VAROL'S PERMUTATION AND ITS GENERALIZATION. Bolian Liu South China Normal University, Guangzhou, P. R. of China {Submitted June 1994)

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1 VROL'S PERMUTTION ND ITS GENERLIZTION Bolian Liu South China Normal University, Guangzhou, P. R. of China {Submitted June 994) INTRODUCTION Let S be the set of n! permutations II = a^x...a n of Z = (,,...,«}. For II es, we write II = a x a...a n, where n(/) = a t. Definition : P : = {U e S \a l+l * a { + for all i,\<i<n), P :={UGP \a n =n}, \P \ = p ; \P \ = p ; P; = {II e S \a i+ *a,-\ for all i, </<»}, P ' = p' n ; P '={ngp ' a =«}, P ' = ^. Definition : T :=<UGP _,aj > J-i - for any i,l<i <n>, % = *; Definitions: T'=\n^pj ^ i(i + l)~... l^oj > for any /, \<i <n>, j=i T{=c/>, \T n \ = t n, \T;\ = f n. ^:=P np ', G :=T nt^, GJ = r, \G \ = g. From Computer Science, Varol first studied T n and obtained the recurrence for t n (see []). In [], R. Luan discussed the enumeration of T and G n. This paper deals with the above problems in a way that is different from [] and []. series of new formulas of enumeration for t n9 t', and g n (Theorems -9) has been derived.. ENUMERTION OF t ND t' n n Lemma : «zj, n - k P = (n-\)\y d {-\f U ^ = D n + D n _ l, (.) Kl k= where D n = «!Zy = -^- is the number of derangement of {,,..., n) (see [3]). Proof: Consider the set S' = {(,); (, 3);...;(«-, ri)}. We say that an element (J, j +) of S' is in a permutation II if IT(/) = y, H(i +) = y + for some z. Let W k be the number of permutations in S containing at least k elements of S'. The number of ways of taking k elements from S' is ("JT ). Suppose the k elements have j digits in common. Then these k elements form (k-j) continuous sequences of natural numbers, each of which is 8 [MY

2 VROL'S PERMUTTION ND ITS GENERLIZTION called a block. Thus, the number of remaining elements in Z n is (n-k + j). The number of permutations of (n-k + j) elements and (k - j) blocks is [in -k + j) + (k- j)] \ = (n-k)\. Hence, W k =("?)(n-k)\. By the principle of inclusion and exclusion (see [3]): k=q V / k= K where D n ~n\ Z = ^-jr k\ is the number of derangement of {,,..., n) (see [3], p. 59). D Lemma : n-l K = l H r w P y, where p = l. (.) Proof: It is easy to see that p n = / V i ~ Pn-\ pplying the above recurrence repeatedly, we get (.). D Let oo PW^Pn*?. (.3) Theorem : '»=I(-ir'-fl^- j= / =! (i-4) Proof: Consider the following subset P of P. P ={(a a...a I...a n ) e P j for some/, \<i <n, i(i + l) u ~.. a +a a / = -, but fori <j<n, ; 7 +) }. If i < w -, then the number of such permutations is pfi^; if i = n -, the number is p n i. Thus, n- Hence, n- ' = Pn ~ I # I = Pn ~ X ft f n /= Substituting (.) into the above formula, we have (.4). D To simplify (.4), we establish a lemma as follows. 7 = -i~pn~^- Lemma 3: If then n- t n= a n-y, h i t n-n "*, z=l 996] 9

3 VROL'S PERMUTTION ND ITS GENERLIZTION L = b, i b \ i h- K-, a «3 a b x - Proof: This follows from the expansion of the determinant along the bottom row. D Hence, we car. write (.4) as (.5) Theorem : P -p + l ft-ft+" ft-ft+ft- + «>, (.6) i +, ( -! ) " Example : ft- + l h = ft ft ft-ft+ft-ft+" Let T(x) = " =? *", 'o =. Then we have = 33. Theorem 3: Hence, Proof: From (.4), P(X)T(X) = YsPn^y T(x) = (l + x y - ^, where P(x) = f> x". oo oo o o / o o «= «= = I I > O O / O O = S(-ir'/',k=S(-ir^,l/' <^,-i=(i+xr l^)-i. w= V= / w= «= T^-^-m n= (.7) Now we shall consider t' n Lemma 4: P'n=Pn- Proof: If (a l a...a ^ ep n, then {a n a n _ x...a a ] ) epj. The above correspondence is one-toone; thus, Pj = i>;. D (.8) no [MY

4 VROL'S PERMUTTION ND ITS GENERLIZTION rguing as in the proof of Theorem, we get Theorem 4: n-\ (.9) By recurrence (.9) and Lemma 4, we have an explicit formula for t' n as follows. Theorem 5: '» = l ft ft ft (.) \Pn-\ Pn- -i Pn Let the generating function for t' be T'(x) = Z" = ^x",? = Theorem 6: r(x) = i- i P(x) ivoo/- By (.9), Z^o C = ft'" ^! Thus > P(x)T(x) = P(x) -, and we have Lemma 5: r(x) = i- P(x) as required. (,) (.) i.e.. Proof: Since T(x) = / ( + x) - / [P(x)], T'(x) = - /[P(x)]; hence, 7(x)-r(x) = - ^, Zft,-^)*" = I(-l)"r". W= «= Comparing the coefficients of x n, we have t n ~f n - (-)", i.e., (.). D ccording to (.) and (.), we have a simple expression for t n as follows: Theorem 7: t = ft ft ft +(-)". (.3) ft-i ft- ft For Example, we can count t 5 by the above formula: 996]

5 VROL'S PERMUTTION ND ITS GENERLIZTION ft ft ft ft ft + (-l) 5 = 3 - = Since the enumeration for p n is simple (we may look it up in a table of values of D n ) counting t n by (.) and (.3) is easier than the method of [].. ENUMERTION OF g n Definition 4: i?v,:= {U = (a l a...a n ) es \H contains either (z,z + l) or (z + l,z), but for V/<z, II contains neither (y, j +) nor (J +,7)}. Let i^. = r.. Lemma 6: r nj = (n-)!, zz > ; r u =. Proof: By definition, i^j is all permutations of S containing either (,) or (,). If we regard (,) as an element, then (,) and the remaining (n-) elements of Z form (n-l)\ permutations. Note that there are two permutations of {,}. Thus, r nl = {n -)!. D Lemma 7: r a = (n -)!- {n - )!. Proof: Let us count the number of permutations containing either (, 3) or (3,) but neither (,) nor (,). rguing as in Lemma 6, we know that the number of permutations containing either (, 3) or (3,) is (n -)!. We have to eliminate those permutations containing (,, 3) or (3,,), the number of which is (^-)! by an argument analogous to Lemma 6. Thus, we have proved Lemma 7. Lemma 8: r nj - r ^ - r ^ ^ - r ^ ^, where <i <n, r n =. (.) Proof: Each permutation of R 7 contains (/', / +) or (/ +, i). If (/', i +) is followed by / - or if (/' +, /) is preceded by / -, then (z +) is removed, and we subtract from every digit that is greater than z +. Thus, we get an element of i^_ t _ x. Conversely, given a permutation of i^_i 5/ _i, we add to each element greater than i and then interpose an / + between (/,/'-l) or (z -, z). This yields an element of R^ i. If (z, z +) is not followed by z - or preceded by (z +, z), we regard (z, z -f ) or (z +, z) as a single element and subtract from every digit greater than z +. This yields an element i r i u u ^ u u - u ^ ) M. Thus, Hence, r n,i ~ r n-\,i-\ r nj =\ + 7- («-l)!-ev u J = 7- (^-l)!-i^-i,y f n-l,i-l> (.) [MY

6 VROL'S PERMUTTION ND ITS GENERLIZTION i.e., r n, i - Vn,i-l ~ r n-\,i-l) ~ V n-\, i-l ^ Using Lemmas 6, 7, and 8 above, we can express r n^k in terms ofr nl : r n,i =r n ~r n _ l = (/!-l)!-(n-)!, ^ = ^ M - 3 r» - u + ^. i = (H-l)!-6(ii-)!+(/i-3)!, In general, let r %k =a k (n-l)\-a k (n-)\ + -+a kk (-i) M ( n -k)\. Obviously a kj is independent of n. It only depends on k. We can prove Lemma 9: a t, ; = a *-i, > + a k-i, j-i + %-a. j-i, < 7 < *, ( - 3 ) ivoo/- Since r >k = y = ia fcy («-./)!(-l>' + by (.), we have: r n,k =r n, k-\ ~~ r n-\,k-\ ~ V n-\,k- y=i M M = %_ u (/i-l)!+ ;W>^^ Comparing the two formulas above, we obtain relations for a nk as follows: %,l = a fc-l,l> t h u s > a k,l = a k-l,l= a k-,l=-~= a l,l a fc,/ = %-l, y + %-l,/-l + %-,,-l> < J < *> = l > %,it = %-i,*-i, t h u s > * *, ^ = a k-i,ik-i = --- = «i.i a Lemma : a kj = a k^k+l _j. (.4) Proof: We prove the lemma by induction on k. For & =,, or 3, this is straightforward. Suppose that (.3) holds for k-l. By (.), a k, j ~ a k-\ j + a -l, j-l + a -, y- ~ a k-l, k-j + a -l, -./+l + a k-, k-j ~ a k, k+l-j By (.3) and (.4), we easily obtain the expression for r nk \ r n = (n-l)\, r n = (n-l)\-3-(n-)\ + (n-3)\, r JIi3 = (/i-l)!-5-(/f-)!+5-(«-3)!-(f!-4)!, r 4 = (/i-l)!-7-(#i-)! + 3-(/i-3)!-7-(#i-4)!+(#i-5)!. 996] 3

7 VROL'S PERMUTTION ND ITS GENERLIZTION For a k j, using u k-j-l + a k-l,j-l +a k-l,j we obtain the above formulas one by one. Now, using (.), we get the table for r nk shown below. TBLE. r ^ ( * < # i ), r n^n: = r n n * ~6~ Setting f n (x) = Z = o ^kx k, and letting / = n - in (.), we have Corollary: r n,n-i = (w-)! + V u - - / U ) (.5) Lemma : 'n ^ " +, w ' w. w - l / ' (.6) Proof: Denote the set of permutations containing neither (n-l,n + l, n) nor («,«+,«-) For any a GR, inserting n + l to the left or right of«in a, we get a' ei^+. Conversely, if a' ei^+ then eliminating w + yields a el^. Hence, r w = ^C-i,«l- Now we count i + i,j- ^ * s sufficient to subtract the number of permutations containing (n -, n +, w) or («,«+, n -) in i^+ n from r w+ w. Regard (n +,«) as a single element. Then ^r n n _ x is the number of permutations containing no (n-l,n +, ri). By a similar argument, the number of permutations containing no (/?, w +, w-) is y ^ ^. Thus, i^+i,j = ^ +i, -i^ -i-y^ -i- Since r n = ^* +> J, we get (.6). Lemma : where -! =. r = (»-)(«-)! +/ _ (l)-/ (l) + r _ ) / =X("-(»-! -/ ), ;= (.7) (.8) 4 [MY

8 VROL'S PERMUTTION ND ITS GENERLIZTION Proof: Substituting (.5) into (.6), we obtain rn=l [. n u r^_ l -f n (l)-(n-l)\-r n _^^ Using (.6), we get (.7). pplying (.7) repeatedly, we have Since r = and / n (l) =, we obtain (.8). D rn = n f(n-i)(n-i)\-f n (l) + r + f (l). By (.6), it is easy to get r n from r n>k. In Table we denote r n>n =r n. Thus, by (.6), the values ofr n on the principal diagonal can be obtained as half the difference between the two adjacent elements on the secondary diagonal. Unfortunately, we cannot count r n until we complete Table. But (.7) and (.8) can do that, namely, both r n k and r are counted without r w+ k. If we set / + (x) = T," k= r +hk x k, we have Lemma 3: Proof: By (.), we have By (.6), we have / (*) = l - x [(n-l)!-r _ ^- -(l + x)/,_ (x)]. (.9) n-\ n-\ n-\ w- ^rn,k I ' k= I - Z^rn,k-\ X ^rn~l,k-l X ^rn-l,k- X^ k= I' k= k= fn( X ) ~ r n, l X = X fni X ) ~ T n, n-l^ ~ X fn-l( X ) ~ ^ fn-l( X ) + V l, W - * " ( - x)/ (x) = {n-)!x -r n _ lx " - x(l + x)f^(x). D Example : Since / 4 (x) = x + 8x +x 3, r 4 =. We get / 5 (x) = - ^ [ - 4! - - x 4 - ( l + x)( + 8x + x 3 )] l - x X [48--x -x 3-6x 4 ] l - x = 48x + 36x + 6x 3 + 6x 4, i.e., r s> j = 48, r 5 = 36, r 53 = 6, r 5>4 = 6. From (.9), we may obtain / (*) = (l-x) n- n- X (-ly-^w - o! x I z '(i+xy-^i - xf -i- n- + x " X (- l )%-i( l +^r'c - *r /_ +(-i)"^_i (i+xfi=l (.) The application of (.) is not as convenient as that of (.9), but it provides the following information: r k must be even. It coincides with the expression of r n^k, i.e., 996] 5

9 VROL'S PERMUTTION ND ITS GENERLIZTION Theorem 7: j=i Proof: By a method similar to Theorem, it is easy to show that 7= n-l «-i j=\ on ~ 'n -a j&n-j ~~ n-l? 7= where T^~ x is the number of permutations in i ^ whose right-most entry is n -. Similar to Theorem, we have Now, substituting (.3) into (.), we obtain (.) as required. D ccording to (.), we can count g by recurrence. g x = g = g 3 =, we get an explicit formula for g n. Theorem 8: n-l 7= (.) (.) (.3) Using (.) and noticing that 8n r r 5 -r 4, «>4. (.4) r _,- + (-TV 4 n-l r _ 4 r _ 5 r r n -r n _ x + + ( - )V 4 Example 3: g 6 = r 4 r 5 -r 4 r r 6 -/- 5 +r Let G(x) = ^g x", R(x) = # *», r =. n= «= - i mr \, i\-i Theorem 9: G(x) = ( + x)" - (i?(x) +)". Proof: By (.), Noticing that g x =, we have thus, G(x) = (l + x)- -(i?(x) + l)-. D n-l I^u^l(-ir^(-i)" 7= 7= G(JC) R(x) + G(x) = ( + xy l R(x) + ( + x)~ l -; 6 [MY

10 VROL'S PERMUTTION ND ITS GENERLIZTION Corollary; CKNOWLEDGMENT The author would like to thank the referee for his/her valuable suggestions and careful corrections. REFERENCES. Y. L. Varol. "On a Subset of all the Permutations of n Marks." The Computer Journal 3.4 (98): Luan Rushu. "Enumeration of Certain Sets of Permutations." Monthly Journal of Science (China) 9. (984). 3. J. Riordan. n Introduction to Combinatorial nalysis. New York: Wiley, 958. MS Classification Number: 55 > > uthor and Title Index T h e UTHOR, TITLE, KEY-WORD, ELEMENTRY PROBLEMS, and DVNCED PROBLEMS indices for t h e first 3 volumes of The Fibonacci Quarterly have been completed by Dr. Charles K. Cook. Publication of the completed indices is on a 3.5-inch, high density disk. The price for a copyrighted version of the disk will be $4. plus postage for non-subscribers, while subscribers to The Fibonacci Quarterly need only pay $. plus postage. For additional information, or to order a disk copy of the indices, write to: PROFESSOR CHRLES K. COOK DEPRTMENT OF MTHEMTICS UNIVERSITY OF SOUTH CROLIN T SUMTER LOUISE CIRCLE SUMTER, SC 95 The indices have been compiled using WORDPERFECT. Should you wish to order a copy of the indices for another wordprocessor or for a non-compatible IBM machine, please explain your situation to Dr. Cook when you place your order and he will try to accommodate you. DO NOT SEND PYMENT WITH YOUR ORDER. You will be billed for the indices and postage by Dr. Cook when he sends you the disk. star is used in the indices to indicate unsolved problems. Furthermore, Dr. Cook is working on a SUBJECT index and will also be classifying all articles by use of the MS Classification Scheme. Those who purchase the indices will be given one free update of all indices when the SUBJECT index and the MS Classification of all articles published in The Fibonacci Quarterly are completed. I 996] 7