398 ENUMERATION OF PERMUTATIONS BY SEQUENCES II [Dec. REFERENCES

Transcription

1 398 ENUMERATION OF PERMUTATIONS BY SEQUENCES II [Dec. Results similar to those obtained for p (x) may be obtained for q*(x). At this stage, it is not certain just how useful a study of q^(x) and r (x) might be. REFERENCES. A. Erdelyi et al. Higher Transcendental Functions. Vol.. New York: McGraw- Hill, A. Erdelyi et at* Tables of Integral Transforms. Vol.. New York: McGraw- Hill, L. Gegenbauer. "Zur Theorie der Functionen C^ix)." Osterreichische Akadamie der Wissenschaften Mathematisch Naturwissen Schaftliche Klasse Denkscriften, 8 (88): A. F. Horadam. "Basic Properties of a Certain Generalized Sequence of Numbers." The Fibonacci Quarterly 3 (965): A. F. Horadam. "Polynomials Associated with Chebyshev Polynomials of the First Kind." The Fibonacci Quarterly 5 (977): D. V. Jaiswal. "On Polynomials Related to Tchebichef Polynomials of the Second Kind." The Fibonacci Quarterly (97): W. Magnus, F. Oberhettinger,& R. P. Soni. Formulas and Theorems for the Special Functions of Mathematical Physics. Berlin: Springer-Verlag, E. D. Rainville. Special Functions. New York: Macmillan, G. Szego. Orthogonal Polynomials. American Mathematical Society Colloquium Publications, 939, Vol. 3. ENUMERATION OF PERMUTATIONS BY S E Q U E N C E S I I L. CARLITZ Duke University, Durham, NC Andre [] discussed the enumeration of permutations by number of sequences; his results are reproduced in Netto's book [5, pp. 05-]. Let P(n 9 s) denote the number of permutations of Z n = {,,..., n} with s ascending or descending sequences. It is convenient to put (.) P(0, s) = P(l, s) = 6 0>s. Andre proved that P(n 9 s) satisfies (.) P(n +, s) = sp(n 9 s) + P(n, s - ) + (n - s + l)p(n, s - ), (n >. ). The following generating function for P(n 9 s) was obtained in []: (.3) ( - x*y»'**ltp( n + l, 8) x n- = l ^ J g / / l - * +_sin JL \ > *-** n!^rt + x\ x - cos z J However, an explicit formula for P(n, s) was not found. In the present note, we shall show how an explicit formula for P(n 9 s) can be obtained. We show first that the polynomial (.) p (x) = Pin +, x)(-x) n - s satisfies (.5) p B (ar) = - _( - x) n -i\ j^(-l) n+k A n + lyk T n. k+ (x) - +, + I k-i

2 98] ENUMERATION OF PERMUTATIONS BY SEQUENCES II 399 and (.6) p B.. l(a r) > - L _ ( l -«) B - E (-l) k - (>l B. k +A n, k+ )T n _ k (x)., Z k = 0 where the A n y k are the Eulerian numbers [3], [7, p. 0] defined by z \r A.JU. \ - x and T n (x) is the Chebychev polynomial of the first kind defined by [6, p. 30] (.8) T n (x) = c o s n<$> 9 x = cos<j>. Making use of (.5) and (.6), explicit formulas for P(n 9 s) are obtained. For the final results, see (3.7), (3.8), and (.), (.3).. In (.3) take x = -cos (j), so that /o (.) i \ \ > ^ / ( s m (b) AN-" ^ n V* > P(n -nt +. i, s ) \ ( / - c o s <b) isn~s = -= + cos (b ±\ / s i n d> j ~ - - s i n s \.. n c o s n o s = o0 ^ \ c o s > + c o s z l We have ( s m (f> - s m s \,_ J-/ J = t a n z l/'o J_ AN (z + A\ - COs(g + ([)) Y d>) cos $ + cos / + cos(s + cj)).* Hence, if we put (.) ( B l n!! s l n * = E / ^cos ^ v / n Y n! \cos (j) + cos zf n = 0 it is clear that /o ON ^ / I N d n - COS (J) (.3) f (cos <J>) - r.n + COS ( aq) To evaluate this derivative, write Then and - cos $ I e^% - l\, + cos y i 0 ^ + w e*i + x (e*t + )S d - cos (t) _ e** i e ^ i 3i _,_ i d$ X + c o s * (e** + l ) (e<* + l ) 3 e** + ( e ^ + l ) ( e ^ + l ) 3 d - cos (j) = i 7i i _ 6i A» + C O S * e** + ( e ^ + l ) (e** + l ) 3 (e** + ) * ' The g e n e r a l formula i s n <T~ cos (j). n - y ( _ r ) f e - i ( ^ l ) 5 (, fe) (. ) (-l)"ii* - c o s 9. ^ - ^ ( - l ) " - ^ - i ;! ^ n > K), ( n > ), where («, k) is the Stirling number of the second kind [7, Ch. ] : Es(«. *); = rr(^ - D*. n = k

3 00 ENUMERATION OF PERMUTATIONS BY SEQUENCES-! I [Dec. The proof of (.) by induction is simple. The derivative of the right-hand side is equal to fc-i (e** + l ) k + fe-i [(«* + l) k (e*t + l ) k + Since fesfa, &) + (n, /c- ) = S(n- l 9 We may rewrite (.) in the following form: " ^ w " E < " ) k (fe "" )! {(*S(n, fc)+- (n, fc- )}.. *-i ( e ^ + l ) k k), this evidently completes the induction. ( ^ _ +cos <j) ( e # i + )» k V In the next place, we require the identity (.6) ^ ( - l ) * - ^ - l)!s(n, k)(x+l)"- k - (-)- k_^. lik a: k. <n >. ), fc- fc-0 where A n _ >k i s the Eulerian number defined by (.7). To prove (.6), take Eff (-D"" ^ ~ DiS(n, fc)(* + I)"" "" ~ " * fc - fc^l n - k - S^fc." ( * + i>"*(** ( * + ) - n* fc.-i = l o g V + ^ + i ] = l o g * + i ',z n (x + l ) n >\ri>9 A.. Differentiating with respect to s 9 we get On the other hand, by <-i>"fir (.7), <-i>**...*** Hence, l + x (~l)*- l (fc - l)!s(n, *)<*+ )"-* = E ^ ) " " " " ^ -!, ^ ' fc- k By (.5) and (*6) we have, on replacing n by n +, I d - COS 0 ( - ) " V *, inn-fc + l, k0i ( U * + COS * - 0, +.» - ^ *» +!. * ' '

4 98] ENUMERATION OF PERMUTATIONS BY SEQUENCES II 0 since A n+ 0 = 0. Moreover, since [3] (3-) A n+<k = A n+>n _ k+ (l<k<n+l), we have ^» i + cos - (" > S-n A n + ^ (eh + i)n+. n+ kn-fc+)*i, n -i(n-fc+)*i -i<-*)" <-D k + X + l.^, / ^ * + Therefore, in view of (.3), we get n + l <n-fc+)<k,. - \(n - k + )<K (3.) fn(cos ) = (-i)" (-l)^x + >, f, ( w + &- ( COS y(j)j It is convenient to consider n even and n odd separately, so that n+ (3.3) / (cos W - -\- n (-l)»***m ll + i f c -i!lzjl±i fc-i (cos <> j and, n s i n ~(n - k + )(() o i i n - j., - " >. k ',. ^ fc-i (cos cp) By (.3), (.), and (.), P B (COS >) = } t cos t S±nn< ) ^ ( c o s +> (3.5),, = " cos n+ ±<j> sin"" j< > / (cos <J>). In particular p n (cos $) = «cos n + c > sin "" <}> f (cos <f>), so that, by (3.3), n + l (3.6) p n (cos <p) = - ^ - ( - cos (p)"" (- )" + f c + l c 0 B +i, 8 k < n " fc + 3 k = l Using (3.) and (.8), (3.6) gives (3-7) p n M - - ^ ( - * ) * Zc- This proves (.5). Next, replacing n by n - in (3.5), we get L^(~~ l^n A n + l t k T n-k + l ^ + A n + l,n + l p ^ C c o s ) = ^cos n + } * sin *" 3 < > / ^ ^ ( c o s (f)) = 8in n - 3 U E (" ) n + ^ n 5 k S i n \^ln ~ k+ *>< fc-l (continued)

5 0 ENUMERATION OF PERMUTATIONS BY SEQUENCES II [ D e c. = sin "- ^^(-D n + k A n k {cos(n --fc)< > - cos(n - fc + l)(f>} n = i_ ( _ c o s M " " ^! ) ' " * ^, + n. fc + >cos(n-fc)< > = ^-( - cos c())? Finally, therefore, by (.8), fc = i n - l E t - l ) n + k W Bi. fc +.n. fc + i)co8(n - m +A nn \. fc = o J (3-8) P ^(x) ~ d - ^ " ^ [ E C - D - ^ a ^,, + ^ W + V, }.. We recall that j - - ' * " - I (-)' (" " f " > W " «, < > ). ' 0<j<_n <J \ J / Ihus (3.7) becomes ~ E ( - ) - E ^ +,, E (-i)( n "^J' + ) (n - k - j \ l / n - \ - fc - j + i S. / n - V x»-s V J - l /IV * / "" n t l ' " + ^ - \n - s) { x) Comparison with (.) gives (.) P(n +. P) - - ^ X X ^,, E (-D J '(r " " J ' + ') ^ & = ; s = n + /c + j ' - t - l l > ^ s=n+ + / n - * - A l / " " l\ n-fe-j + l + _ W «~ \ t7 / Similarly, it follows from (3.8) that w-i (.3) P(n, a) = - L - M + ) E (-)' fc = 0 ^ n, «+ l s = n + fc + j-t-l {( n -y j ) + ( n -y-i-% n -*y- _J_( n - \ k-j

6 98] ENUMERATION OF PERMUTATIONS BY SEQUENCES II For numerical checks of the above results, it is probably easier to use (3.7) and (3.8) rather than the explicit formulas (.) and (.3). It is convenient to recall the following tables for P(n 9 s) and A n k, respectively: TABLE \v S n \ TABLE We f i r s t take (3.7) with n =. Then P l tte) - ( - x){^5jl T (x) - ^^(3?) + ^ 5, 3 } = ( - ^r){(^ - ) + 5# + 66} = x 3-8tf + 58; - 3.

7 0 ENUMERATION OF PERMUTATIONS BY SEQUENCES II [Dec. Taking n = 3 in (3.7), we get p s (x) - ( - x) {-A 7A T 3 (x) + A 7a T (x) - U^ T^x)-+ A^} = - ( - a;) {-(a; 3-3x) + 0(a; - ) - 9a; + 6} = ( - a;) (5-88a; + 0a; - a? 3 ) = 5-68a; + 85a; - 836a; 3 + a; * - a; 5. Next, taking n = in (3.8), we get fc = 0 = A ha T (x). - W lfjl + ^ )T (a;) + ^> = (a; - ) - a; + = a; - a? + 0. Similarly, taking n = 3 in (3.8), we get p E ( - D 3 + k 5 (x) = y U - a;) W 6tfc +A 6>k + )T s _ k (x) +A B>3 \ [k = o J = ( - a;){-a 6jl T 3 (x) + G 6sl + A 6> )T (x) - (A sa + A 6t3 )T ± (x) + A 6t3 ] = ( - a;){-(a; 3-3a;) + 58(a; - ) - 359a; + 30} = x h - 60a; 3-36a; - 300a; +. Another partial check is furnished by taking x = - in (3.7) and (3.8). Since T n (-l) = cos rm = (-l) n, it is easily verified that (3.7) and (3.8) reduce to and n Pin (- ) = E W B+. k + A n+lw n + ) = ^ n + _ fc = (n + )! fc- fc- n - n P B - < - + > - E ^. k +, ^ + l ) n,k = Y, A n, k = ( ">! > fc = 0 fc = l respectively. On the other hand, for x =, it is evident from (3.7) and (3.8) that (5.) p () = 0 (n > ). L n Moreover, since T n (l) =, it follows from (3.7) and (3.) that p ( w + ) ' m = r-n n - l ( n ~ )! \i "T(-i) n+k+ A +A [ *-i J n+l By (.7), we have fc-i

8 98] ENUMERATION OF PERMUTATIONS BY SEQUENCES II»-i fc-i e z + n=o in the notation of Norlund [5, p. 7]. Hence <*.> pr i, < i >- i Fi il w For example p^'(l) = = 36; since C 7 = 7, this is in agreement with (5.). As for p _ (^r), it follows from (3.8) that so that p : > = (-i>"-,i5^{i:(-i)" + *w I. k + ^,, + ) + **,.,.} (n - " ^ [ fc- fc«l = (w - ) 0 n - Z ^ ^ ) ^ n, fc» fc-l (5.3) PzVA'd) = («> ). p<-_i) () = (. D - ^ i l l f "f (-i)-* k w li>k + *,,,. k+> p I fc = o By (. 8 ), mt, N n s i n neb,. * W ) - s i. n, y (a: = cos <(,), which gives T n '(l) = n. Thus fe After some m a n i p u l a t i o n, we get z = o (5.) P» " " ( ) = ^ n - ^ ' E ( - D ' : " ( " -?C + l ), k fc- n (n - ) n 9 n - fc- Making use of (.7), it can be proved that (5.5) ( - x)a^(x) = A n + (x) - (n + l)aca n (ar), where Hence fe = ^n(-l) - ^ n + i(-l) + (w + l)a n (-i) - C n +, where C n+ has the same meaning as above. Thus (5.) reduces to

9 06 HOW TO FIND THE "GOLDEN NUMBER" WITHOUT REALLY TRYING [Dec. (5.6) p ( " _) (i) - ( w ~ )! e,,. For example, in agreement with (5.6). (Please turn to page 65.) p'^cl) = = 36, REFERENCES. D. Andre. "Etude sur les maxima, minima et sequences des permutations." Annates soientifiques de l f Eoole Normale Superieure (3) (89):-3.. L. Carlitz. "Enumeration of Permutations by Sequences." The Fibonacci Quarterly 6 (978):59-68, 3. L. Carlitz. "Eulerian Numbers and Polynomials." Math, Mag, 3 (959): E. Netto. Lehrbuch der Combinatorik, Leipzig: Teubner, N. E. Norlund. Vorlesungen uber Differenzenrechnung. Berlin: Springer Verlag, E..D-. Rainville. Special Functions, New York: Macmillan, J. Riordan. An Introduction to Combinatorial Analysis, New York: Wiley, 958. HOW TO FIND THE "GOLDEN NUMBER" WITHOUT REALLY TRYING ROGER FISCHLER Carleton University, Ottawa, Canada K55B6 ".... I WAJbh,.. to potnt out that the, aae ol thu qold<in A&ction... hcu> appaxtntly bu/a>t out -into a &uddm and d&voatattng dluzaaz whtch hah &hou)n no AtgnA o& stopping... f? [, p. 5] Most of the papers involving claims concerning the "golden number" deal with distinct items such as paintings, basing their assertions on measurements of these individual objects. As an example, we may cite the article by Hedian [3]. However measurements, no matter how accurate, cannot be used to reconstruct the original system of proportions used to design an object, for many systems may give rise to approximately the same set of numbers; see [6, 7] for an example of this. The only valid way of determining the system of proportions used by an artist is by means of documentation. A detailed investigation of three cases [8, 9, 0, ] for which it had been claimed in the literature that the artist in question had used the "golden number" showed that these assertions were without any foundation whatsoever. There is, however, another class of papers that seeks to convince the reader via statistical data applied to a whole class of related objects. The earliest examples of these are Zeising's morphological works, e.g., [7]- More recently we have Duckworth's book [5] on Vergil's Aeneid and a series of papers by Benjafield and his coauthors involving such things as interpersonal relationships (see e.g. [], which gives a partial listing of some of these papers). Mathematically we may approach the question in the following way. Suppose we have a certain length which is split into two parts, the larger being M and the smaller m. If the length is divided according to the golden section, then it does not matter which of the quantities, m/m or M/(M + m), we use, for they are equal. But now suppose we have a collection of lengths and we are trying to determine statistically if the data are consistent with a partition according to the golden