Òèïîâîé ðàñ÷¼ò 1
Íà÷í¼ì ñ ïóíêòà À
Âûïèøåì íèæíèå ñòðîêè ïîäñòàíîâîê g è h:
g=(4,18,13,5,8,12,14,1,17,10,6,11,15,3,9,16,19,2,7)
h=(3,10,12,7,11,17,16,6,5,4,18,8,1,14,2,13,15,19,9)
Äëÿ ïåðåñòàíîâîê ñïðàâåäëèâû óòâåðæäåíèÿ:
Mob g={1,2,3,4,5,6,7,8,9,11,12,13,14,15,17,18,19}
Fix g={10,16}
Mob h={1,2,3,4,5,6,7,8,9,10,11,12,13,15,16,17,18,19}
Fix h={14}
Ïåðåéä¼ì ê ïóíêòó Á
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 s a n y a s u c k s !
s a n y a s u c k s !
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g·h=(1,12,4,5,10,2,7,9)(3,14,17,8,13,11)(6,18,16,15)
h·g=(1,14,15,3)(2,5,17,4,12,18)(6,10,9,13,19,16,7,11)
g −1 =(18,17,11,10,7,5,13,1)(8,19,12,14,16,4,3,6,2)
h−1 =(9,10,7,6,15,14,4,19,8,18,3,17,12,13,2,11,5,16,1)
Ïåðåéä¼ì ê ïóíêòó Â
g=(1,13,5,7,10,11,17,18)(2,6,3,4,16,14,12,19,8)
h=(1,16,5,11,2,13,12,17,3,18,8,19,4,14,15,6,7,10,9)
[g] = [81 , 91 , 12 ] ⇒ ord g=72
[h] = [191 ] ⇒ ord h=19
Ïåðåéä¼ì ê ïóíêòó Ã
Ðàçëîæèì íàøè ïîäñòàíîâêè íà ïðîèçâåäåíèå òðàíñïîçèöèé:
g=(1,18)(13,18)(5,18)(7,18)(10,18)(11,18)(17,18)(2,8)(6,8)(3,8)(4,18)(16,8)(14,8)(12,8)(19,8)
Âèäèì, ÷òî ó íàñ íå÷¼òíîå êîëè÷åñòâî òðàíñïîçèöèé ⇒ ïåðåñòàíîâêà g íå÷¼òíàÿ
h=(1,9)(16,9)(5,9)(11,9)(2,9)(13,9)(12,9)(17,9)(3,9)(18,9)(19,9)(4,9)(14,9)(15,9)(6,9)(7,9)(10,9)(8,9)
Âèäèì, ÷òî ó íàñ ÷¼òíîå êîëè÷åñòâî òðàíñïîçèöèé ⇒ ïåðåñòàíîâêà h ÷¼òíàÿ
1
Òèïîâîé ðàñ÷¼ò 2
Íà÷í¼ì ñ ïóíêòà À
Âûïèøåì íèæíèå ñòðîêè ïîäñòàíîâîê g è h:
g=(1,2,6,3,9,4,5,7,8)
h=(6,2,8,4,9,5,3,7,1)
Ðàçëîæèì ïîäñòàíîâêè g è h íà öèêëîâûå ñòðóêòóðû:
g=(7,5,9,8)(3,6,4), ãäå [g] = [41 , 31 , 12 ]
h=(1,6,9,5)(3,8,7), ãäå [h] = [41 , 31 , 12 ]
Ïåðåéä¼ì ê ïóíêòó Á
×èñëî ðåøåíèé äëÿ óðàâíåíèé x−1 gx = h è y −1 hy = g îäèíàêîâî òàê êàê â g è h
îäèíàêîâûå öèêëîâûå ñòðóêòóðû. Ìû ìîæåì íàéòè
ýòèõ
Q2 ÷èñëîkðåøåíèé
i
óðàâíåíèé ïî ôîðìóëå: [Nsn (g)] = i=1 (ki )! · li
⇒ ÷èñëî ðåøåíèé ðàâíî (1! · 41 ) · (1! · 31 ) · (2! · 12 ) = 24
Ïåðåéä¼ì ê ïóíêòó Â
2
Ñîñòàâèì òàáëèöó äëÿ óðàâíåíèÿ x−1 gx = h ñî âñåìè åãî ðåøåíèÿìè:

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Ñîñòàâèì òàáëèöó äëÿ óðàâíåíèÿ y −1 hy = g ñî âñåìè åãî ðåøåíèÿìè:
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Ïåðåéä¼ì ê ïóíêòó Ã
Ïðîâåä¼ì ïðîâåðêó äâóõ ïðîèçâîëüíûõ ðåøåíèé íàøèõ óðàâíåíèé:
x=
x
−1
7 5 9 8 3 6 4 1 2
1 6 5 9 3 8 7 4 2
=
= (1, 4, 7)(5, 6, 8, 9)
1 6 5 9 3 8 7 4 2
7 5 9 8 3 6 4 1 2
= (7, 4, 1)(9, 8, 6, 5)
x−1 gx = (7, 4, 1)(9, 8, 6, 5) · (7, 5, 9, 8)(3, 6, 4) · (1, 4, 7)(5, 6, 8, 9) = (7, 3, 8)(1, 6, 5, 9) = h
x=
x
−1
7 5 9 8 3 6 4 1 2
1 6 5 9 8 7 3 4 2
=
1 6 5 9 8 7 3 4 2
7 5 9 8 3 6 4 1 2
4
= (7, 1, 4, 3, 8, 9, 5, 6)
= (1, 7, 6, 5, 9, 8, 3, 4)
x−1 gx = (1, 7, 6, 5, 9, 8, 3, 4) · (7, 5, 9, 8)(3, 6, 4) · (7, 1, 4, 3, 8, 9, 5, 6) = (1, 6, 5, 9)(3, 8, 7) =
h
y=
y
−1
1 6 5 9 3 8 7 4 2
7 5 9 8 3 6 4 1 2
=
= (1, 7, 4)(6, 5, 9, 8)
7 5 9 8 3 6 4 1 2
1 6 5 9 3 8 7 4 2
= (4, 7, 1)(8, 9, 5, 6)
y −1 hy = (7, 1, 4)(5, 6, 8, 9) · (1, 6, 5, 9)(3, 8, 7) · (1, 7, 5)(6, 5, 9, 8) = (7, 5, 9, 8)(4, 3, 6) = g
y=
y
−1
1 6 5 9 3 8 7 4 2
7 5 9 8 6 4 3 1 2
=
7 5 9 8 6 4 3 1 2
1 6 5 9 3 8 7 4 2
= (1, 7, 3, 6, 5, 9, 8, 4)
= (4, 8, 9, 5, 6, 3, 7)
y −1 hy = (4, 8, 9, 5, 6, 3, 7) · (1, 6, 5, 9)(3, 8, 7) · (1, 7, 3, 6, 5, 9, 8, 4) = (7, 5, 9, 8)(4, 3, 6) = g
5
Òèïîâîé ðàñ÷¼ò 3
Èìååì ïîðÿäîê k = 8 â ãðóïïå S17 , ò.ê. ïîðÿäîê ãðóïïû ýòî ÍÎÊ äëèí âñåõ
öèêëîâ íàøåé ïåðåñòàíîâêè, òî â ëþáîì èç ïðåäñòàâëåíèé äàííîé ïåðåñòàíîâêè
áóäåò ôèãóðèðîâàòü õîòÿ áû îäèí öèêë äëèíû 8 è âñå îñòàëüíûå íå áîëüøå 8.
Èíà÷å ãîâîðÿ, ordg = [(g1 , ..., gk )(gt , ..., gn )...(ge , ..., gj )]
Ìû ìîæåì íàéòè ÷èñëî ðåøåíèé
Q ýòèõ óðàâíåíèé ïî ôîðìóëå:
2
i=1 (ki )!
[Nsn (g)] =
· lik
i
Ñîñòàâèì ñïèñîê âñåõ âîçìîæíûõ öèêëîâ â âèäå êîòîðûõ ìîæåò áûòü
ïðåäñòàâëåíà íàøà ïîäñòàíîâêà:
1) 8 + 8 + 1 = 17 ⇒ (2! · 82 )(1! · 11 ) = 128
2) 8 + 4 + 4 + 1 = 17 ⇒ (1! · 81 )(2! · 42 )(1! · 11 ) = 256
3) 8 + 4 + 2 + 2 + 1 = 17 ⇒ (1! · 81 )(1! · 41 )(2! · 22 )(1! · 11 ) = 256
4) 8 + 2 + 2 + 2 + 2 + 1 = 17 ⇒ (1! · 81 )(4! · 24 )(1! · 11 ) = 3072
5) 8 + 4 + 2 + 1 + 1 + 1 = 17 ⇒ (1! · 81 )(1! · 41 )(1! · 21 )(3! · 13 ) = 384
6) 8 + 4 + 1 + 1 + 1 + 1 + 1 = 17 ⇒ (1! · 81 )(1! · 41 )(5! · 15 ) = 3840
7) 8 + 2 + 2 + 1 + 1 + 1 + 1 + 1 = 17 ⇒ (1! · 81 )(2! · 22 )(5! · 15 ) = 7680
8) 8 + 2 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 17 ⇒ (1! · 81 )(1! · 21 )(7! · 17 ) = 80640
9) 8 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 17 ⇒ (1! · 81 )(9! · 19 ) = 2903040
 èòîãå ó íàñ ïîëó÷àåòñÿ 2999296 ðàçëè÷íûõ ïîäñòàíîâîê.
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Òèïîâîé ðàñ÷¼ò 4
7