Question

Compute the sign of the following permutations: (a) (1, 4, 3, 6, 7)(5, 8, 9, 10). (b) σ ∈ Sn, i 7→ n + 1 − i. (c) Show that this initial configuration of the 15 puzzle is not solvable 3 1 15 13 2 8 5 9 12 7 11 4 14 6 10 − (d) Suppose that (i, j) ∈ Sn is a transposition and that i < j. Find an expression (in terms of i and j for the number of flips of (i, j).

Answer 1

we ca nwrite above рест, utanonintncy c les ar ( 1.7) ( 1,C ) (1,5) ( ,,4) С 5,0) ( s'o») cs.r) Sa There ave t" an sp。"hon So sisn of iven pemuton is C- 규 . odd Pemum hon. There is direch methd Irke nd a cycle type Here s 4 cycle type. 6 This queshon is not cleaγ please oonc ushat do yo u me n bi ;구→n+1-1 in Comment I will update answer了aft- Given 3 IS 13 23 912 4 G to Alooithun to Selve Such poblc m O Cal culate a totol no. of Inrerion ② use folloぃ,ngYule to decide solvahlih, to decide selvasbili ty O If gi d İS width is odd , then no. of inversion in Solvable Case is even Cese In case grid widt, is even su,c as ca):- Empty blockis on oddmu fnm bottom he Last, third at se forth then no. of i Case even invesin in selvable

t2t ax odd lat) then in solvoble Ce no of inver sion In our case evid is 4x4, he will calc니 che no, ofinvern on Ovo,of inver s on ave nothing but (i,j) . if ,リ theause skir then and count tota one we har to swipe 12 times 12 inve,ste 13 lo invesio invssio 2 Inv es

Please check the dat once is not last which I as ca/ c cted with, p lease updote gues hm ④ Suppose that Cij)cSn is hans rosironmj we win swap衔吴leaves all 뉴 oheγ elements unchan ged Let alb best. element. ぃ ach ( ij) ca) > (j) (b) ,fb rn w lies be twecnifi also then one more care when, Jea 2 lj-i-1 )

Note: Please Update question I will update solution As b is not clear . And What is position of empty block in 15 puzzle?