MA THEMATICS ON A CONJECTURE OF RYSElt AND MINC BY ALBERT NIJE~HUIS*) AND HERBERT S. WILF *) (Communicatd at th mting of January 31, 1970) 1. Introduction Lt A b an n x n matrix of zros and ons, and suppos that A has r, ons in its i t h row (i= 1,2,..., n). It has bn conjcturd [2] that for th prmannt of A w hav th inquality (1).. Pr A.;; II (r,!)l/ri. Th purpos of this papr is to prov Thorm 1: Thr is a univrsal constant r=.136708... such that.. (2) PrA.;; II {(rt!) l /ri+r}. (3) Prvious work on inqualitis of th form n Pr A.;; II q;(r,) has bn don by MIN [2] who showd that q;(x) = Hx+ 1) is admissibl in (3), MIN [3] who found q;(x) = (1 +V2)-1(x -;- V2) for (3), JURKAT and RYSER [4], WILF [5], and othrs, but no prvious q;(x) has givn vn th corrct first trm of th asymptotic bhaviour of (4) x! llx ~ ~ + log x + log ~ +o(1) 2 (x --+ x:» whras our thorm 1 abov corrctly givs th first two trms. 2. Th function q;(n) Lt q; b a fixd function of th positiv intgrs, and suppos (3) holds for all matrics A of zros and ons of ordr <:n-l. Now lt A b an n x n matrix of this typ, with row sums Tl,..., Tn. If any T, = 1, thn (3) holds for A providd only that q;(i) = 1. Othrwis, suppos th rows and columns of A hav bn prmutd, if ncssary, so that th *) Rsarch supportd in part by th National Scinc Foundation.
152 ons in th first column occur in th first c rows. Expanding by minors down th first column w find Pr A = L Pr (AH) c C n < L {II lfj(rk- 1) II lfj(rk)} i=1 k=i k~c+1 k*i 1 c n = L II lfj(rk - 1) II lfj(rk)' lfj(ri- 1) k-i k-c+l Comparing this xprssion with th right sid of (3) w s that in ordr to prov (3) for A it is nough to xhibit a function lfj such that (5) c 1 c lfj(rk - 1) 1 L )II < lfj(ri- 1 k-i lfj(rk) for all positiv intgrs c, rl,..., Tc- Considr th function lfj which is rcursivly dfind by (6) l(a) (b) lfj(i)=1 lfj(n+l)=lfj(n)1f'p(n). For this lfj, th lft sid of (5) is c L 1 c II -1/'P(rk-1) i= I lfj(ri - 1) k-i = { ± I} xp {_ ~ ± I} i= I lfj(ri- 1) k-i lfj(rk -1) =x-x / <max x- x / ",;;;'0 =1. It follows that th function lfj of (6) is admissibl for th inquality (3). Th rmaindr of this papr is dvotd to a clos study of th rcurranc (6) with a viw to stablishing th rlations n log n A (7) lfj(n) = + ~ + +0(1) (n ~ (0) and (8) A-Iog~ lfj(n) < n!i/n + (all n;» 1). 3. Asymptotic bhaviour ot lfj. W rmark first that putting bn=(lfj(n))-l in (6) (b) yilds bn +1 = b n-bn = F(bn)
153 whr P(x) = x:», Th asymptotic rlation (7) thn follows from wllknown thorms about th succssiv itrats of functions F which hav th form F(x)-=x-ax 2 +... (a> 0) nar th origin (s,.g., [6]). Nonthlss, w prov (7) indpndntly of thos rsults. First, from (6b), 1 cp(n+ 1);;;,cp(n)i and so (a) Nxt, from (6b), n cp(n»-. cp(n t-l) <; cp(n )+ t 2cp(n) which, with (10) (9), yilds n cp(n) <; - + O(log n) (n-)-oo). Now writ li(x)=x {1/x-l- ~ 1 } x 2x (11) j 2, = O(x- 2 ) (;l: -)- 0). Thn, with Yn=cp(n), w hav Summing, 1 Yn+1=Yn+ - -2 +H(Yn). Yn (lla) Yk ) n-] ] ] Ylc 1 n-l 1 n-l (k- ~n+-1+i I -k + t L -k- + I H(YI) =n+-1+!logn+'i..+! l (k- YIc) + ~H(YIc)+O(l) 2 1 kyl I =n+i log n+a +0(1) which provs (7), with r 00 (k - YI) 00 (llb) A=-1+2+tt kyl +fh(yi).
154 4. Proof of (8) W claim that th diffrncs incras monotonically. In viw of (7) and (4), this will stablish (8). W hav first, (12) n+l {I r'(x+1) 1,.' } =! I'( x + 1)1/% X F(x + 1) - x2log 1 (x 1 I) dx. From [1], p. 18, q. (27), (13) F'(x+ 1) 1 l'(x+i) <: log(x+i) - 2(x +I) <: log x+ 2x + 2x(x+ 1)' From [I], p. 22, q. (9), (14) ( log l'(x -) ;;;' (x+~) log(x -; I)-x- I+O ) (0 = log V2n) I :> x log X-X -i- -log x +C--. 2 4x 2 Th quantity in bracs in (12) is thrfor { } <:.!. _ log,x +.~ - 0 + 2. X 2x 2 x 2 x 3 (x ;;;. 2). Nxt, from [1], p. 21, q. (8), log P(x -t-i) <: (X 'l ~) log(x -l-i)-x-i+o +~ whr I I 11 K = max T t - 1 - t + '2 1>0 and aftr som calculation, on finds that (15) lf( '1)11 X log x 0 (log X)2 0 log X A 2 XT x <: - + -- + - + + + - 2 8x 2x x (x>xo)
155 whr Xo, A 2 ar any pair such that A A I. 0 (log X)2 Al log X Al (log x)2 "';;''''0 { 8x 2x 4x 2;> max -, + + 2 and A I = 2K + 1. Substituting (15), (14) in (12), (16) \ I n+1 {I 1 (log X)2 (n+l)!l/n+l n!l/n,,;; f +- " 2x - 8x2 (log x)3 [ 0 AI]} +-+- 8x x x2 (O-k)logx 2x2 A3}d - +- x x2 whr Xl, A 3 ar any pair such that A 3;> max {(A2 + ~) + C(~ - 0)1 (log X)2 + A 2 + 0 log X} "';;''''1 x 8x 2 x 2 2x 2 Thn (16) givs (n t-l)!l/n+l-n!l/n.;;; -1 -I- -1 - (log n)2 - (0-- -!) (17) 2n 8n j 2 2 logn A 4 --+n 2 n 2 whr n2, A 4 ar any pair such that n ;»n2 implis [log (n ;-1)]2 0 - t log (n +- 1), A 3 ---,- 8(n+ 1)2 2 (nl 1)2 n2 <; _ (0 - t) log n,a4 (log n )2 2 ----:n;2 T 1i2-8n2. Now from (6b), and sinc w find that qj(n-t-l);>qj(n)+ t 2 2qJ(n) n log n qj(n)<; - + -2- +A5 (18) log n A 5 qj(n+ 1)-qJ(n);> - + - +---- 2n 4n 2 2n 2
Subtracting (17) from (18), 156 [Ip(n+ 1)-(n f- l )!l /n+l ] _ [Ip(n ) - n!l /n] (log n)2 2C -1 log n As -I- 2A4 ;;> 8n2 -~ ~- n 2 for n ;;>n*. By simpl stimations, on can tak, succssivly, K = 1/12, (xo,a2)c=(1,1.5), (Xl, A s)=(i, 1.9), (n2,a 4)-=(1,2), As=l, and finally n* = l 2< 5000, which provs th monotonicity of th squnc for n ;;> 5000. By actual calculation l) on obsrvs th monotonicity for n «; 5000 also. complting th proof. 5. Conclusion Sinc {Ip(n)-n!lln} t w hav from (7) and (4) th dsird inquality (8). By computation w found, for xampl q;(09,000) = 36,422.65517926.... (!H},OOO)!l /99,OOO = 36,422.5186529... W rmark that th constant A can b xhibitd as th solution of a crtain functional quation, as is charactristic of problms of functional itration. Suppos w lt Ip(n,;) dnot th solution of th rcurrnc (6) (b) with th starting valu Ip(I, ;) =;, in plac of (6) (a). On has again th formula (7), n log n A(;) rp(n,;) ~ +2'--- +0(1) (n -)- CXJ). Now, sinc thr follows (19) A (;1M ) = 1+A(;). Thus, our constant A is A(I), whr A(;) is a solution of (19). W rmark finally that th mthod will not prov th conjctur in th sns that th function Ip(X) =X!l/x dos not satisfy th rlation (5), for xampl whn rl=... =rm=2m; r m+l= =rm+n=2n. Th constant A = 1.2905502 was computd from (llb) in which th sris shown ar rapidly convrgnt. Univrsity of Pnnsylvania 1) W gratfully acknowldg th assistanc of th Univrsity of Pnnsylvania Computr Cntr in providing tho tim for numrous calculations rlatd to this papr.
157 REFERENCES 1. BATEMAN, H. t al, Highr Transcndntal Functions, vol. 1, McGraw Hill, 1953. 2. MINc, H., Uppr Bounds for Prmannts of (O,I)-matrics, Bull. Amr. Math. Soc. 69, 789-791 (1963). 3., An inquality for prmannts of (0,l j-matrics, J. Combirratorial Thory, 2, 312-6 (1967). 4. JURKAT, W. B. and H. J. RYSER, Matrix Factorizations of dtrminants and prmannts, J. Algbra 3, 1-27 (1966). 5. WILF, H. S., A mchanical counting mthod and combinatorial applications, J. Combinatorial Thory 4, 246-258 (1968). 6. BRUIJN, N. G. DE, Asymptotic mthods in analysis, North-Holland Intrscinc, 1958.