Yiziphi ezinye izinhlelo zokusebenza ze-lemma kaBurnside ku-combinatorics ezisekelwe kulokhu?
Context:I-lemma ka-Burnside, ngezinye izikhathi ebizwa nangokuthi i-theorem ka-Burnside yokubala, i-Cauchy<unk>Frobenius lemma, i-orbit-counting theorem, noma i-lemma engeyona ka-Burnside, ingumphumela wombono weqembu ovame ukuba wusizo ekubhekeni ukufana lapho kubalwa izinto zezibalo. Amagama ayo ahlukahlukene asekelwe kuWilliam Burnside, uGeorge Pólya, u-Augustine Louis Cauchy, noFerdinand Georg Frobenius. Umphumela awubangelwa uBurnside ngokwakhe, ovele awucaphune encwadini yakhe ethi 'On the Theory of Groups of Finite Order', esikhundleni salokho awunikeze uFrobenius (1887). I-Burnside's Lemma ibala "ama-orbits", okuyinto efanayo nokubala izinto ezihlukene kucatshangelwa ukufana. Ezinye izindlela zokusho lokhu zibala izinto ezihlukene kuze kube sebuhlotsheni bokulingana R, noma ukubala izinto ezisesimweni esivumelekile.  Kulokhu okulandelayo, ake sithi u-G yiqembu eliphelele elisebenza kusethi X. Ku-g ngayinye ku-G, ake u-Xg akhombise isethi yezakhi ku-X ezihlelwe ngu-g (futhi kuthiwa zishiywe zingashintshi ngu-g), okungukuthi, i-Xg = { x ∈ X │ g.x = x }. I-lemma kaBurnside ithi le formula elandelayo yenani lemigudu, elibhalwe ngokuthi X/G:  Ukubamba  /  Ukubamba = 1 1 Ukubamba  Ukubamba <unk> <unk>  ∈ Isikhathi  Ukubamba   Ukubamba . futhi. {\displaystyle X/G Berry ={\frac {1}{Berry Berry}}\sum _{g\in G} Berry X^{g} Berry.} Ngakho-ke inani le-orbit (inombolo engokwemvelo noma +∞) lilingana nenani eliphakathi lamaphoyinti alungiswe yi-element ye-G (okuyinombolo engokwemvelo noma engapheli). Uma u-G engenamkhawulo, ukuhlukaniswa ngo-G kungahle kungachazeki kahle; kulokhu isitatimende esilandelayo ku-arithmetic cardinal sigcina:  Ukubamba  Ukubamba Ukubamba  /  Ukubamba = <unk> <unk>  ∈ Isikhathi  Ukubamba   Ukubamba . futhi. {\displaystyle \GgadgadgadgadgadgadX/Ggadgadgad=\sum _{g\in G}gadgadgadX^{g}gadgad.} Izibonelo zezicelo zokubalwa Izindandatho Kukhona ama-vector angu-8 okungenzeka abe nobude obuyi-3, kodwa amane kuphela ama-collar ahlukene anemibala emibili obude obuyi-3: 000, 001, 011 no-111, ngoba u-100 no-010 balingana no-001 ngokujikeleza; ngokufanayo u-110 no-101 balingana no-011. Ifomula isekelwe enanini lokujikeleza, okuyinto kulokhu kuyinto 3 (kuhlanganise ukujikeleza null), kanye nenani bit vectors kwesokunxele unchanged by ukujikeleza ngamunye. Wonke ama-vector angu-8 awashintshi ngokujikeleza okungu-null, futhi amabili (000 no-111) awashintshi ngokujikeleza ngakunye kokunye okubili, anikeze: 4 Izinketho = 1 1 3 Izinketho (Umthetho 8 Izinketho + 2 - 2 - 2 - 3 + 2 - 2 - 2 - 3 ) Izindawo {\displaystyle 4={\frac {1}{3}}(8+2+2)}.  Ukuze ubude 4, zikhona 16 bit vectors kungenzeka; 4 ukujikeleza; ukujikeleza null kushiya zonke 16 bit vectors engashintshiwe; 1-Ukujikeleza futhi 3-Ukujikeleza ngamunye ushiye ezimbili bit vectors engashintshiwe (0000 futhi 1111); 2-Ukujikeleza kushiya 4 bit vectors engashintshiwe (0000, 0101, 1010, futhi 1111); enikeza: 6 izinga = 1 1 4 Izinketho (Umthetho 16 izinyanga + 2 - 2 - 2 - 3 + 4 Izinketho + 2 - 2 - 2 - 3 ) Izindawo {\displaystyle 6={\frac {1}{4}}(16+2+4+2)}. Lezi yilezi: 0000, 0001, 0011, 0101, 0111, kanye no-1111.  Imibala yekhibhodi Inani lemibala ehlukene ngokuguquguquka kobuso becube kusetshenziswa imibala emithathu lingatholakala ngale ndlela elandelayo.  Ake sithi u-X uyisethi yezinhlanganisela zombala zobuso ezingama-36 ezingaba khona ezingasetshenziswa ku-cube ekuqondeni okukodwa, futhi ake iqembu lokujikeleza G le-cube lisebenze ku-X ngendlela engokwemvelo. Khona-ke izakhi ezimbili ze-X ziyingxenye yomjikelezo ofanayo ngokunembile lapho enye imane nje ingukuzungeza kwenye. Ngakho-ke, inani lemibala ehlukene ngokujikeleza lilingana nenani lemigudu futhi lingatholakala ngokubala ubukhulu beqoqo elizinzile lezinto ezingu-24 ze-G.   Icube enezinhlangothi ezinemibala isici esisodwa sobunikazi esishiya zonke izakhi ezingu-36 ze-X zingashintshiwe ukuzungezisa ubuso obungama-degree angu-90 okuyisithupha, ngayinye eshiya izakhi ezingu-33 ze-X zingashintshiwe ukuphenduka kobuso okungu-180 degrees kathathu, ngakunye okushiya izinto ezingu-34 ze-X zingashintshiwe ukuzungeza ama-vertex angu-120 degrees ayisishiyagalombili, ngayinye eshiya izinto ezingu-32 ze-X zingashintshiwe ukuzungeza emaphethelweni ama-degree angu-180, ngayinye eshiya 33 wezakhi ze-X zingashintshiwe Ukuhlolwa okuningiliziwe kwalezi automorphisms kungatholakala lapha.  Ngakho-ke ubukhulu obujwayelekile bokulinganisa  1 1 24 izinyanga (Umthetho 3 Izinketho 6 izinga + 6 izinga ⋅ Izindawo 3 Izinketho 3 Izinketho + 3 Izinketho ⋅ Izindawo 3 Izinketho 4 Izinketho + 8 Izinketho ⋅ Izindawo 3 Izinketho 2 - 2 - 2 - 3 + 6 izinga ⋅ Izindawo 3 Izinketho 3 Izinketho ) Izindawo = Amaphesenti angu-57. I-{frac {1}{24}}\left{3^{6}+6\cdot 3^{3}+3\cdot 3^{4}+8\cdot 3^{2}+6\cdot 3^{3}\right) =57. Ngakho-ke kunemibala engu-57 ehlukene ngokuguquguquka kobuso bekhibhodi ngemibala emithathu. Ngokuvamile, inombolo imibala ehlukene ngokujikeleza kobuso becube ngemibala n inikezwa  1 1 24 izinyanga (Umthetho  6 izinga + 3 Izinketho  4 Izinketho + 12 izinyanga  3 Izinketho + 8 Izinketho  2 - 2 - 2 - 3 ) Izindawo . futhi. I-{frac {1}{24}}\left(n^{6}+3n^{4}+12n^{3}+8n^{2}\right). Iphutha Labonkulunkulukazi Abangu-8 Kulesi sixazululo samakhosikazi ayisishiyagalombili kunezixazululo ezingu-92, okuvela kuzo izixazululo eziyisisekelo ezingu-12 ezihlukile kuze kube ukujikeleza nokubonakaliswa kwebhodi. Kunezinhlanganisela ezingu-8 zokujikeleza nokuzibonakalisa, kuhlanganise nesenzo esingenalutho. Isenzo esingenalutho sishiya zonke izixazululo ezingu-92 zingashintshiwe. Izixazululo ezine kwezingu-92 ziyalingana, azishintshiwe ngokuzungeza ngo-180°. Lokho kunikeza: 12 izinyanga = 1 1 8 Izinketho (Umthetho 92 izinyanga + 4 Izinketho ) Izindawo {\displaystyle 12={\frac {1}{8}}(92+4)}.
* Inkinga yokubala imigexo: ukubala imigexo ehlukene ngokujikeleza yobude obuhlukahlukene lapho ubuhlalu ngabunye bungaba ngomunye wemibala emibili. Lesi sihloko sibala imigexo enobude obungu-3 no-4 futhi impendulo ingu-4 no-6 ngokulandelana. * Imibala yecube inkinga: bala imibala ehlukene ngokuguquguquka kobuso becube. Lesi sihloko sixoxa ngesimo lapho kusetshenziswa imibala emithathu bese impendulo ingu-57. * 8 amakhosikazi puzzle: inguqulo yakudala yenkinga ukubala izindlela ezahlukene ukubeka 8 amakhosikazi chess ku 8x8 ibhodi ukuze akukho indlovukazi ukuhlasela omunye (impendulo 92). Ngokusebenzisa i-lemma kaBurnside lesi sihloko sibala inani lezikhundla ezehlukene zamakhosikazi kuze kube ukujikeleza nokucabanga, sifika ku-12 njengempendulo.