[glashan] reduction OF CIRCULANTS 43 



Examples.- — ^For n = 7 , ei = 2, 66 = 2 and .'. ;;? = 4 = 2- 



i^ f I o o, 7(3!) ^ 7^(1!)^ „/ 3 . 7\ „^„ 



2! 2! 2! \ 2 2/ 



If n = 7, ei = 62= €3= ^4= ^5= ^6= 1 and .'. ^j? = 6 = 4-|-2 = 3+3 = 



2 + 2 + 2 



Kof bi>'23'33'43'53'6| = -7(5 !) + 7H3(3!l!) + (2!2!)}-7'ni!l!l!) = -7X15 



If n = ll, 61 = 2, 62 = 1, 63 = 64= . .•. . . . =68=0, 69=1, 610 = 2 



and .-. m = 6 = 4 + 2=(2x2)+2 = 3 + 3 = 2 + 2 + 2. 



T. f, o ,| 11(5!) , ,,oLm, , 3!1! , 2!2!\ 113(l!)2(i!) 



Kof vr>'23'93'-io = — • + 1 1" < 3 ! 1 ! + + > — 



' -^ ' 2!2! I 2!2! 2! 2\} 2! 



= ll(-30 + 66+16,Î2 + ll-603^) = llX3. 



If W = ll, 61 = 62 = 63= =610= 1, 



and .-.^ = 10 = 8 + 2 = 7 + 3 = 6+4 = 5 + 5 



=6+2+2=5+3+2=4+4+2=4+3+3=4+2+2 

 = 3+3+2 + 2 = 2 + 2+2 + 2 + 2. 

 Kof |3'iy23'33'43'53'6>'73'8>'9>'ioI = -ll(9!) + lP{5(7!l!) + 10(6!2!) 

 +20(5!3!) + ll(4!4!)}-lP{l0(5!l!l!)+20(4!2!l!) 

 +20(3!3!1!) + 15(3!2!2!)} + 114{10(3!1!1!1!)+5(2!2!1!1!)} 

 -1P(1!1!1!1!1!) = 11X615. 



Theorem IV.^ — ^By the cyclosymmetry of the circulant C{n) 



Kof I .r^o^i*i3'2^2 3'^!lriM == Kof | .t^«-i ji^o y2^^ >'f/L~i^ 1 



Hence 



Kof [0^01^12^2 (w-1 )«»-!] = Kof bi^i3'2^2 3,363 3,e„_-i| 



Examples 



C (7) = [0^] - 7 { [on 6] - [Oq^s] _ 2[0n24] + [0^34] + [0n223] - 2[0n262] 



- [0n256] + [On^-326] +2[02p246] - 5[0n2345] + 15[0123456] } 

 Arranged according to powers of x this becomes 



C(7) =x-'-7{ (xnQ) - (xn^5) -2(xn24) + (rc='P4)+3(%n-^23) 

 -2{xn^Q^) - (xn256) - {xH'S) -2(xn^2^) -Sixn^5Q) 

 + (x2p3^6) +2(;x:n2246) - 5(x2p'345) + (xP2) - (:«1*50 

 - 2(:\;1446) + (xlW)-\-3{xlW5) - (.'v:P236) +2(:x:P2235) 

 - 5(:k123224) + (.vl-2242) + 15(xl23456) + (P36) - (1^226) 

 -2(1^235) + (P2^5) +3(P2234) -2(P3-6^) - (1^3456) + (1-23623) 

 +2(123^256) - 5 (r22456) } + (F) . 



C(ll) = [0'i]-ll{[0nj]-[08p9]_2[08128] + [0U38]+3[0U227+07p36 

 + 0n-45]-4[0n-j-] + 6[0n235]-5[071297 + 0U38j] 

 - ]0«1''7] - 2[0''P4-] - 4[06p26 + 0n^S5] + 7[0n^9j] 



