
1276:
1268:
1264:
1258:
1254:
1248:
1243:
1240:
1237:
1233:
1227:
1224:
1221:
1217:
1205:
1196:
1191:
1185:
1179:
1172:
1171:
1170:
1168:
1164:
1160:
1155:
1152:
1148:
1141:
1134:
1110:
1102:
1098:
1092:
1088:
1082:
1077:
1074:
1071:
1067:
1063:
1053:
1044:
1038:
1035:
1032:
1028:
1020:
1019:
1018:
1017:the monomial
1016:
997:
994:
991:
985:
977:
973:
968:
963:
958:
955:
952:
948:
934:
930:
926:
920:
914:
906:
902:
898:
895:
888:
887:
886:
884:
880:
876:
871:
867:
862:
860:
856:
849:
843:
837:
833:
829:
825:
821:
818:
814:
809:
807:
803:
799:
796:
792:
782:
777:
771:
765:
758:
752:
746:
742:
716:
708:
704:
698:
694:
688:
683:
680:
677:
673:
665:
664:
663:
662:
661:
660:
656:
652:
648:
643:
639:
635:
630:
626:
621:
617:
613:
608:
604:
597:
590:
583:
579:
574:
572:
568:
547:
541:
532:
523:
517:
513:
506:
501:
496:
491:
486:
481:
474:
469:
464:
459:
454:
449:
442:
434:
433:
432:
431:
430:
428:
427:
421:
419:
414:
410:
409:
404:
400:
378:
371:
366:
361:
356:
351:
344:
339:
334:
329:
324:
317:
309:
308:
307:
306:
305:
303:
300:= {1,2, ...,
299:
289:
287:
283:
279:
275:
271:
267:
262:
260:
256:
255:
250:
246:
242:
238:
234:
230:
226:
222:
218:
214:
210:
206:
202:
198:
194:
190:
186:
182:
178:
174:
170:
166:
162:
158:
148:
146:
142:
138:
134:
130:
126:
121:
119:
115:
111:
107:
102:
98:
94:
87:
80:
77:in variables
76:
72:
68:
64:
60:
55:
53:
49:
45:
41:
37:
34:
30:
26:
22:
19:
18:combinatorial
7954:
7950:
7927:
7923:
7910:
7902:
7878:
7860:
7857:Tucker, Alan
7839:
7819:
7799:
7781:
7759:
7752:Cameron 1994
7747:
7735:
7723:
7714:
7705:
7697:
7693:
7689:
7684:
7672:
7663:
7655:
7651:
7646:
7637:
7630:Cameron 1994
7625:
7618:Cameron 1994
7613:
7601:
7580:
7568:
7557:
7552:
7548:
7544:
7540:
7536:
7532:
7528:
7524:
7520:
7516:
7512:
7508:
7504:
7500:
7496:
7492:
7488:
7484:
7480:
7476:
7472:
7470:
7334:
7330:
7324:
7178:
7174:
7168:
7160:
7153:
7149:
7140:
7136:
7131:
7127:
7123:
7119:
7115:
7107:
7103:
7099:
7098:-subsets of
7095:
7091:
7087:
7083:
7081:
6977:
6975:
6972:Applications
6855:
6568:
6378:
6374:
6330:
6081:
5591:
5299:
5264:
5233:
5231:
5074:
4846:
4842:
4837:
4831:
4827:
4749:
4467:
4459:
4178:cyclic group
4176:is like the
4171:
4165:
4161:
4045:
4041:
4037:
4033:
4028:
4024:
4020:
4012:
4004:
4000:
3996:
3989:
3982:
3978:
3972:
3965:
3961:
3958:cyclic group
3955:
3949:
3945:
3879:
3873:
3869:
3674:
3672:
3404:
3398:
3221:
3217:
3215:
3187:
2942:
2935:
2933:
2861:
2857:
2850:
2835:
2833:
2710:
2706:
2704:
2573:
2569:
2562:
2547:
2545:
2539:
2532:
2394:
2362:
2360:
2355:
2347:
2343:
2335:
2331:
2325:
2320:
2316:
2312:
2308:
2304:
2300:
2296:
2292:
2288:
2284:
2280:
2276:
2274:
2145:
2035:
2033:
2028:
2022:
2018:
1878:
1871:
1867:
1863:
1859:
1855:
1851:
1847:
1846:would send (
1843:
1839:
1832:
1830:
1682:
1678:
1674:
1670:
1666:
1662:
1658:
1654:
1650:
1646:
1642:
1638:
1634:
1631:
1627:
1624:
1621:
1618:
1611:
1604:
1603:= {1,3} and
1600:
1596:
1592:
1588:
1584:
1580:
1576:
1572:
1568:
1564:
1560:
1556:
1552:
1548:
1544:
1540:
1536:
1532:
1528:
1527:would send {
1524:
1520:
1513:
1511:
1385:
1378:
1371:
1364:
1360:
1353:
1351:cyclic group
1346:
1342:
1330:
1326:
1322:
1306:
1304:
1169:is given by
1166:
1162:
1158:
1156:
1150:
1146:
1139:
1132:
1130:
1014:
1012:
882:
878:
874:
869:
865:
863:
858:
854:
847:
841:
835:
831:
827:
823:
819:
812:
810:
805:
801:
797:
790:
788:
775:
769:
763:
756:
750:
744:
740:
738:
658:
654:
650:
646:
641:
637:
633:
628:
624:
619:
615:
611:
606:
602:
595:
588:
581:
575:
570:
566:
564:
426:fixed points
424:
422:
417:
412:
406:
402:
398:
396:
301:
297:
295:
281:
277:
273:
269:
268:acts on set
265:
263:
258:
252:
249:group action
244:
240:
236:
232:
224:
220:
208:
204:
200:
192:
184:
180:
168:
164:
160:
154:
141:differential
122:
117:
113:
109:
105:
100:
96:
85:
78:
70:
56:
44:coefficients
24:
15:
7930:: 133–156.
7656:composition
7102:and on the
4433: even.
3977:, that is,
2845:to its own
2622:reflections
2338:is regular
2334:on the set
2279:on the set
1661:) and e = (
1555:). The set
822:and degree
791:cycle index
173:composition
127:due to the
118:cycle index
61:of objects
25:cycle index
21:mathematics
7966:Categories
7774:References
7696:acting on
7572:integrated
7555: |.
7491:. For any
6424:There are
4015:which are
4003:, where φ(
2847:line graph
2843:isomorphic
2559:symmetries
2350: |.
2307:such that
2285:transitive
1831:The group
1512:The group
881:of length
861: |.
785:Definition
93:cycle type
73:of π is a
63:partitions
59:finite set
29:polynomial
7679:, pg. 473
7442:…
7352:∑
7282:…
7196:∑
7046:…
6945:−
6900:∑
6833:−
6806:−
6776:−
6754:−
6743:−
6711:∑
6641:∏
6625:∈
6618:∑
6541:−
6490:−
6455:−
6444:−
6402:≤
6396:≤
6270:≥
6263:∑
6254:
6185:
6174:≥
6167:∏
6027:
6016:≥
6009:∏
5919:≥
5905:∑
5896:≥
5889:∏
5785:∏
5757:⋯
5708:∑
5699:≥
5692:∑
5651:≥
5644:∑
5524:∏
5467:∏
5409:∏
5359:∏
5180:−
5168:…
5021:∏
4967:∏
4933:⋯
4884:∑
4693:∏
4639:∏
4631:⋯
4594:−
4555:⋯
4506:∑
4373:−
4289:−
4109:φ
4092:∑
3992:for each
1599:= {1,4},
1595:= {3,4},
1591:= {2,3},
1587:= {1,2},
1543:} (where
1335:rotations
1234:∏
1225:∈
1218:∑
1068:∏
1036:∈
1029:∏
949:∑
938:⌋
924:⌊
921:≤
899:≤
674:∏
223:into Sym(
157:bijective
48:algebraic
7547:) where
7515:. Thus,
7112:#Example
6381:, where
4048:, thus:
3971:regular
3462:parallel
2553:with an
2025:identity
1583:} where
1559:is now {
1339:identity
885:, where
864:Now let
645: (
632: (
189:subgroup
75:monomial
7652:product
7543:, ...,
7325:b) The
7169:a) The
3994:divisor
2317:regular
1655:A B C D
1317:in the
1301:Example
1145:, ...,
286:kernels
219:φ from
215:with a
191:of Sym(
7885:
7867:
7846:
7828:
7806:
7788:
7146:orbits
7126:. Let
2620:Three
2328:finite
1884:(with
1384:, and
1315:square
211:be an
201:degree
133:action
108:. The
69:; the
67:cycles
7593:Notes
7110:(see
3987:order
2346:| = |
1886:loops
1854:) → (
1535:} → {
1313:of a
1165:) of
817:order
793:of a
578:dummy
229:image
129:group
38:on a
27:is a
7883:ISBN
7865:ISBN
7844:ISBN
7826:ISBN
7804:ISBN
7786:ISBN
7479:and
7471:Let
7329:for
7173:for
7135:and
7082:Let
4172:The
4036:has
3975:-gon
3401:cube
3227:is:
3208:Six
2291:and
1653:), (
1637:), (
789:The
594:and
231:, φ(
36:acts
7932:doi
7928:263
7551:= |
7527:to
7511:to
7495:in
7487:to
7319:and
7148:of
6856:or
6251:exp
6182:exp
6024:exp
4019:to
3999:of
3677:is
3220:of
2926:).
2841:is
2716:is
2568:of
2303:in
2295:in
2283:is
2113:132
2101:123
1659:E F
1643:B D
1639:A C
1309:of
830:in
804:in
542:456
257:of
199:of
179:of
131:'s
40:set
16:In
7968::
7953:.
7949:.
7926:.
7922:.
7589:.
7566:.
7539:,
7535:;
7499:,
7159:=
7122:≤
7090:.
6109::
5079::
4822:.
3960:,
3956:A
3708:24
3258:24
2326:A
2311:=
2125:13
2089:12
2077:23
2031:.
1948:16
1866:,
1858:,
1681:)(
1677:)(
1673:)(
1669:)(
1665:)(
1657:)(
1649:)(
1645:)(
1641:)(
1630:)(
1579:,
1575:,
1571:,
1567:,
1563:,
1539:,
1377:,
1370:,
1154:.
1138:,
808:.
781:.
587:,
573:.
524:12
155:A
147:.
120:.
84:,
54:.
23:a
7957:.
7955:1
7940:.
7934::
7698:X
7694:G
7690:G
7553:Y
7549:b
7545:b
7541:b
7537:b
7533:G
7529:Y
7525:X
7521:Y
7517:G
7513:Y
7509:X
7505:x
7503:(
7501:h
7497:G
7493:g
7489:Y
7485:X
7481:h
7477:X
7473:G
7454:.
7451:)
7448:1
7445:,
7439:,
7436:1
7433:,
7430:1
7427:,
7424:t
7421:+
7418:1
7415:;
7412:G
7409:(
7406:Z
7403:=
7400:!
7397:k
7393:/
7387:k
7383:t
7377:k
7373:F
7367:n
7362:0
7359:=
7356:k
7335:k
7331:F
7307:,
7304:)
7299:n
7295:t
7291:+
7288:1
7285:,
7279:,
7274:2
7270:t
7266:+
7263:1
7260:,
7257:t
7254:+
7251:1
7248:;
7245:G
7242:(
7239:Z
7236:=
7231:k
7227:t
7221:k
7217:f
7211:n
7206:0
7203:=
7200:k
7179:k
7175:f
7164:0
7161:F
7157:0
7154:f
7150:G
7141:k
7137:F
7132:k
7128:f
7124:n
7120:k
7116:k
7108:X
7104:k
7100:X
7096:k
7092:G
7088:X
7084:G
7065:.
7062:)
7057:n
7053:a
7049:,
7043:,
7038:2
7034:a
7030:,
7025:1
7021:a
7017:;
7014:G
7011:(
7008:Z
7005:=
7002:)
6999:G
6996:(
6993:Z
6978:G
6956:.
6953:)
6948:l
6942:n
6938:S
6934:(
6931:Z
6925:l
6921:a
6915:n
6910:1
6907:=
6904:l
6894:n
6891:1
6886:=
6883:)
6878:n
6874:S
6870:(
6867:Z
6841:)
6836:l
6830:n
6826:S
6822:(
6819:Z
6815:!
6812:)
6809:l
6803:n
6800:(
6794:l
6790:a
6785:!
6782:)
6779:1
6773:l
6770:(
6763:)
6757:1
6751:l
6746:1
6740:n
6734:(
6726:n
6721:1
6718:=
6715:l
6704:!
6701:n
6697:1
6692:=
6687:)
6684:g
6681:(
6676:k
6672:j
6666:k
6662:a
6656:n
6651:1
6648:=
6645:k
6633:n
6629:S
6622:g
6611:!
6608:n
6604:1
6599:=
6596:)
6591:n
6587:S
6583:(
6580:Z
6550:!
6547:)
6544:1
6538:l
6535:(
6532:=
6527:l
6523:!
6520:l
6493:1
6487:l
6464:)
6458:1
6452:l
6447:1
6441:n
6435:(
6408:.
6405:n
6399:l
6393:1
6379:n
6375:l
6361:1
6358:=
6355:)
6350:0
6346:S
6342:(
6339:Z
6312:.
6308:)
6302:k
6296:k
6292:y
6286:k
6282:a
6273:1
6267:k
6258:(
6248:]
6243:n
6239:y
6235:[
6232:=
6221:)
6216:k
6210:k
6206:y
6200:k
6196:a
6189:(
6177:1
6171:k
6163:]
6158:n
6154:y
6150:[
6147:=
6140:)
6135:n
6131:S
6127:(
6124:Z
6095:n
6091:S
6067:,
6063:)
6058:k
6052:k
6048:y
6042:k
6038:a
6031:(
6019:1
6013:k
6005:=
5999:!
5994:k
5990:j
5982:k
5978:j
5973:k
5963:k
5959:j
5954:)
5948:k
5944:y
5938:k
5934:a
5930:(
5922:0
5914:k
5910:j
5899:1
5893:k
5885:=
5879:!
5874:k
5870:j
5862:k
5858:j
5853:k
5843:k
5839:j
5835:k
5831:y
5823:k
5819:j
5813:k
5809:a
5800:n
5795:1
5792:=
5789:k
5779:n
5776:=
5771:n
5767:j
5763:n
5760:+
5754:+
5749:3
5745:j
5741:3
5738:+
5733:2
5729:j
5725:2
5722:+
5717:1
5713:j
5702:1
5696:n
5688:=
5685:)
5680:n
5676:S
5672:(
5669:Z
5664:n
5660:y
5654:1
5648:n
5620:y
5600:n
5577:.
5571:!
5566:k
5562:j
5554:k
5550:j
5545:k
5539:n
5534:1
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79:a
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