2568:
5398:
4970:
3255:
2091:
2083:
5393:{\displaystyle {\begin{aligned}\sum _{\omega }x^{\omega }\left|(Y^{X})_{\omega ,g}\right|&=\prod _{q{\text{ cycle of }}g}f\left(x_{1}^{|q|},x_{2}^{|q|},x_{3}^{|q|},\ldots \right)\\&=f(x_{1},x_{2},\ldots )^{j_{1}(g)}f\left(x_{1}^{2},x_{2}^{2},\ldots \right)^{j_{2}(g)}\cdots f\left(x_{1}^{n},x_{2}^{n},\ldots \right)^{j_{n}(g)}\end{aligned}}}
3973:
1278:
3424:
The Polya enumeration theorem translates the recursive structure of rooted ternary trees into a functional equation for the generating function F(t) of rooted ternary trees by number of nodes. This is achieved by "coloring" the three children with rooted ternary trees, weighted by node number, so
3090:
1739:
782:
2786:
2486:
4555:
3243:
vertices of degree at most 3 (by ignoring the leaves). In general, two rooted trees are isomorphic when one can be obtained from the other by permuting the children of its nodes. In other words, the group that acts on the children of a node is the symmetric group
1977:
with a fixed number of vertices, or the generating function of these graphs according to the number of edges they have. For the latter purpose, we can say that a black or present edge has weight 1, while an absent or white edge has weight 0. Thus
3742:
1463:
1009:
2258:
1906:
3262:
One can view a rooted, ternary tree as a recursive object which is either a leaf or a node with three children which are themselves rooted ternary trees. These children are equivalent to beads; the cycle index of the symmetric group
3419:
4086:. The weighted version of the theorem has essentially the same proof, but with a refined form of Burnside's lemma for weighted enumeration. It is equivalent to apply Burnside's lemma separately to orbits of different weight.
1468:
In the celebrated application of counting trees (see below) and acyclic molecules, an arrangement of "colored beads" is actually an arrangement of arrangements, such as branches of a rooted tree. Thus the generating function
4325:
3720:
2801:
360:
4951:
4764:
1515:
998:
3577:
575:
3258:
Rooted ternary trees on 0, 1, 2, 3 and 4 nodes (=non-leaf vertices). The root is shown in blue, the leaves are not shown. Every node has as many leaves as to make the number of its children equal to 3.
3211:
429:
In the more general and more important version of the theorem, the colors are also weighted in one or more ways, and there could be an infinite number of colors provided that the set of colors has a
2588:
521:
4975:
3747:
4648:
2273:
4406:
3235:
consists of rooted trees where every node (or non-leaf vertex) has exactly three children (leaves or subtrees). Small ternary trees are shown at right. Note that rooted ternary trees with
3582:
as the generating function for rooted ternary trees, weighted by one less than the node number (since the sum of the children weights does not take the root into account), so that
2559:
4374:
4133:
3968:{\displaystyle {\begin{aligned}t_{0}&=1\\t_{n+1}&={\frac {1}{6}}\left(\sum _{a+b+c=n}t_{a}t_{b}t_{c}+3\sum _{a+2b=n}t_{a}t_{b}+2\sum _{3a=n}t_{a}\right)\end{aligned}}}
1964:
4808:
1305:
4204:
1273:{\displaystyle F(t_{1},t_{2},\ldots )=Z_{G}(f(t_{1},t_{2},\ldots ),f(t_{1}^{2},t_{2}^{2},\ldots ),f(t_{1}^{3},t_{2}^{3},\ldots ),\ldots ,f(t_{1}^{n},t_{2}^{n},\ldots )).}
2098:
The eight graphs on three vertices (before identifying isomorphic graphs) are shown at the right. There are four isomorphism classes of graphs, also shown at the right.
3467:
2053:
2017:
4394:
4228:
4177:
2114:
240:
4072:
4018:
399:
1761:
4157:
2264:
2792:
1970:= {black, white} corresponds to edges that are present (black) or absent (white). The Pólya enumeration theorem can be used to calculate the number of graphs
3276:
3085:{\displaystyle F(t)=Z_{G}\left(t+1,t^{2}+1,t^{3}+1,t^{4}+1\right)={\frac {(t+1)^{6}+9(t+1)^{2}(t^{2}+1)^{2}+8(t^{3}+1)^{2}+6(t^{2}+1)(t^{4}+1)}{24}}}
4236:
3588:
2067:
of possible edges: a permutation φ turns the edge {a, b} into the edge {φ(a), φ(b)}. With these definitions, an isomorphism class of graphs with
260:
4836:
1734:{\displaystyle Z_{C}(t_{1},t_{2},t_{3},t_{4})={\frac {1}{24}}\left(t_{1}^{6}+6t_{1}^{2}t_{4}+3t_{1}^{2}t_{2}^{2}+8t_{3}^{2}+6t_{2}^{3}\right)}
4656:
871:
777:{\displaystyle Z_{G}(t_{1},t_{2},\ldots ,t_{n})={\frac {1}{|G|}}\sum _{g\in G}t_{1}^{c_{1}(g)}t_{2}^{c_{2}(g)}\cdots t_{n}^{c_{n}(g)}}
4031:
3475:
3101:
2781:{\displaystyle Z_{G}(t_{1},t_{2},t_{3},t_{4})={\frac {1}{24}}\left(t_{1}^{6}+9t_{1}^{2}t_{2}^{2}+8t_{3}^{2}+6t_{2}t_{4}\right)}
1749:
2481:{\displaystyle F(t)=Z_{G}\left(t+1,t^{2}+1,t^{3}+1\right)={\frac {1}{6}}\left((t+1)^{3}+3(t+1)(t^{2}+1)+2(t^{3}+1)\right),}
439:
4550:{\displaystyle F(x_{1},x_{2},\ldots )={\frac {1}{|G|}}\sum _{g\in G,\omega }x^{\omega }\left|(Y^{X})_{\omega ,g}\right|.}
4567:
70:, who then greatly popularized the result by applying it to many counting problems, in particular to the enumeration of
5636:
5626:
5584:
5556:
2267:). Thus, according to the enumeration theorem, the generating function of graphs on 3 vertices up to isomorphism is
544:≥ 0. In the multivariate case, the weight of each color is a vector of integers and there is a generating function
51:
5473:
5431:
5580:
5416:
2497:
4333:
4092:
5631:
125:
55:
1458:{\displaystyle \left|Y^{X}/G\right|=F(0)=Z_{G}(m,m,\ldots ,m)={\frac {1}{|G|}}\sum _{g\in G}m^{c(g)}.}
1930:
4780:
1491:
175:
145:
4182:
1509:
of the cube acts on the six sides of the cube, which are equivalent to beads. Its cycle index is
5610:
4047:, which says that the number of orbits of colorings is the average of the number of elements of
3251:. We define the weight of such a ternary tree to be the number of nodes (or non-leaf vertices).
2253:{\displaystyle Z_{G}(t_{1},t_{2},t_{3})={\frac {1}{6}}\left(t_{1}^{3}+3t_{1}t_{2}+2t_{3}\right)}
2090:
2079:
of colored arrangements; the number of edges of the graph equals the weight of the arrangement.
3428:
2567:
78:
2022:
1981:
5595:
4379:
4213:
4162:
82:
19:"Polya theorem" redirects here. For Pólya's theorem for positive polynomials on simplex, see
219:
5566:
5502:
5460:
4050:
3996:
1901:{\displaystyle F(0)=Z_{C}(m,m,m,m)={\frac {1}{24}}\left(m^{6}+3m^{4}+12m^{3}+8m^{2}\right)}
368:
24:
8:
4044:
3232:
430:
183:
153:
106:
47:
20:
5611:
Enumerating alcohols and other classes of chemical molecules, an example of Pólya theory
5541:
5490:
5448:
4142:
3414:{\displaystyle Z_{S_{3}}(t_{1},t_{2},t_{3})={\frac {t_{1}^{3}+3t_{1}t_{2}+2t_{3}}{6}}.}
410:
63:
2082:
5592:
5552:
2263:(obtained by inspecting the cycle structure of the action of the group elements; see
1974:
71:
59:
5512:"Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen"
565:
The enumeration theorem employs another multivariate generating function called the
5525:
5516:
5482:
5440:
5562:
5548:
5498:
5456:
2056:
2019:
is the generating function for the set of colors. The relevant symmetry group is
1477:
for arrangements, and the Pólya enumeration theorem becomes a recursive formula.
67:
191:
121:
4320:{\displaystyle {\frac {1}{|G|}}\sum _{g\in G}\left|(Y^{X})_{\omega ,g}\right|}
562:, ...) that tabulates the number of colors with each given vector of weights.
5620:
43:
3715:{\displaystyle F(t)=1+t\cdot {\frac {F(t)^{3}+3F(t)F(t^{2})+2F(t^{3})}{6}}.}
1501:
How many ways are there to color the sides of a three-dimensional cube with
5468:
355:{\displaystyle \left|Y^{X}/G\right|={\frac {1}{|G|}}\sum _{g\in G}m^{c(g)}}
161:
118:
4946:{\displaystyle f\left(x_{1}^{|q|},x_{2}^{|q|},x_{3}^{|q|},\ldots \right).}
4024:
1, 1, 1, 2, 4, 8, 17, 39, 89, 211, 507, 1238, 3057, 7639, 19241 (sequence
5429:
Redfield, J. Howard (1927). "The Theory of Group-Reduced
Distributions".
4956:
It follows that the generating function by weight of the points fixed by
566:
110:
1923:
vertices can be interpreted as an arrangement of colored beads. The set
1744:
which is obtained by analyzing the action of each of the 24 elements of
5530:
5511:
5494:
5452:
4759:{\displaystyle t_{1}^{j_{1}(g)}t_{2}^{j_{2}(g)}\cdots t_{n}^{j_{n}(g)}}
98:
3725:
This is equivalent to the following recurrence formula for the number
5600:
5486:
5444:
140:
may be a chosen group of permutations of the beads. For example, if
5543:
Combinatorial
Enumeration of Groups, Graphs, and Chemical Compounds
993:{\displaystyle F(t)=Z_{G}(f(t),f(t^{2}),f(t^{3}),\ldots ,f(t^{n}))}
526:
is the generating function of the set of colors, so that there are
5590:
4043:
The simplified form of the Pólya enumeration theorem follows from
250:. The Pólya enumeration theorem counts the number of orbits under
849:
of such an arrangement is defined as the sum of the weights of φ(
538:
1283:
To reduce to the simplified version given earlier, if there are
3254:
433:
with finite coefficients. In the univariate case, suppose that
208:
is a finite set of colors — the colors of the beads — so that
3572:{\displaystyle {\frac {F(t)^{3}+3F(t)F(t^{2})+2F(t^{3})}{6}}}
1971:
865:
of the number of colored arrangements by weight is given by:
1755:
We take all colors to have weight 0 and find that there are
212:
is the set of colored arrangements of beads (more formally:
5471:; Ed Palmer (1967). "The Enumeration Methods of Redfield".
4026:
254:
of colored arrangements of beads by the following formula:
3206:{\displaystyle F(t)=t^{6}+t^{5}+2t^{4}+3t^{3}+2t^{2}+t+1.}
4810:
if and only if the function φ is constant on every cycle
77:
The Pólya enumeration theorem has been incorporated into
1505:
colors, up to rotation of the cube? The rotation group
1473:
for the colors is derived from the generating function
4400:. If we then sum over all possible weights, we obtain
66:
in 1927. In 1937 it was independently rediscovered by
5467:
4973:
4839:
4783:
4659:
4570:
4409:
4382:
4336:
4239:
4216:
4185:
4165:
4145:
4095:
4053:
3999:
3745:
3591:
3478:
3431:
3279:
3104:
2804:
2591:
2500:
2276:
2117:
2025:
1984:
1933:
1764:
1518:
1308:
1012:
874:
578:
442:
371:
263:
222:
4960:
is the product of the above term over all cycles of
1914:
516:{\displaystyle f(t)=f_{0}+f_{1}t+f_{2}t^{2}+\cdots }
821:A colored arrangement is an orbit of the action of
5540:
5538:
5392:
4945:
4802:
4758:
4643:{\displaystyle j_{1}(g),j_{2}(g),\ldots ,j_{n}(g)}
4642:
4549:
4388:
4368:
4319:
4222:
4198:
4171:
4151:
4127:
4066:
4012:
3967:
3714:
3571:
3461:
3413:
3205:
3084:
2780:
2553:
2480:
2252:
2071:vertices is the same as an orbit of the action of
2047:
2011:
1958:
1900:
1733:
1457:
1272:
992:
861:. The theorem states that the generating function
776:
515:
393:
354:
234:
46:that both follows from and ultimately generalizes
23:. For the recurrence of lattice random walks, see
1950:
1937:
88:
5618:
4210:. Applying Burnside's lemma to orbits of weight
2564:Thus there is one graph each with 0 to 3 edges.
5581:Applying the Pólya-Burnside Enumeration Theorem
5407:yields the substituted cycle index as claimed.
3425:that the color generating function is given by
2571:Isomorphism classes of graphs on four vertices.
4826:| identical colors from the set enumerated by
16:Formula for number of orbits of a group action
4135:be the variables of the generating function
4206:denote the corresponding monomial term of
3239:nodes are equivalent to rooted trees with
1485:
5529:
4230:, the number of orbits of this weight is
424:
136:may represent a finite set of beads, and
5509:
5428:
3253:
2566:
2089:
2081:
1966:possible edges, while the set of colors
3469:which by the enumeration theorem gives
3219:
5619:
5583:by Hector Zenil and Oleksandr Pavlyk,
5403:Substituting this in the sum over all
4822:the generating function by weight of |
2094:Nonisomorphic graphs on three vertices
5591:
3216:These graphs are shown at the right.
2554:{\displaystyle F(t)=t^{3}+t^{2}+t+1.}
62:. The theorem was first published by
2108:acting on the set of three edges is
2063:letters. This group acts on the set
837:denotes the set of all functions φ:
417:when considered as a permutation of
25:Random walk § Higher dimensions
4369:{\displaystyle (Y^{X})_{\omega ,g}}
4128:{\displaystyle x_{1},x_{2},\ldots }
4038:
1287:colors and all have weight 0, then
13:
5585:The Wolfram Demonstrations Project
4376:is the set of colorings of weight
1941:
14:
5648:
5574:
1915:Graphs on three and four vertices
182:beads in a circle, rotations and
2582:acting on the set of 6 edges is
1748:on the 6 sides of the cube, see
1496:
5474:American Journal of Mathematics
5432:American Journal of Mathematics
1959:{\displaystyle {\binom {m}{2}}}
5381:
5375:
5303:
5297:
5228:
5222:
5208:
5175:
5145:
5137:
5117:
5109:
5089:
5081:
5017:
5003:
4923:
4915:
4895:
4887:
4867:
4859:
4803:{\displaystyle \phi \in Y^{X}}
4751:
4745:
4717:
4711:
4686:
4680:
4637:
4631:
4609:
4603:
4587:
4581:
4524:
4510:
4466:
4458:
4445:
4413:
4351:
4337:
4297:
4283:
4255:
4247:
3700:
3687:
3675:
3662:
3656:
3650:
3632:
3625:
3601:
3595:
3560:
3547:
3535:
3522:
3516:
3510:
3492:
3485:
3456:
3450:
3441:
3435:
3336:
3297:
3114:
3108:
3073:
3054:
3051:
3032:
3017:
2997:
2982:
2962:
2953:
2940:
2925:
2912:
2814:
2808:
2654:
2602:
2510:
2504:
2467:
2448:
2439:
2420:
2417:
2405:
2390:
2377:
2286:
2280:
2167:
2128:
1994:
1988:
1814:
1790:
1774:
1768:
1581:
1529:
1447:
1441:
1410:
1402:
1389:
1365:
1349:
1343:
1264:
1261:
1219:
1204:
1162:
1153:
1111:
1102:
1070:
1064:
1048:
1016:
987:
984:
971:
956:
943:
934:
921:
912:
906:
900:
884:
878:
769:
763:
735:
729:
704:
698:
655:
647:
634:
589:
452:
446:
387:
379:
347:
341:
310:
302:
226:
89:Simplified, unweighted version
1:
5539:G. Pólya; R. C. Read (1987).
5422:
3732:of rooted ternary trees with
2575:The cycle index of the group
2101:The cycle index of the group
1003:or in the multivariate case:
810:-cycles of the group element
791:is the number of elements of
5417:Labelled enumeration theorem
4159:. Given a vector of weights
2086:All graphs on three vertices
401:is the number of colors and
7:
5596:"Polya Enumeration Theorem"
5410:
4199:{\displaystyle x^{\omega }}
1480:
10:
5653:
4560:Meanwhile a group element
4089:For clearer notation, let
3990:are nonnegative integers.
1489:
18:
5637:Theorems in combinatorics
5627:Enumerative combinatorics
4650:will contribute the term
4074:fixed by the permutation
3462:{\displaystyle f(t)=F(t)}
1927:of "beads" is the set of
833:is the set of colors and
32:Pólya enumeration theorem
3993:The first few values of
2048:{\displaystyle G=S_{m},}
2012:{\displaystyle f(t)=1+t}
1492:Necklace (combinatorics)
216:is the set of functions
152:beads in a circle, then
4818:. For every such cycle
4396:that are also fixed by
4389:{\displaystyle \omega }
4223:{\displaystyle \omega }
4172:{\displaystyle \omega }
1486:Necklaces and bracelets
204:. Suppose further that
5394:
4947:
4804:
4769:to the cycle index of
4760:
4644:
4551:
4390:
4370:
4321:
4224:
4200:
4173:
4153:
4129:
4082:over all permutations
4068:
4014:
3969:
3716:
3573:
3463:
3415:
3259:
3207:
3086:
2782:
2572:
2555:
2482:
2254:
2095:
2087:
2049:
2013:
1960:
1902:
1735:
1459:
1274:
994:
778:
517:
425:Full, weighted version
395:
356:
236:
235:{\displaystyle X\to Y}
79:symbolic combinatorics
36:Redfield–Pólya theorem
5395:
4948:
4805:
4761:
4645:
4564:with cycle structure
4552:
4391:
4371:
4322:
4225:
4201:
4174:
4154:
4130:
4069:
4067:{\displaystyle Y^{X}}
4015:
4013:{\displaystyle t_{n}}
3970:
3717:
3574:
3464:
3416:
3270:that acts on them is
3257:
3208:
3087:
2783:
2570:
2556:
2483:
2255:
2093:
2085:
2050:
2014:
1961:
1911:different colorings.
1903:
1736:
1460:
1275:
995:
779:
518:
413:of the group element
396:
394:{\displaystyle m=|Y|}
357:
237:
83:combinatorial species
5054: cycle of
4971:
4837:
4781:
4657:
4568:
4407:
4380:
4334:
4237:
4214:
4183:
4163:
4143:
4093:
4051:
3997:
3743:
3589:
3476:
3429:
3277:
3220:Rooted ternary trees
3102:
3095:which simplifies to
2802:
2589:
2498:
2491:which simplifies to
2274:
2115:
2023:
1982:
1931:
1762:
1516:
1306:
1010:
872:
814:as a permutation of
576:
440:
369:
261:
220:
34:, also known as the
5352:
5334:
5274:
5256:
5150:
5122:
5094:
4928:
4900:
4872:
4755:
4721:
4690:
3359:
2746:
2725:
2710:
2689:
2202:
1725:
1704:
1683:
1668:
1637:
1616:
1254:
1236:
1197:
1179:
1146:
1128:
806:) is the number of
773:
739:
708:
431:generating function
409:) is the number of
154:rotational symmetry
21:Positive polynomial
5593:Weisstein, Eric W.
5531:10.1007/BF02546665
5390:
5388:
5338:
5320:
5260:
5242:
5126:
5098:
5070:
5061:
4987:
4943:
4904:
4876:
4848:
4800:
4756:
4725:
4691:
4660:
4640:
4547:
4494:
4386:
4366:
4317:
4277:
4220:
4196:
4169:
4149:
4125:
4064:
4010:
3965:
3963:
3945:
3900:
3839:
3712:
3569:
3459:
3411:
3345:
3260:
3231:of rooted ternary
3203:
3082:
2778:
2732:
2711:
2696:
2675:
2573:
2551:
2478:
2250:
2188:
2096:
2088:
2045:
2009:
1956:
1898:
1731:
1711:
1690:
1669:
1654:
1623:
1602:
1455:
1432:
1270:
1240:
1222:
1183:
1165:
1132:
1114:
990:
774:
743:
709:
678:
677:
513:
391:
352:
332:
242:.) Then the group
232:
81:and the theory of
72:chemical compounds
64:J. Howard Redfield
42:, is a theorem in
5632:Graph enumeration
5510:G. Pólya (1937).
5055:
5044:
4978:
4777:fixes an element
4473:
4471:
4262:
4260:
4152:{\displaystyle Y}
3927:
3876:
3812:
3805:
3707:
3567:
3406:
3080:
2668:
2370:
2181:
1948:
1828:
1752:for the details.
1595:
1417:
1415:
662:
660:
533:colors of weight
317:
315:
50:on the number of
5644:
5609:Frederic Chyzak
5606:
5605:
5570:
5546:
5535:
5533:
5517:Acta Mathematica
5506:
5464:
5399:
5397:
5396:
5391:
5389:
5385:
5384:
5374:
5373:
5363:
5359:
5351:
5346:
5333:
5328:
5307:
5306:
5296:
5295:
5285:
5281:
5273:
5268:
5255:
5250:
5232:
5231:
5221:
5220:
5200:
5199:
5187:
5186:
5165:
5161:
5157:
5149:
5148:
5140:
5134:
5121:
5120:
5112:
5106:
5093:
5092:
5084:
5078:
5060:
5056:
5053:
5036:
5032:
5031:
5030:
5015:
5014:
4997:
4996:
4986:
4952:
4950:
4949:
4944:
4939:
4935:
4927:
4926:
4918:
4912:
4899:
4898:
4890:
4884:
4871:
4870:
4862:
4856:
4809:
4807:
4806:
4801:
4799:
4798:
4765:
4763:
4762:
4757:
4754:
4744:
4743:
4733:
4720:
4710:
4709:
4699:
4689:
4679:
4678:
4668:
4649:
4647:
4646:
4641:
4630:
4629:
4602:
4601:
4580:
4579:
4556:
4554:
4553:
4548:
4543:
4539:
4538:
4537:
4522:
4521:
4504:
4503:
4493:
4472:
4470:
4469:
4461:
4452:
4438:
4437:
4425:
4424:
4395:
4393:
4392:
4387:
4375:
4373:
4372:
4367:
4365:
4364:
4349:
4348:
4326:
4324:
4323:
4318:
4316:
4312:
4311:
4310:
4295:
4294:
4276:
4261:
4259:
4258:
4250:
4241:
4229:
4227:
4226:
4221:
4205:
4203:
4202:
4197:
4195:
4194:
4178:
4176:
4175:
4170:
4158:
4156:
4155:
4150:
4134:
4132:
4131:
4126:
4118:
4117:
4105:
4104:
4073:
4071:
4070:
4065:
4063:
4062:
4045:Burnside's lemma
4039:Proof of theorem
4029:
4019:
4017:
4016:
4011:
4009:
4008:
3974:
3972:
3971:
3966:
3964:
3960:
3956:
3955:
3954:
3944:
3920:
3919:
3910:
3909:
3899:
3869:
3868:
3859:
3858:
3849:
3848:
3838:
3806:
3798:
3789:
3788:
3759:
3758:
3721:
3719:
3718:
3713:
3708:
3703:
3699:
3698:
3674:
3673:
3640:
3639:
3620:
3578:
3576:
3575:
3570:
3568:
3563:
3559:
3558:
3534:
3533:
3500:
3499:
3480:
3468:
3466:
3465:
3460:
3420:
3418:
3417:
3412:
3407:
3402:
3401:
3400:
3385:
3384:
3375:
3374:
3358:
3353:
3343:
3335:
3334:
3322:
3321:
3309:
3308:
3296:
3295:
3294:
3293:
3212:
3210:
3209:
3204:
3190:
3189:
3174:
3173:
3158:
3157:
3142:
3141:
3129:
3128:
3091:
3089:
3088:
3083:
3081:
3076:
3066:
3065:
3044:
3043:
3025:
3024:
3009:
3008:
2990:
2989:
2974:
2973:
2961:
2960:
2933:
2932:
2910:
2905:
2901:
2894:
2893:
2875:
2874:
2856:
2855:
2829:
2828:
2787:
2785:
2784:
2779:
2777:
2773:
2772:
2771:
2762:
2761:
2745:
2740:
2724:
2719:
2709:
2704:
2688:
2683:
2669:
2661:
2653:
2652:
2640:
2639:
2627:
2626:
2614:
2613:
2601:
2600:
2560:
2558:
2557:
2552:
2538:
2537:
2525:
2524:
2487:
2485:
2484:
2479:
2474:
2470:
2460:
2459:
2432:
2431:
2398:
2397:
2371:
2363:
2358:
2354:
2347:
2346:
2328:
2327:
2301:
2300:
2259:
2257:
2256:
2251:
2249:
2245:
2244:
2243:
2228:
2227:
2218:
2217:
2201:
2196:
2182:
2174:
2166:
2165:
2153:
2152:
2140:
2139:
2127:
2126:
2054:
2052:
2051:
2046:
2041:
2040:
2018:
2016:
2015:
2010:
1965:
1963:
1962:
1957:
1955:
1954:
1953:
1940:
1907:
1905:
1904:
1899:
1897:
1893:
1892:
1891:
1876:
1875:
1860:
1859:
1844:
1843:
1829:
1821:
1789:
1788:
1740:
1738:
1737:
1732:
1730:
1726:
1724:
1719:
1703:
1698:
1682:
1677:
1667:
1662:
1647:
1646:
1636:
1631:
1615:
1610:
1596:
1588:
1580:
1579:
1567:
1566:
1554:
1553:
1541:
1540:
1528:
1527:
1464:
1462:
1461:
1456:
1451:
1450:
1431:
1416:
1414:
1413:
1405:
1396:
1364:
1363:
1336:
1332:
1328:
1323:
1322:
1295:) =
1279:
1277:
1276:
1271:
1253:
1248:
1235:
1230:
1196:
1191:
1178:
1173:
1145:
1140:
1127:
1122:
1095:
1094:
1082:
1081:
1063:
1062:
1041:
1040:
1028:
1027:
999:
997:
996:
991:
983:
982:
955:
954:
933:
932:
899:
898:
783:
781:
780:
775:
772:
762:
761:
751:
738:
728:
727:
717:
707:
697:
696:
686:
676:
661:
659:
658:
650:
641:
633:
632:
614:
613:
601:
600:
588:
587:
522:
520:
519:
514:
506:
505:
496:
495:
480:
479:
467:
466:
400:
398:
397:
392:
390:
382:
361:
359:
358:
353:
351:
350:
331:
316:
314:
313:
305:
296:
291:
287:
283:
278:
277:
241:
239:
238:
233:
186:are relevant so
48:Burnside's lemma
5652:
5651:
5647:
5646:
5645:
5643:
5642:
5641:
5617:
5616:
5577:
5559:
5549:Springer-Verlag
5487:10.2307/2373127
5445:10.2307/2370675
5425:
5413:
5387:
5386:
5369:
5365:
5364:
5347:
5342:
5329:
5324:
5319:
5315:
5314:
5291:
5287:
5286:
5269:
5264:
5251:
5246:
5241:
5237:
5236:
5216:
5212:
5211:
5207:
5195:
5191:
5182:
5178:
5163:
5162:
5144:
5136:
5135:
5130:
5116:
5108:
5107:
5102:
5088:
5080:
5079:
5074:
5069:
5065:
5052:
5048:
5037:
5020:
5016:
5010:
5006:
5002:
4998:
4992:
4988:
4982:
4974:
4972:
4969:
4968:
4922:
4914:
4913:
4908:
4894:
4886:
4885:
4880:
4866:
4858:
4857:
4852:
4847:
4843:
4838:
4835:
4834:
4794:
4790:
4782:
4779:
4778:
4739:
4735:
4734:
4729:
4705:
4701:
4700:
4695:
4674:
4670:
4669:
4664:
4658:
4655:
4654:
4625:
4621:
4597:
4593:
4575:
4571:
4569:
4566:
4565:
4527:
4523:
4517:
4513:
4509:
4505:
4499:
4495:
4477:
4465:
4457:
4456:
4451:
4433:
4429:
4420:
4416:
4408:
4405:
4404:
4381:
4378:
4377:
4354:
4350:
4344:
4340:
4335:
4332:
4331:
4300:
4296:
4290:
4286:
4282:
4278:
4266:
4254:
4246:
4245:
4240:
4238:
4235:
4234:
4215:
4212:
4211:
4190:
4186:
4184:
4181:
4180:
4164:
4161:
4160:
4144:
4141:
4140:
4113:
4109:
4100:
4096:
4094:
4091:
4090:
4058:
4054:
4052:
4049:
4048:
4041:
4025:
4004:
4000:
3998:
3995:
3994:
3962:
3961:
3950:
3946:
3931:
3915:
3911:
3905:
3901:
3880:
3864:
3860:
3854:
3850:
3844:
3840:
3816:
3811:
3807:
3797:
3790:
3778:
3774:
3771:
3770:
3760:
3754:
3750:
3746:
3744:
3741:
3740:
3730:
3694:
3690:
3669:
3665:
3635:
3631:
3621:
3619:
3590:
3587:
3586:
3554:
3550:
3529:
3525:
3495:
3491:
3481:
3479:
3477:
3474:
3473:
3430:
3427:
3426:
3396:
3392:
3380:
3376:
3370:
3366:
3354:
3349:
3344:
3342:
3330:
3326:
3317:
3313:
3304:
3300:
3289:
3285:
3284:
3280:
3278:
3275:
3274:
3269:
3250:
3230:
3222:
3185:
3181:
3169:
3165:
3153:
3149:
3137:
3133:
3124:
3120:
3103:
3100:
3099:
3061:
3057:
3039:
3035:
3020:
3016:
3004:
3000:
2985:
2981:
2969:
2965:
2956:
2952:
2928:
2924:
2911:
2909:
2889:
2885:
2870:
2866:
2851:
2847:
2834:
2830:
2824:
2820:
2803:
2800:
2799:
2767:
2763:
2757:
2753:
2741:
2736:
2720:
2715:
2705:
2700:
2684:
2679:
2674:
2670:
2660:
2648:
2644:
2635:
2631:
2622:
2618:
2609:
2605:
2596:
2592:
2590:
2587:
2586:
2581:
2533:
2529:
2520:
2516:
2499:
2496:
2495:
2455:
2451:
2427:
2423:
2393:
2389:
2376:
2372:
2362:
2342:
2338:
2323:
2319:
2306:
2302:
2296:
2292:
2275:
2272:
2271:
2239:
2235:
2223:
2219:
2213:
2209:
2197:
2192:
2187:
2183:
2173:
2161:
2157:
2148:
2144:
2135:
2131:
2122:
2118:
2116:
2113:
2112:
2107:
2057:symmetric group
2036:
2032:
2024:
2021:
2020:
1983:
1980:
1979:
1949:
1936:
1935:
1934:
1932:
1929:
1928:
1917:
1887:
1883:
1871:
1867:
1855:
1851:
1839:
1835:
1834:
1830:
1820:
1784:
1780:
1763:
1760:
1759:
1720:
1715:
1699:
1694:
1678:
1673:
1663:
1658:
1642:
1638:
1632:
1627:
1611:
1606:
1601:
1597:
1587:
1575:
1571:
1562:
1558:
1549:
1545:
1536:
1532:
1523:
1519:
1517:
1514:
1513:
1499:
1494:
1488:
1483:
1437:
1433:
1421:
1409:
1401:
1400:
1395:
1359:
1355:
1324:
1318:
1314:
1313:
1309:
1307:
1304:
1303:
1249:
1244:
1231:
1226:
1192:
1187:
1174:
1169:
1141:
1136:
1123:
1118:
1090:
1086:
1077:
1073:
1058:
1054:
1036:
1032:
1023:
1019:
1011:
1008:
1007:
978:
974:
950:
946:
928:
924:
894:
890:
873:
870:
869:
800:
757:
753:
752:
747:
723:
719:
718:
713:
692:
688:
687:
682:
666:
654:
646:
645:
640:
628:
624:
609:
605:
596:
592:
583:
579:
577:
574:
573:
561:
554:
531:
501:
497:
491:
487:
475:
471:
462:
458:
441:
438:
437:
427:
386:
378:
370:
367:
366:
337:
333:
321:
309:
301:
300:
295:
279:
273:
269:
268:
264:
262:
259:
258:
221:
218:
217:
198:
168:
156:is relevant so
91:
28:
17:
12:
11:
5:
5650:
5640:
5639:
5634:
5629:
5615:
5614:
5607:
5588:
5576:
5575:External links
5573:
5572:
5571:
5557:
5536:
5524:(1): 145–254.
5507:
5481:(2): 373–384.
5465:
5439:(3): 433–455.
5424:
5421:
5420:
5419:
5412:
5409:
5401:
5400:
5383:
5380:
5377:
5372:
5368:
5362:
5358:
5355:
5350:
5345:
5341:
5337:
5332:
5327:
5323:
5318:
5313:
5310:
5305:
5302:
5299:
5294:
5290:
5284:
5280:
5277:
5272:
5267:
5263:
5259:
5254:
5249:
5245:
5240:
5235:
5230:
5227:
5224:
5219:
5215:
5210:
5206:
5203:
5198:
5194:
5190:
5185:
5181:
5177:
5174:
5171:
5168:
5166:
5164:
5160:
5156:
5153:
5147:
5143:
5139:
5133:
5129:
5125:
5119:
5115:
5111:
5105:
5101:
5097:
5091:
5087:
5083:
5077:
5073:
5068:
5064:
5059:
5051:
5047:
5043:
5040:
5038:
5035:
5029:
5026:
5023:
5019:
5013:
5009:
5005:
5001:
4995:
4991:
4985:
4981:
4977:
4976:
4954:
4953:
4942:
4938:
4934:
4931:
4925:
4921:
4917:
4911:
4907:
4903:
4897:
4893:
4889:
4883:
4879:
4875:
4869:
4865:
4861:
4855:
4851:
4846:
4842:
4797:
4793:
4789:
4786:
4773:. The element
4767:
4766:
4753:
4750:
4747:
4742:
4738:
4732:
4728:
4724:
4719:
4716:
4713:
4708:
4704:
4698:
4694:
4688:
4685:
4682:
4677:
4673:
4667:
4663:
4639:
4636:
4633:
4628:
4624:
4620:
4617:
4614:
4611:
4608:
4605:
4600:
4596:
4592:
4589:
4586:
4583:
4578:
4574:
4558:
4557:
4546:
4542:
4536:
4533:
4530:
4526:
4520:
4516:
4512:
4508:
4502:
4498:
4492:
4489:
4486:
4483:
4480:
4476:
4468:
4464:
4460:
4455:
4450:
4447:
4444:
4441:
4436:
4432:
4428:
4423:
4419:
4415:
4412:
4385:
4363:
4360:
4357:
4353:
4347:
4343:
4339:
4328:
4327:
4315:
4309:
4306:
4303:
4299:
4293:
4289:
4285:
4281:
4275:
4272:
4269:
4265:
4257:
4253:
4249:
4244:
4219:
4193:
4189:
4168:
4148:
4124:
4121:
4116:
4112:
4108:
4103:
4099:
4061:
4057:
4040:
4037:
4036:
4035:
4007:
4003:
3976:
3975:
3959:
3953:
3949:
3943:
3940:
3937:
3934:
3930:
3926:
3923:
3918:
3914:
3908:
3904:
3898:
3895:
3892:
3889:
3886:
3883:
3879:
3875:
3872:
3867:
3863:
3857:
3853:
3847:
3843:
3837:
3834:
3831:
3828:
3825:
3822:
3819:
3815:
3810:
3804:
3801:
3796:
3793:
3791:
3787:
3784:
3781:
3777:
3773:
3772:
3769:
3766:
3763:
3761:
3757:
3753:
3749:
3748:
3728:
3723:
3722:
3711:
3706:
3702:
3697:
3693:
3689:
3686:
3683:
3680:
3677:
3672:
3668:
3664:
3661:
3658:
3655:
3652:
3649:
3646:
3643:
3638:
3634:
3630:
3627:
3624:
3618:
3615:
3612:
3609:
3606:
3603:
3600:
3597:
3594:
3580:
3579:
3566:
3562:
3557:
3553:
3549:
3546:
3543:
3540:
3537:
3532:
3528:
3524:
3521:
3518:
3515:
3512:
3509:
3506:
3503:
3498:
3494:
3490:
3487:
3484:
3458:
3455:
3452:
3449:
3446:
3443:
3440:
3437:
3434:
3422:
3421:
3410:
3405:
3399:
3395:
3391:
3388:
3383:
3379:
3373:
3369:
3365:
3362:
3357:
3352:
3348:
3341:
3338:
3333:
3329:
3325:
3320:
3316:
3312:
3307:
3303:
3299:
3292:
3288:
3283:
3267:
3248:
3228:
3221:
3218:
3214:
3213:
3202:
3199:
3196:
3193:
3188:
3184:
3180:
3177:
3172:
3168:
3164:
3161:
3156:
3152:
3148:
3145:
3140:
3136:
3132:
3127:
3123:
3119:
3116:
3113:
3110:
3107:
3093:
3092:
3079:
3075:
3072:
3069:
3064:
3060:
3056:
3053:
3050:
3047:
3042:
3038:
3034:
3031:
3028:
3023:
3019:
3015:
3012:
3007:
3003:
2999:
2996:
2993:
2988:
2984:
2980:
2977:
2972:
2968:
2964:
2959:
2955:
2951:
2948:
2945:
2942:
2939:
2936:
2931:
2927:
2923:
2920:
2917:
2914:
2908:
2904:
2900:
2897:
2892:
2888:
2884:
2881:
2878:
2873:
2869:
2865:
2862:
2859:
2854:
2850:
2846:
2843:
2840:
2837:
2833:
2827:
2823:
2819:
2816:
2813:
2810:
2807:
2789:
2788:
2776:
2770:
2766:
2760:
2756:
2752:
2749:
2744:
2739:
2735:
2731:
2728:
2723:
2718:
2714:
2708:
2703:
2699:
2695:
2692:
2687:
2682:
2678:
2673:
2667:
2664:
2659:
2656:
2651:
2647:
2643:
2638:
2634:
2630:
2625:
2621:
2617:
2612:
2608:
2604:
2599:
2595:
2579:
2562:
2561:
2550:
2547:
2544:
2541:
2536:
2532:
2528:
2523:
2519:
2515:
2512:
2509:
2506:
2503:
2489:
2488:
2477:
2473:
2469:
2466:
2463:
2458:
2454:
2450:
2447:
2444:
2441:
2438:
2435:
2430:
2426:
2422:
2419:
2416:
2413:
2410:
2407:
2404:
2401:
2396:
2392:
2388:
2385:
2382:
2379:
2375:
2369:
2366:
2361:
2357:
2353:
2350:
2345:
2341:
2337:
2334:
2331:
2326:
2322:
2318:
2315:
2312:
2309:
2305:
2299:
2295:
2291:
2288:
2285:
2282:
2279:
2261:
2260:
2248:
2242:
2238:
2234:
2231:
2226:
2222:
2216:
2212:
2208:
2205:
2200:
2195:
2191:
2186:
2180:
2177:
2172:
2169:
2164:
2160:
2156:
2151:
2147:
2143:
2138:
2134:
2130:
2125:
2121:
2105:
2044:
2039:
2035:
2031:
2028:
2008:
2005:
2002:
1999:
1996:
1993:
1990:
1987:
1952:
1947:
1944:
1939:
1916:
1913:
1909:
1908:
1896:
1890:
1886:
1882:
1879:
1874:
1870:
1866:
1863:
1858:
1854:
1850:
1847:
1842:
1838:
1833:
1827:
1824:
1819:
1816:
1813:
1810:
1807:
1804:
1801:
1798:
1795:
1792:
1787:
1783:
1779:
1776:
1773:
1770:
1767:
1742:
1741:
1729:
1723:
1718:
1714:
1710:
1707:
1702:
1697:
1693:
1689:
1686:
1681:
1676:
1672:
1666:
1661:
1657:
1653:
1650:
1645:
1641:
1635:
1630:
1626:
1622:
1619:
1614:
1609:
1605:
1600:
1594:
1591:
1586:
1583:
1578:
1574:
1570:
1565:
1561:
1557:
1552:
1548:
1544:
1539:
1535:
1531:
1526:
1522:
1498:
1495:
1490:Main article:
1487:
1484:
1482:
1479:
1466:
1465:
1454:
1449:
1446:
1443:
1440:
1436:
1430:
1427:
1424:
1420:
1412:
1408:
1404:
1399:
1394:
1391:
1388:
1385:
1382:
1379:
1376:
1373:
1370:
1367:
1362:
1358:
1354:
1351:
1348:
1345:
1342:
1339:
1335:
1331:
1327:
1321:
1317:
1312:
1281:
1280:
1269:
1266:
1263:
1260:
1257:
1252:
1247:
1243:
1239:
1234:
1229:
1225:
1221:
1218:
1215:
1212:
1209:
1206:
1203:
1200:
1195:
1190:
1186:
1182:
1177:
1172:
1168:
1164:
1161:
1158:
1155:
1152:
1149:
1144:
1139:
1135:
1131:
1126:
1121:
1117:
1113:
1110:
1107:
1104:
1101:
1098:
1093:
1089:
1085:
1080:
1076:
1072:
1069:
1066:
1061:
1057:
1053:
1050:
1047:
1044:
1039:
1035:
1031:
1026:
1022:
1018:
1015:
1001:
1000:
989:
986:
981:
977:
973:
970:
967:
964:
961:
958:
953:
949:
945:
942:
939:
936:
931:
927:
923:
920:
917:
914:
911:
908:
905:
902:
897:
893:
889:
886:
883:
880:
877:
798:
785:
784:
771:
768:
765:
760:
756:
750:
746:
742:
737:
734:
731:
726:
722:
716:
712:
706:
703:
700:
695:
691:
685:
681:
675:
672:
669:
665:
657:
653:
649:
644:
639:
636:
631:
627:
623:
620:
617:
612:
608:
604:
599:
595:
591:
586:
582:
559:
552:
529:
524:
523:
512:
509:
504:
500:
494:
490:
486:
483:
478:
474:
470:
465:
461:
457:
454:
451:
448:
445:
426:
423:
389:
385:
381:
377:
374:
363:
362:
349:
346:
343:
340:
336:
330:
327:
324:
320:
312:
308:
304:
299:
294:
290:
286:
282:
276:
272:
267:
231:
228:
225:
196:
192:dihedral group
166:
122:symmetry group
90:
87:
40:Pólya counting
15:
9:
6:
4:
3:
2:
5649:
5638:
5635:
5633:
5630:
5628:
5625:
5624:
5622:
5612:
5608:
5603:
5602:
5597:
5594:
5589:
5586:
5582:
5579:
5578:
5568:
5564:
5560:
5558:0-387-96413-4
5554:
5550:
5545:
5544:
5537:
5532:
5527:
5523:
5519:
5518:
5513:
5508:
5504:
5500:
5496:
5492:
5488:
5484:
5480:
5476:
5475:
5470:
5466:
5462:
5458:
5454:
5450:
5446:
5442:
5438:
5434:
5433:
5427:
5426:
5418:
5415:
5414:
5408:
5406:
5378:
5370:
5366:
5360:
5356:
5353:
5348:
5343:
5339:
5335:
5330:
5325:
5321:
5316:
5311:
5308:
5300:
5292:
5288:
5282:
5278:
5275:
5270:
5265:
5261:
5257:
5252:
5247:
5243:
5238:
5233:
5225:
5217:
5213:
5204:
5201:
5196:
5192:
5188:
5183:
5179:
5172:
5169:
5167:
5158:
5154:
5151:
5141:
5131:
5127:
5123:
5113:
5103:
5099:
5095:
5085:
5075:
5071:
5066:
5062:
5057:
5049:
5045:
5041:
5039:
5033:
5027:
5024:
5021:
5011:
5007:
4999:
4993:
4989:
4983:
4979:
4967:
4966:
4965:
4963:
4959:
4940:
4936:
4932:
4929:
4919:
4909:
4905:
4901:
4891:
4881:
4877:
4873:
4863:
4853:
4849:
4844:
4840:
4833:
4832:
4831:
4829:
4825:
4821:
4817:
4813:
4795:
4791:
4787:
4784:
4776:
4772:
4748:
4740:
4736:
4730:
4726:
4722:
4714:
4706:
4702:
4696:
4692:
4683:
4675:
4671:
4665:
4661:
4653:
4652:
4651:
4634:
4626:
4622:
4618:
4615:
4612:
4606:
4598:
4594:
4590:
4584:
4576:
4572:
4563:
4544:
4540:
4534:
4531:
4528:
4518:
4514:
4506:
4500:
4496:
4490:
4487:
4484:
4481:
4478:
4474:
4462:
4453:
4448:
4442:
4439:
4434:
4430:
4426:
4421:
4417:
4410:
4403:
4402:
4401:
4399:
4383:
4361:
4358:
4355:
4345:
4341:
4313:
4307:
4304:
4301:
4291:
4287:
4279:
4273:
4270:
4267:
4263:
4251:
4242:
4233:
4232:
4231:
4217:
4209:
4191:
4187:
4166:
4146:
4138:
4122:
4119:
4114:
4110:
4106:
4101:
4097:
4087:
4085:
4081:
4077:
4059:
4055:
4046:
4033:
4028:
4023:
4022:
4021:
4005:
4001:
3991:
3989:
3985:
3981:
3957:
3951:
3947:
3941:
3938:
3935:
3932:
3928:
3924:
3921:
3916:
3912:
3906:
3902:
3896:
3893:
3890:
3887:
3884:
3881:
3877:
3873:
3870:
3865:
3861:
3855:
3851:
3845:
3841:
3835:
3832:
3829:
3826:
3823:
3820:
3817:
3813:
3808:
3802:
3799:
3794:
3792:
3785:
3782:
3779:
3775:
3767:
3764:
3762:
3755:
3751:
3739:
3738:
3737:
3735:
3731:
3709:
3704:
3695:
3691:
3684:
3681:
3678:
3670:
3666:
3659:
3653:
3647:
3644:
3641:
3636:
3628:
3622:
3616:
3613:
3610:
3607:
3604:
3598:
3592:
3585:
3584:
3583:
3564:
3555:
3551:
3544:
3541:
3538:
3530:
3526:
3519:
3513:
3507:
3504:
3501:
3496:
3488:
3482:
3472:
3471:
3470:
3453:
3447:
3444:
3438:
3432:
3408:
3403:
3397:
3393:
3389:
3386:
3381:
3377:
3371:
3367:
3363:
3360:
3355:
3350:
3346:
3339:
3331:
3327:
3323:
3318:
3314:
3310:
3305:
3301:
3290:
3286:
3281:
3273:
3272:
3271:
3266:
3256:
3252:
3247:
3242:
3238:
3234:
3227:
3217:
3200:
3197:
3194:
3191:
3186:
3182:
3178:
3175:
3170:
3166:
3162:
3159:
3154:
3150:
3146:
3143:
3138:
3134:
3130:
3125:
3121:
3117:
3111:
3105:
3098:
3097:
3096:
3077:
3070:
3067:
3062:
3058:
3048:
3045:
3040:
3036:
3029:
3026:
3021:
3013:
3010:
3005:
3001:
2994:
2991:
2986:
2978:
2975:
2970:
2966:
2957:
2949:
2946:
2943:
2937:
2934:
2929:
2921:
2918:
2915:
2906:
2902:
2898:
2895:
2890:
2886:
2882:
2879:
2876:
2871:
2867:
2863:
2860:
2857:
2852:
2848:
2844:
2841:
2838:
2835:
2831:
2825:
2821:
2817:
2811:
2805:
2798:
2797:
2796:
2794:
2774:
2768:
2764:
2758:
2754:
2750:
2747:
2742:
2737:
2733:
2729:
2726:
2721:
2716:
2712:
2706:
2701:
2697:
2693:
2690:
2685:
2680:
2676:
2671:
2665:
2662:
2657:
2649:
2645:
2641:
2636:
2632:
2628:
2623:
2619:
2615:
2610:
2606:
2597:
2593:
2585:
2584:
2583:
2578:
2569:
2565:
2548:
2545:
2542:
2539:
2534:
2530:
2526:
2521:
2517:
2513:
2507:
2501:
2494:
2493:
2492:
2475:
2471:
2464:
2461:
2456:
2452:
2445:
2442:
2436:
2433:
2428:
2424:
2414:
2411:
2408:
2402:
2399:
2394:
2386:
2383:
2380:
2373:
2367:
2364:
2359:
2355:
2351:
2348:
2343:
2339:
2335:
2332:
2329:
2324:
2320:
2316:
2313:
2310:
2307:
2303:
2297:
2293:
2289:
2283:
2277:
2270:
2269:
2268:
2266:
2246:
2240:
2236:
2232:
2229:
2224:
2220:
2214:
2210:
2206:
2203:
2198:
2193:
2189:
2184:
2178:
2175:
2170:
2162:
2158:
2154:
2149:
2145:
2141:
2136:
2132:
2123:
2119:
2111:
2110:
2109:
2104:
2099:
2092:
2084:
2080:
2078:
2074:
2070:
2066:
2062:
2058:
2042:
2037:
2033:
2029:
2026:
2006:
2003:
2000:
1997:
1991:
1985:
1976:
1973:
1969:
1945:
1942:
1926:
1922:
1912:
1894:
1888:
1884:
1880:
1877:
1872:
1868:
1864:
1861:
1856:
1852:
1848:
1845:
1840:
1836:
1831:
1825:
1822:
1817:
1811:
1808:
1805:
1802:
1799:
1796:
1793:
1785:
1781:
1777:
1771:
1765:
1758:
1757:
1756:
1753:
1751:
1747:
1727:
1721:
1716:
1712:
1708:
1705:
1700:
1695:
1691:
1687:
1684:
1679:
1674:
1670:
1664:
1659:
1655:
1651:
1648:
1643:
1639:
1633:
1628:
1624:
1620:
1617:
1612:
1607:
1603:
1598:
1592:
1589:
1584:
1576:
1572:
1568:
1563:
1559:
1555:
1550:
1546:
1542:
1537:
1533:
1524:
1520:
1512:
1511:
1510:
1508:
1504:
1497:Colored cubes
1493:
1478:
1476:
1472:
1452:
1444:
1438:
1434:
1428:
1425:
1422:
1418:
1406:
1397:
1392:
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1360:
1356:
1352:
1346:
1340:
1337:
1333:
1329:
1325:
1319:
1315:
1310:
1302:
1301:
1300:
1298:
1294:
1290:
1286:
1267:
1258:
1255:
1250:
1245:
1241:
1237:
1232:
1227:
1223:
1216:
1213:
1210:
1207:
1201:
1198:
1193:
1188:
1184:
1180:
1175:
1170:
1166:
1159:
1156:
1150:
1147:
1142:
1137:
1133:
1129:
1124:
1119:
1115:
1108:
1105:
1099:
1096:
1091:
1087:
1083:
1078:
1074:
1067:
1059:
1055:
1051:
1045:
1042:
1037:
1033:
1029:
1024:
1020:
1013:
1006:
1005:
1004:
979:
975:
968:
965:
962:
959:
951:
947:
940:
937:
929:
925:
918:
915:
909:
903:
895:
891:
887:
881:
875:
868:
867:
866:
864:
860:
856:
852:
848:
844:
840:
836:
832:
828:
824:
819:
817:
813:
809:
805:
801:
794:
790:
766:
758:
754:
748:
744:
740:
732:
724:
720:
714:
710:
701:
693:
689:
683:
679:
673:
670:
667:
663:
651:
642:
637:
629:
625:
621:
618:
615:
610:
606:
602:
597:
593:
584:
580:
572:
571:
570:
568:
563:
558:
551:
547:
543:
540:
536:
532:
510:
507:
502:
498:
492:
488:
484:
481:
476:
472:
468:
463:
459:
455:
449:
443:
436:
435:
434:
432:
422:
420:
416:
412:
408:
404:
383:
375:
372:
344:
338:
334:
328:
325:
322:
318:
306:
297:
292:
288:
284:
280:
274:
270:
265:
257:
256:
255:
253:
249:
245:
229:
223:
215:
211:
207:
203:
199:
193:
189:
185:
181:
177:
173:
169:
163:
159:
155:
151:
147:
143:
139:
135:
131:
127:
123:
120:
116:
112:
108:
104:
100:
96:
86:
84:
80:
75:
73:
69:
65:
61:
57:
53:
49:
45:
44:combinatorics
41:
37:
33:
26:
22:
5599:
5547:. New York:
5542:
5521:
5515:
5478:
5472:
5469:Frank Harary
5436:
5430:
5404:
5402:
4961:
4957:
4955:
4827:
4823:
4819:
4815:
4811:
4774:
4770:
4768:
4561:
4559:
4397:
4329:
4207:
4136:
4088:
4083:
4079:
4075:
4042:
3992:
3987:
3983:
3979:
3977:
3733:
3726:
3724:
3581:
3423:
3264:
3261:
3245:
3240:
3236:
3225:
3223:
3215:
3094:
2790:
2576:
2574:
2563:
2490:
2262:
2102:
2100:
2097:
2076:
2072:
2068:
2064:
2060:
1967:
1924:
1920:
1918:
1910:
1754:
1745:
1743:
1506:
1502:
1500:
1474:
1470:
1467:
1296:
1292:
1288:
1284:
1282:
1002:
862:
858:
854:
850:
846:
842:
838:
834:
830:
826:
822:
820:
815:
811:
807:
803:
796:
792:
788:
786:
564:
556:
549:
545:
541:
534:
527:
525:
428:
418:
414:
406:
402:
364:
251:
247:
243:
213:
209:
205:
201:
194:
187:
179:
171:
164:
162:cyclic group
157:
149:
141:
137:
133:
129:
114:
111:permutations
102:
94:
92:
76:
68:George Pólya
56:group action
39:
35:
31:
29:
2075:on the set
1975:isomorphism
1919:A graph on
853:) over all
825:on the set
567:cycle index
184:reflections
170:, while if
132:). The set
5621:Categories
5423:References
200:of order 2
99:finite set
5601:MathWorld
5357:…
5309:⋯
5279:…
5205:…
5155:…
5046:∏
5022:ω
4994:ω
4984:ω
4980:∑
4933:…
4788:∈
4785:ϕ
4723:⋯
4616:…
4529:ω
4501:ω
4491:ω
4482:∈
4475:∑
4443:…
4384:ω
4356:ω
4302:ω
4271:∈
4264:∑
4218:ω
4192:ω
4167:ω
4123:…
3929:∑
3878:∑
3814:∑
3617:⋅
2795:.) Hence
1426:∈
1419:∑
1381:…
1259:…
1211:…
1202:…
1151:…
1100:…
1046:…
963:…
741:⋯
671:∈
664:∑
619:…
537:for each
511:⋯
326:∈
319:∑
227:→
5411:See also
3224:The set
1481:Examples
246:acts on
176:bracelet
146:necklace
101:and let
5567:0884155
5503:0214487
5495:2373127
5461:1506633
5453:2370675
4964:, i.e.
4030:in the
4027:A000598
3736:nodes:
845:). The
829:(where
539:integer
190:is the
160:is the
5565:
5555:
5501:
5493:
5459:
5451:
4330:where
4179:, let
3978:where
847:weight
787:where
411:cycles
365:where
119:finite
117:(or a
52:orbits
5491:JSTOR
5449:JSTOR
3233:trees
2791:(see
1972:up to
174:is a
144:is a
124:that
107:group
105:be a
97:be a
58:on a
54:of a
5553:ISBN
4032:OEIS
4020:are
3986:and
2793:here
2265:here
2055:the
1750:here
1299:and
795:and
126:acts
93:Let
38:and
30:The
5526:doi
5483:doi
5441:doi
4830:is
4814:of
4139:of
4078:of
2059:on
857:in
818:.
178:of
148:of
128:on
113:of
109:of
60:set
5623::
5598:.
5563:MR
5561:.
5551:.
5522:68
5520:.
5514:.
5499:MR
5497:.
5489:.
5479:89
5477:.
5457:MR
5455:.
5447:.
5437:49
5435:.
4820:q,
4034:).
3982:,
3201:1.
3078:24
2666:24
2549:1.
1865:12
1826:24
1593:24
569::
555:,
421:.
85:.
74:.
5613:.
5604:.
5587:.
5569:.
5534:.
5528::
5505:.
5485::
5463:.
5443::
5405:g
5382:)
5379:g
5376:(
5371:n
5367:j
5361:)
5354:,
5349:n
5344:2
5340:x
5336:,
5331:n
5326:1
5322:x
5317:(
5312:f
5304:)
5301:g
5298:(
5293:2
5289:j
5283:)
5276:,
5271:2
5266:2
5262:x
5258:,
5253:2
5248:1
5244:x
5239:(
5234:f
5229:)
5226:g
5223:(
5218:1
5214:j
5209:)
5202:,
5197:2
5193:x
5189:,
5184:1
5180:x
5176:(
5173:f
5170:=
5159:)
5152:,
5146:|
5142:q
5138:|
5132:3
5128:x
5124:,
5118:|
5114:q
5110:|
5104:2
5100:x
5096:,
5090:|
5086:q
5082:|
5076:1
5072:x
5067:(
5063:f
5058:g
5050:q
5042:=
5034:|
5028:g
5025:,
5018:)
5012:X
5008:Y
5004:(
5000:|
4990:x
4962:g
4958:g
4941:.
4937:)
4930:,
4924:|
4920:q
4916:|
4910:3
4906:x
4902:,
4896:|
4892:q
4888:|
4882:2
4878:x
4874:,
4868:|
4864:q
4860:|
4854:1
4850:x
4845:(
4841:f
4828:f
4824:q
4816:g
4812:q
4796:X
4792:Y
4775:g
4771:G
4752:)
4749:g
4746:(
4741:n
4737:j
4731:n
4727:t
4718:)
4715:g
4712:(
4707:2
4703:j
4697:2
4693:t
4687:)
4684:g
4681:(
4676:1
4672:j
4666:1
4662:t
4638:)
4635:g
4632:(
4627:n
4623:j
4619:,
4613:,
4610:)
4607:g
4604:(
4599:2
4595:j
4591:,
4588:)
4585:g
4582:(
4577:1
4573:j
4562:g
4545:.
4541:|
4535:g
4532:,
4525:)
4519:X
4515:Y
4511:(
4507:|
4497:x
4488:,
4485:G
4479:g
4467:|
4463:G
4459:|
4454:1
4449:=
4446:)
4440:,
4435:2
4431:x
4427:,
4422:1
4418:x
4414:(
4411:F
4398:g
4362:g
4359:,
4352:)
4346:X
4342:Y
4338:(
4314:|
4308:g
4305:,
4298:)
4292:X
4288:Y
4284:(
4280:|
4274:G
4268:g
4256:|
4252:G
4248:|
4243:1
4208:f
4188:x
4147:Y
4137:f
4120:,
4115:2
4111:x
4107:,
4102:1
4098:x
4084:g
4080:G
4076:g
4060:X
4056:Y
4006:n
4002:t
3988:c
3984:b
3980:a
3958:)
3952:a
3948:t
3942:n
3939:=
3936:a
3933:3
3925:2
3922:+
3917:b
3913:t
3907:a
3903:t
3897:n
3894:=
3891:b
3888:2
3885:+
3882:a
3874:3
3871:+
3866:c
3862:t
3856:b
3852:t
3846:a
3842:t
3836:n
3833:=
3830:c
3827:+
3824:b
3821:+
3818:a
3809:(
3803:6
3800:1
3795:=
3786:1
3783:+
3780:n
3776:t
3768:1
3765:=
3756:0
3752:t
3734:n
3729:n
3727:t
3710:.
3705:6
3701:)
3696:3
3692:t
3688:(
3685:F
3682:2
3679:+
3676:)
3671:2
3667:t
3663:(
3660:F
3657:)
3654:t
3651:(
3648:F
3645:3
3642:+
3637:3
3633:)
3629:t
3626:(
3623:F
3614:t
3611:+
3608:1
3605:=
3602:)
3599:t
3596:(
3593:F
3565:6
3561:)
3556:3
3552:t
3548:(
3545:F
3542:2
3539:+
3536:)
3531:2
3527:t
3523:(
3520:F
3517:)
3514:t
3511:(
3508:F
3505:3
3502:+
3497:3
3493:)
3489:t
3486:(
3483:F
3457:)
3454:t
3451:(
3448:F
3445:=
3442:)
3439:t
3436:(
3433:f
3409:.
3404:6
3398:3
3394:t
3390:2
3387:+
3382:2
3378:t
3372:1
3368:t
3364:3
3361:+
3356:3
3351:1
3347:t
3340:=
3337:)
3332:3
3328:t
3324:,
3319:2
3315:t
3311:,
3306:1
3302:t
3298:(
3291:3
3287:S
3282:Z
3268:3
3265:S
3249:3
3246:S
3241:n
3237:n
3229:3
3226:T
3198:+
3195:t
3192:+
3187:2
3183:t
3179:2
3176:+
3171:3
3167:t
3163:3
3160:+
3155:4
3151:t
3147:2
3144:+
3139:5
3135:t
3131:+
3126:6
3122:t
3118:=
3115:)
3112:t
3109:(
3106:F
3074:)
3071:1
3068:+
3063:4
3059:t
3055:(
3052:)
3049:1
3046:+
3041:2
3037:t
3033:(
3030:6
3027:+
3022:2
3018:)
3014:1
3011:+
3006:3
3002:t
2998:(
2995:8
2992:+
2987:2
2983:)
2979:1
2976:+
2971:2
2967:t
2963:(
2958:2
2954:)
2950:1
2947:+
2944:t
2941:(
2938:9
2935:+
2930:6
2926:)
2922:1
2919:+
2916:t
2913:(
2907:=
2903:)
2899:1
2896:+
2891:4
2887:t
2883:,
2880:1
2877:+
2872:3
2868:t
2864:,
2861:1
2858:+
2853:2
2849:t
2845:,
2842:1
2839:+
2836:t
2832:(
2826:G
2822:Z
2818:=
2815:)
2812:t
2809:(
2806:F
2775:)
2769:4
2765:t
2759:2
2755:t
2751:6
2748:+
2743:2
2738:3
2734:t
2730:8
2727:+
2722:2
2717:2
2713:t
2707:2
2702:1
2698:t
2694:9
2691:+
2686:6
2681:1
2677:t
2672:(
2663:1
2658:=
2655:)
2650:4
2646:t
2642:,
2637:3
2633:t
2629:,
2624:2
2620:t
2616:,
2611:1
2607:t
2603:(
2598:G
2594:Z
2580:4
2577:S
2546:+
2543:t
2540:+
2535:2
2531:t
2527:+
2522:3
2518:t
2514:=
2511:)
2508:t
2505:(
2502:F
2476:,
2472:)
2468:)
2465:1
2462:+
2457:3
2453:t
2449:(
2446:2
2443:+
2440:)
2437:1
2434:+
2429:2
2425:t
2421:(
2418:)
2415:1
2412:+
2409:t
2406:(
2403:3
2400:+
2395:3
2391:)
2387:1
2384:+
2381:t
2378:(
2374:(
2368:6
2365:1
2360:=
2356:)
2352:1
2349:+
2344:3
2340:t
2336:,
2333:1
2330:+
2325:2
2321:t
2317:,
2314:1
2311:+
2308:t
2304:(
2298:G
2294:Z
2290:=
2287:)
2284:t
2281:(
2278:F
2247:)
2241:3
2237:t
2233:2
2230:+
2225:2
2221:t
2215:1
2211:t
2207:3
2204:+
2199:3
2194:1
2190:t
2185:(
2179:6
2176:1
2171:=
2168:)
2163:3
2159:t
2155:,
2150:2
2146:t
2142:,
2137:1
2133:t
2129:(
2124:G
2120:Z
2106:3
2103:S
2077:Y
2073:G
2069:m
2065:X
2061:m
2043:,
2038:m
2034:S
2030:=
2027:G
2007:t
2004:+
2001:1
1998:=
1995:)
1992:t
1989:(
1986:f
1968:Y
1951:)
1946:2
1943:m
1938:(
1925:X
1921:m
1895:)
1889:2
1885:m
1881:8
1878:+
1873:3
1869:m
1862:+
1857:4
1853:m
1849:3
1846:+
1841:6
1837:m
1832:(
1823:1
1818:=
1815:)
1812:m
1809:,
1806:m
1803:,
1800:m
1797:,
1794:m
1791:(
1786:C
1782:Z
1778:=
1775:)
1772:0
1769:(
1766:F
1746:C
1728:)
1722:3
1717:2
1713:t
1709:6
1706:+
1701:2
1696:3
1692:t
1688:8
1685:+
1680:2
1675:2
1671:t
1665:2
1660:1
1656:t
1652:3
1649:+
1644:4
1640:t
1634:2
1629:1
1625:t
1621:6
1618:+
1613:6
1608:1
1604:t
1599:(
1590:1
1585:=
1582:)
1577:4
1573:t
1569:,
1564:3
1560:t
1556:,
1551:2
1547:t
1543:,
1538:1
1534:t
1530:(
1525:C
1521:Z
1507:C
1503:m
1475:F
1471:f
1453:.
1448:)
1445:g
1442:(
1439:c
1435:m
1429:G
1423:g
1411:|
1407:G
1403:|
1398:1
1393:=
1390:)
1387:m
1384:,
1378:,
1375:m
1372:,
1369:m
1366:(
1361:G
1357:Z
1353:=
1350:)
1347:0
1344:(
1341:F
1338:=
1334:|
1330:G
1326:/
1320:X
1316:Y
1311:|
1297:m
1293:t
1291:(
1289:f
1285:m
1268:.
1265:)
1262:)
1256:,
1251:n
1246:2
1242:t
1238:,
1233:n
1228:1
1224:t
1220:(
1217:f
1214:,
1208:,
1205:)
1199:,
1194:3
1189:2
1185:t
1181:,
1176:3
1171:1
1167:t
1163:(
1160:f
1157:,
1154:)
1148:,
1143:2
1138:2
1134:t
1130:,
1125:2
1120:1
1116:t
1112:(
1109:f
1106:,
1103:)
1097:,
1092:2
1088:t
1084:,
1079:1
1075:t
1071:(
1068:f
1065:(
1060:G
1056:Z
1052:=
1049:)
1043:,
1038:2
1034:t
1030:,
1025:1
1021:t
1017:(
1014:F
988:)
985:)
980:n
976:t
972:(
969:f
966:,
960:,
957:)
952:3
948:t
944:(
941:f
938:,
935:)
930:2
926:t
922:(
919:f
916:,
913:)
910:t
907:(
904:f
901:(
896:G
892:Z
888:=
885:)
882:t
879:(
876:F
863:F
859:X
855:x
851:x
843:Y
841:→
839:X
835:Y
831:Y
827:Y
823:G
816:X
812:g
808:k
804:g
802:(
799:k
797:c
793:X
789:n
770:)
767:g
764:(
759:n
755:c
749:n
745:t
736:)
733:g
730:(
725:2
721:c
715:2
711:t
705:)
702:g
699:(
694:1
690:c
684:1
680:t
674:G
668:g
656:|
652:G
648:|
643:1
638:=
635:)
630:n
626:t
622:,
616:,
611:2
607:t
603:,
598:1
594:t
590:(
585:G
581:Z
560:2
557:t
553:1
550:t
548:(
546:f
542:w
535:w
530:w
528:f
508:+
503:2
499:t
493:2
489:f
485:+
482:t
477:1
473:f
469:+
464:0
460:f
456:=
453:)
450:t
447:(
444:f
419:X
415:g
407:g
405:(
403:c
388:|
384:Y
380:|
376:=
373:m
348:)
345:g
342:(
339:c
335:m
329:G
323:g
311:|
307:G
303:|
298:1
293:=
289:|
285:G
281:/
275:X
271:Y
266:|
252:G
248:Y
244:G
230:Y
224:X
214:Y
210:Y
206:Y
202:n
197:n
195:D
188:G
180:n
172:X
167:n
165:C
158:G
150:n
142:X
138:G
134:X
130:X
115:X
103:G
95:X
27:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.