A FIBONACCI MATRIX AND THE PERMANENT FUNCTION

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1 A FIBONACCI MATRIX AND THE PERMANENT FUNCTION BRUCE W. KING Burt Hiils-Ballsto Lake High School, Ballsto Lake, New York ad FRANCIS D. PARKER The St. Lawrece Uiversity, Cato, New York The permaet of a -square matrix [a..] is defied to be a, " S i=l J i where a = <Ji»J2»*»3 ) is a member of the symmetric group S of permutatios o distict objects. For example, the permaet of the matrix a i i a l 2 a i 3 a 2 i a22 a 2 3 is a 3 a 32 a 33 a l i a 2 2 a a li a 23 a 32 + a i2 a 2 a 33 + a i2 a 23 a 3 + a i3 a 2i a 32 + a 3 a 22 a 3 This is similar to the defiitio of the determiat of [a.. ], which is ^ ii ^. a e S i=l where e^. is or - depedig upo whether " is a eve or a odd permutatio. There are other similarities betwee the permaet ad the determiat fuctios, amog them; (a) iterchagig two rows, or two colums, of a matrix chages the sig of the determiat but it does ot chage the permaet at all. Thus, the permaet of a matrix remais ivariat uder arbitrary permutatios of its rows ad colums; ad J i I this otatio, (j 9 j2? w > 3 ) \Ji 32 3 / i s a abbreviatio for the permutatio 539

2 54 A FIBONACCI MATRIX AND THE PERMANENT FUNCTION [Dec. (b) there is a Laplace expasio for the permaet of a matrix as well as for the determiat. I particular, there is a row or colum expasio for the permaet. For example, if we use "per [ a.. ] " for the permaet of the matrix [ a.. ], the expasio alog the first colum yields that per a l i a 2. a 3 a i 2 a 22 a 32 a 3* a 23 a 33. = a iiper [a 22 [ a 32 a 2 3 ] + a p e r [ a i 2 a i3l + a 3 l P e r r a i 2 a 2 F 33j [ a 32 a 3 F 33j (j* 22 a i3l ^ J For further iformatio o properties of the permaet, the reader should see [, p. 578] ad [3, pp ], Ufortuately, oe of the most useful prpperties of the determiat its ivariace uder the additio of a multiple of a row (or colum) to aother row (or colum) is false for the permaet fuctio. As a result, evaluatig the permaet of a matrix i s, geerally, a much more difficult problem tha evaluatig the correspodig determiat. Let P be the -square matrix whose etries are alio, except that each etry alog the first diagoal above the mai diagoal is equal to, ad the etry i the row ad first colum also is. (P is a "permutatio matrix. ") For example, PR = ' ". The reader ca verify that ad Pi = P 3 5 = "" _ ro Lo *

3 969] A FIBONACCI MATRIX AND THE PERMANENT FUNCTION We ow defie the matrix Q(,r) to be 54 L«J 3= For example, Q(5,2) P B + Ps =. It is ot difficult to see that perq(,l) = per P =, per Q(,2) = 2, ad that perq(,) = L It has bee show [2] that () ^^^.(i-l^^i^iy The strategy used i the derivatio of () was to use techiques for the solutio of a liear differece equatio o a certai recurrece ivolvig perq(,3). There are, also, expressios available for perq(,4), p e r Q (, - ) ad p e r Q (, - 2 ). (See [3], [ 3, pp ] ad [3, pp. 3-35], respectively.) However, per Q(,r) has ot bee determied for 5 < r < - 3. The objectives of this paper are to use a "Fiboacci matrix" to derive (), ad to derive a explicit expressio for perq(,3) other tha that provided by (). (By J a "Fiboacci matrix" we mea a matrix M for which M = per ^ M p e r M _ 2. ) Let F be the matrix ff..], where f.. = if li - jl < ad f.. = L 3 3 J - r j otherwise. The, by startig with a expasio alog the first colum, we fid that F is a Fiboacci m a t r i x / Sice per F 2 = 2 ad per F 3 = 3, per F yields the ( + ) well kow that the term of the Fiboacci sequece,, 2, 3, 5, a. It is Fiboacci umber is give by *There are other Fiboacci matrices. See problem E553 i the 962 volume of the America Mathematical Mothly, for example.

4 542 A FIBONACCI MATRIX AND THE PERMANENT FUNCTION [Dec. ( + ^ ) - ( - */5) 2 -v/5 This, it follows that i (2) 9 \ per F v = J + V5 - - ^ 5 ), V 5 by It is ot quite as well kow that the Fiboacci umber is also give EH- ). where is the greatest iteger i M - (See [4, pp. 3-4] for a proof.) From this it follows that (3) * "»= ("; k ) Now let U (i, j) be the -square matrix all of whose etries are except the etry i row i ad colum j, which is. If we let R = F + U (,l), by expasio alog the first row of R we fid that

5 969] A FIBONACCI MATRIX AND THE PERMANENT FUNCTION 543 p e r R = per F + p e r f F ^ - ^(2,) + U ^ -,)]. But, by expadig alog the first colum, (4) per [ F _ - ^(2,) + U ^ ^ -,)] = per F _ 2 +. Thus, per R = per F + per F + = + per F K ^ ^ - ^ -2 If we ow let S = R +U (l,), by expasio alog the first row of S we fid that per S = per F ^ + per [ F ^ - ^(2,) + U ^ f o -,)] (5) + per [Q( -,2) - U ( - 2,) - u - ( - i. 2 ) + p ; : J ]. If we substitute from (4) ad use Z for the matrix of the third term of the right member of (5), we have per S = per F - + per F «+ + per Z = per F + + per Z. Now expad Z alog its first colum to obtai per Z = + per F 9. The ^ per S = 2 + per F + per F ~. Sice per S = perq(,3) (because S ca be obtaied from Q(,3) by a permutatio of colums), it follows that per Q(,3) = 2 + per F + per F. & By usig (2), we obtai a expressio for per Q(,3) which reduces to that give by Mie i [ ]. By usig (3), we obtai:

6 544 A FIBONACCI MATRIX AND THE PERMANENT FUNCTION Dec. 969 k= k= REFERENCES. Marvi Marcus ad Heryk Mie, " P e r m a e t s, f f The America Mathematical Mothly, Vol. 72 (965), No. 6, pp Heryk Mie,, " P e r m a e t s of (,)-Circulats, " Caadia Mathematical Bulleti, Vol. 7 (964), pp H e r b e r t J, R y s e r, Combiatorial Mathematics, MAA Carus Moograph No. 4, J. Wiley ad Sos, New York, N. N. Vorob f ev, Fiboacci N u m b e r s, Blaisdell Publishig Compay, New York, 96. [Cotiued from page 538. ] SOLUTIONS TO PROBLEMS T ^ = 5T + 2T - - 9T Q - 5T Q ' T + = 5 T " 4 T - l " 9 T T T T ^ = 5T - 7T - + 3T T ^ = 4 T ^Q - 2T ^ - 5T _ + 2T T j C = 2 T l C + 4T, - 4T, - 6T l + T * * * * *