Download pdf - Permutations with Prescribed Pattern. II. Applications

Math. Xachr. 83,101-126 (1978)

Permutations with Prescribed Pattern

11. Applications

By L. CARLITZ of Durham (U.S.A.)

(Eingegangen am 5.1.1976)

1. Introduction and summary. LetZM = { 1,2, . . . , n} and let 1~ = (ai, a2, . . . , a,) denote an arbitrary permutation of 2,. Let k, , k,, . . . , k, be positive integers such that

(1.1) k l + k 2 + . . .+k,=n.

Thepermutation ~d will be said to have the pattern [k,, k’, . . . , k,] if the following conditions are satisfied :

(1 4 al-=al<. * e a k , ; a k i + l < a k 1 + 2 < ’ * < a k l + k 2 * * ;

Bk~+...+k,-~+l“ “ < a ? 8

and

(1.3) akl’akl+19 a k l + k 2 w a k i + k ? + l , ‘ * .

This is best, visualized graphically. For example the graph

represents the pattern [a, 1, 4, 1, 1, 21; the graph

with pattern [2, 2 , 2 , 2 , 2 , 11 represents an up-down permutation, while the graph

with pattern [I , 2, 2, 2, 2, 21 represents a down-up permutation. Let A(k i , k2, . . . , km) = A,(kl, k2, . . . , k,) denote the number of permutations

of 8, with pattern [k,, k2, . . . , k,], where the kj satisfy (1.1). In [2] it is shown that the enumemnt A ( k , , k:, . . . , k,) can be evaluated in the following way.

Supported in part by NSF grant 67 - 37924X.

102 Carlitz, Permutations with Prescribed Pattern

For brevity put

(k i , k2, . (k l+ka+. - - + k m ) ! - 9 k,) = k l ! kz! . . . k,! *

Then m

r = i A ( k i , kq, . . , k,)= C 8, 9 (1.4)

where

(1.5) ST = 2 ( s i , ~ 2 , * * * 9 8,)

and

(1.6) s , = k l + . .+kj i , sz=kji+i+- - *+kji+jz 3

. - 3 S r = k j i s ...+ j , - i+ i+* . .+ kii+ ...+ i, and the summation in the right member of (1.5) is over all ji, . . . , j, such that

(1.7) j,+j,+. . .+j,=rn, j I > O , j2>0 , . , . , j , > O .

This result theoretically enables one to compute A(ki, k2, . . . , kr) for arbitrary ki. However i t is rather complicated and so not really satisfactory. Thus it seems of interest to see what can be done in special case?,. A number of examples are worked out in [Z]. In the first place it is shown that if

f(n, m)= c A(k1, * - - f km) , E l + ...+ km=n

kp- 0

(1.8)

then f (n, m) is equal to the Eulerian number An,, which enumerates tho number of nEZ, with m rises.

(1.9)

Next put

g(n, m)= A(kL, - . . , k,) . k1+ ...+ k,=n

R p l

It was proved that

where a, /3 are the roots of x%--z+y=O. The EuLERian number An,, satisfies [4]

ex - e” - x‘yS r.s=O (r+si- i ) ! xeu-yex’

m c A ( ? , 8 )

where A (‘9 8) = A , +s+i,s+i =A, +s+i,r + I = A (8, ) *

This suggests defining the array of numbers A(?, s) by means of

Carlite, Permutations with Prescribed Pattern 103

It t,hen follows from (1.10) that ffl

g (n,m)= E ( - I ) ~ - ~ - - -

g(2m, m) = C ( - I ) " - ~ A(% -8, s) .

A(n-s, 8) (2rnK.n) s=O

ffl

e = o

(1.11)

Finally it was proved in [2] that if A,(mk)=A,(k,k, . . . , k ) ,

then x d 1 (1.12) i A k ( m k ) - -

f f l = O (mk)! E',(z) ' where

Moreover if

A, (mk+t)=A,+,(k, k , . . . , k , t ) ( t z l ) , then

( t Z 1 ) ' F,,t (4 - x7nk +t

(1.13) A , ( m k f t ) -- - =--- f f l =o (mk + t ) ! FJT)

For another proof of (1.12) and ( 1.13) see [ 11.

types. In the present paper we give some additional applications. These are of two

I. Let t be a fixed integer z2 and put

ft(n, m ) = C A(k,, h, . . - hffl) , k l +... +t, =n

kd Zt

so that ft(n, m) is the number of permutations of 2, with m inclines and the number of nodes in each incline zt; also put

- xn I?,(% y) = 1 + c -- 2 ft(n, m ) y" *

,=t n! o c t m s n

We show that

where d(x, y), D(z, y) are determinants of order t defined by

A(z,y)=lai- ' l ( i , j = ~ , z , . . . , t ) ,

104 Carlitz, Permut$tions with Prescribed Pattern

D(x, Y) =

e 9 x . . . UI uz . . . ut a: u; ...lg . . . . . . . . . . . . &' ut- l ut - 1

t 1 '' . . . and ul, uz, . . . , ut are the ro0t.s of

z L z t - i + y = o .

For t = 2 , it is easily verified that (1.14) reduces to (1.10).

[ k j , k2, . . . , k,], where Also if fJn,m) denotes the number of permutations of 2, with pattern

k i s t , . . . , k m - l S t , kmSS ( t ~ 2 , S Z ~ ) ,

we evaluate the generating function

m

The results are contained in Theorems 2 and 3 below. 11. P u t

9 t h m)= z k2, * * 9 km) 7

kl+ ...+ k , =n k i ~ 0 ( 1 ~ 1 o d t ) , k i > 0

so that ga(n, m ) denotes the number of permutations of 2, with m inclines and tho number of nodes on each incline is a multiple of t ; also put

We shall show that

( t z 2 ) , I-Y (1.15) Gt(z , y) 1 -!I%@ (1 -Y)I")

where

W e also evaluate the generating function - ,nt-j n

where g l t - j (nt - j , m ) denotes the number of permutations of Zn2-+ with pattern [k,, k2, . . . , k,], where

ki=O (modt) ( l ~ i - = m ) ; k , r -j ( m o d t ) .

See Theorem 5 .

Carlitz, Perniutations with Prescribed Pattern 105

Xote that for t = l , the generating function on the right of (1.15) reduces to

where, as above, the An,m are the EuLERian numbers.

in $3 6, 7 below. Let

We show that f(n, m ) , g(n, m ) satisfy the mixed recurrences

The case t=2 of I1 is of special interest and is examined in greater detail

f(n, m) = g h m), g(n, m ) =g2,J2n-- 1, m) .

f(n, m)'=(n-m+1)g(n,m- l )+mg(n,rn) 1 g ( n f 1, m)= (n-m+ 2) f(n, - 2) + (n+ 1) f(n, m - 1) +mf(n, m ) , by means of which the enumerants can be computed. We show also that

m

2 = O m ! g(n, n-m)= c (-l)Q&) A ( 2 n + m - j - l ) ,

X" - where

2 A(?%) -=see z+tan x n=O a !

and Pm,?(n), Q,n,j(n) are polynomials in n of degree j. Finally (Theorem 8) we obtain explicit formulas for f(n, m ) and g(n, m).

2. kj z t. As above let t z 2 and put

(2.1) ft(n, m)= 2 kZ, - . , km) where the summation is over all ki satisfying (2.2) k l + k 2 + . . . + k m = n ; k i z t ( i = l , 2 , . . . , m ) . We also define

(2.3) - Xn

pt't(z, Y) = 1 + c 7 c f t h m ) I" n=t ~ < t m s n

In order to evaluate Ft (2, y) we apply (1.4) and (1.5). Since the number of solutions k l , k,, . . . , kj of

is equal to kl+k?+. . .+k j=s , k i z t , . . . , k i s t

s - ( t - 1 ) j - 1 ( j - 1 i t follows that

m

106

where

Carlitz, Permutations with Prescribed Pattern

j-1 The double sum

where

It follows from (2.8) that

Now let ui, u2, . . , , ut denote the roots of

(2.7) zt-zI-1 +y=O, so that

1 -z+y2=(1 -u& (1 -u22) . . . (1 -utz) . If we put

where the A, are independent of z , then

Alternatively ~~ a;- I

(2.10) A.= 3 (q--J * * * ( U j - a j - { ) (cxi-uai+l). . . bj-4 -

By (2.6) and (2.8),


80 that

(2.14) d(xi, ... , x t ) =

t

j = 1 (2.11) @:'(y)= C Aja!.

Substituting from (2.11) in (2.5) we get

1 1 . . . 1 ai a2 ... at . . . . . . . . . . . . ort-2 ut-2 '2 p . . . 1 I

d (x,, . . . . xt) = c,zi + Cgx, + ... + c tx t . where cj denotes the cofactor of xi in r3(xt, . . . . xt). Then by (2.10) we have

(2.15) A j = a : - ' C j d t ' , where A, is defined by (2.13).

It follows from (2.11), (2.13) and (2.15) that

Also

and

where

(2.16)

107

t 1 ( l s k e t ) .

j = 1 j = i

. . . . . . . . . . . ... at-' I

3 08 Carlitz, Permutations with Prescribed Pattern

Hence (2.12) becomes

Substituting from (2.17) in (2.4)' we get

= r=O i ( 1 """)Y')L-&. At

To justify the last step note that A, =D,(O, y) and indeed

This completes the proof of the following

Theorem 1. The generating function

i s evaluated by

where D,(x, y) is defined by (2.16) and A,=Dt(O, y). Thus for example, for t = 2 , a, = u , a2=/3, (2.18) reduces to

where a , f l are the roots of 22-2 +y = O . For t = 3 , ai=a, uz=B, u3=y, we get

Carlitz. Permutations with Prescribed Pattern

1 1 1 Xi= z y 2

x2 y' 22

109

1 1 1 1 1 1 . N z = ex ey ez , N 3 = x y x

x2 y' 22 ex er er

The result for t = 3 suggests that the generat,ing functions

D = x ez ey e'

z2 y2 2 2 y z

Consider the effect of removing the maximal element from 3t. If this element is not on the extreme right, n breaks into two pieces, of which the one on the left has pattern [kl, . . . , k i - l , ki- 11 and the one on the right has pattern [kj+i, . . . , km], for some j, 1 z j -=m. If however the maximal element of 3t is on the extreme right, the resulting permutation has pattern [k,, km-i, k,- 11.

Now let f,,,-,(n, m ) denote the number of permutations of 2, with pattern [k,, k?, . . . km], where

Also define (3.1) k i s t , . . . , k q n - l Z t , k , Z t - l .

f , ( O , O ) = l , f& m)=O (m=-O) - We then have the following recurrence:


Hence, if we put


Q t - , ( x , y ) = (3.7)

we get

(3.4)

1 1 . . . 1 a1 u2 . . . ut

1 u, ... . . . . . . . . . . . . . at-a t - 2 -2

t

-. . ea2z

Substituting in (3.4), we get

It is easily verified, using (3.5), that

Continuing this process, we remove the largest element from a permutation of Zn with pattern satisfying (3.1) and get

+ f t , & 2 ( % - L m ) (t=-.S) 9

where ft,Jn, m) denotes the number of permutations of 2, with pattern [k , , 4,. . , , km] such that

(3.10) k i s t , . . . , k , - , ~ t , k m S s (1 5 s - d ) . By (3.8)’

w,t - 1(x, Y) (3.11) - { q - A x , Y ) P + q t - Z ( X , Y) ($=-a ’ ax

where


(3.17) ot,t-j(~, y)=

111

. . . . . . . . . . . . . , . . . a:--? a;-2 ... at t - 2

a j - i ea12: j - Je=2Z . . . pests i a2

it follows from (3.5) and (3.11) that

Then, by (3.14),

Since (3.15) holds for j = 2 and j = 3, it follows that

(3.16)

It follows from (3.6) and (3.7) that

where 1 1 ... 1 I "i a2 ... at

(1 Sj4) .

Thus (3.16) becomes

This completes the proof of Theorem 2 . Let f,,,(n, m ) denote the number of permutations of ZN with pattern

[ki, k2, . . . , k,], where

Put ki z f, . . . , kwtk.,,_, ~ t , k, z s .

Then, for 1 Ss-= t ,

where D,(z, y), Dt,Jx, y) are defined by (2.16) and (3.17).

112 Carlita, Permutations with Prescribed Pattern

1 1 ... 1 GI1 GI2 . . . Glt A , = . . . . . . . . . . . (.J-l or#--l &-i

t ’ ., . . . t

t t

(3.20) A,=C C j , D,(x, y ) = c Cieix j=1. j = j

and, by (3.17),

(3.21) Dt, t -e(z , y)=-- Cjc(eiXi” (1 s s - = t ) . Thus, in place of (3.19), we may write

l t

Y ,=1

(3.22) F J x , ~ ) = j = ~ (1 s s c t ) .

To remove the restriction s< t , we note first that (3.14) is in fact valid for all 8 ~ 1 . Then, by (3.5) and (3.14) with s = t - 1 ,

2-Q

It follows easily that

as might have been anticipated. Generally, by (3.14) and (3.5),

a (3.24) & (DtPt,8+J=DtFt,8 ( sg l ) ’

We assume that

where 8 - t ,r

I n view of (3.23), (3.25) holds for s=t .

C'arlitz, Permutations wit11 Prescribed Pattern 113

By (3.24) and (3.26), t

Q+

DtD; , ,+ j=C Cj {Pm+l(jr)-e

ft,,+l(n,.m)=O (n-=r+l) 9

} x:-'-' t K , ( y ) , j=i

where K J y ) is independent of x. Since

it follows that K,(y)=O. Therefore (3.25) holds for all s ~ t .

Theorem 3. With the notation of Theorem 2, we kace. for all s g t

This proves the following theorein complementary to Theorem 2.

whew Cj i s the cofactor of the element in the first row and j-th column of A , and (qx) is defined by (3.26).

4. kj=O (mod t ) . Let t z - 2 and put

(4-1) gt(n. ~ L ) = C kzt . . , kN2) t

where the summation is over all positive kl such that

(4.2) k l + k z + . . .+km=n, k j=O(modt) ( j = 1 , 2 , . . . , m ) .

Thus gJn, m) is equal to the number of permutations of 2, with m inclines and the number of nodes on each incline is a multiple of t . Put

Then, by (4.1), kit +... +kmt

X - - A(kf f , . . . , k,t) - -

(kit+. . . t k , t ) ! ' G,(IL., y) = 1 + c y"

m = l Ep..,km=l

We now apply (1.4) and (1.5). Since the number of solutions of

k , + . . .+Tcj=s, k l ~ 1 , . . . , k j Z 1

is equal to

(;I:) '

it follows that M sit +... +s,t

where

8 Math. Bachr. Bd. 83


r = I m=O

We have

Hence (4.3) becomes

Put

Then

so that

We may state

Theorem 4. The generating function - x"t -

G&, ~ ) = 1 + ,.JJ __ C g t ( m t , m) 9" *=i (w!m=i ( t ~ 2 ) ,

is evaluated by (4.5) and (4.4). The result is in fact valid for t z 1.

Cnrlitz, Permutations with Prescribed Pattern 115

5. kj=O (mod t)-continuation. Let n denot,e a permutation with psttern [ k , , k2, . . . , kml, where kj=O (mod t ) , j = 1 ,2 , . . . , m, and consider the effect of removing the largest element of n.

lt.31.

~

If this element is not on Dhe extreme right, the given permutation breaks into two pieces, of which the one on the left has j t - I nodes and the one on t,he right has (m -j) t nodes, for some j, 1 sj sn.

Let yt,t-I (nt- 1, m) denote the number of permutations of ZM-l with m inclines, in which the number of nodes in each incline, except the last, is divisible by t , while t.he last contains rt - 1 , for some r z 1. Also define

g1(0, 0)=1, gt(0, m)=O (m=-O) *

Then we have the recurrence


Hence, if we put

we get


Continuing this process, we remove the largest element from an admissible permutation of Zn, - and get,

1 , nt-k)

+g t , ,_ ,W-% m) (t=-2) , where gt, t Jnt - r , m), 1 57- -=t, denotes the number of permutations of Z,, --r,

with m inclines, in which the number of nodes in each incline is a multiple of t , while the last contains kt - T nodes.


where of course

It is convenient to put

Then (5.7) gives

60 that

Generally we have

Crtl.litz. Permutations with Prescribed Pattern 117

This is equivalent to

The left hand side of (5.13) is equal to

p"(z) f ' (z) f ( t -1 ) f'(z"'t-'' (4 - Y __ (1 -Y) vt(x (1 - Y P t ) f(4 f(.) f b ) f ( x ) f ( 4 1 -YV& (1 - 9 ) Y

+- -

1-Y f (4

--(1-y)+-. - 1-Y - - ( I - y ) + ___ 1 - y'Pt(x ( 1 - y)l't)

Thus we have verified (5.13). To sum up the results of this section we state

Theorem 6. The generating function - ,,at - j n

satisfies (5.1 1 ) for 1 sj-=t.

6. The case t = 2. We shall now examine the case t = 2 in greater detail. It will be convenient to put

F=P(x , y)=G,(x,y) G =G(x , 9 ) =G2,1(z, y) . (6.1) {

It follows that

(6.2) , -- y) 1 -y sinh (XI=) I G(z, y) = I 1 - y cosh (XI=)


This is equivalent to

To facilitate the computation of partial derivatives we put

(6.7)

- y (1 -y) sinh x F, = -~ = P G ,

G, = + - 1 F = - +

(1 -y Gosh x)2

1 - _. =-1+--F+@,

- y cosh x y2 sinh2 x I -y cash x (1 -y cash x)Z 1 - Y

(1 -y) cash x - P + F * - - 1 -y cosh x sinh x y sinh x cosh x FG

I - -ycoshx+ (l-ycoshx)2 y ( 1 - y ) ’

y sinh x

(I -y cosh x)’ y ( I -y) ’ II

- 0, = - -~

Moreover, since y cosh x

we have

This gives

1 -y - 2P + (1 + y) P2= (1 - y) G2 ,

It follows from these relations that -

(6.9) P z = Y (1 -9) q/ and

(6.10) ( l - y ) Q z - y ( I - Y ~ ) F g = y F . In view of (6.7), (6.9) is equivalent to

(6.11) f m = ~ (1 4 &(Y)

Carlits, Permutations with Prescribed Pattern 119

while (6.10) is equivalent to

16-12)

Hence, by ( 6 4 , (6.11) and (6.12) become

(6.13)

and

(6.14)

respectively.

(1 -Y) gfls.1 -Y (1 --Y*)f;(Y)=Yf?l(Y) -

f n ( Y ) = ~ Y g , ( Y ) +Y (1 -9) 92Y)

S,+l(Y) = ( ( n + 1) Y + W 2 ) f ,(Y) +Y ( 1 -Y2) f 2 Y ) 9

Comparing coefficients of powers of y in (6.13) and (6.14), we get

Theorem 6. The enumerants f (n, k ) , g(n, k ) satisfy the mixed recurrences

(6.15) f ( n , k ) = ( n - k + l ) g ( n , k - l ) + k g ( n , k), (6.16) g ( n + l , k ) = ( n - k + 2 ) f ( n , k - 2 ) + ( n f l ) f ( n , k - l ) + k f ( n , E).

It follows from (6.15) and (6.16) that

fl

It. is evident from the definitions that

(6.19) f(n, n)=A(2n), g(n, % ) = A (212-1) ,

where A ( n ) denote8 the number of up-down perrnutat.ions of 2%. This is also implied by (6.2), as we shall now show.


1

Replace y by y-4 and x by q T i n P(x, y) . Then

For y=O this reduces to

A

(6.21) see x= 1 + f(n, n)- n=l (2n) !

in agreement with the first of (6.19).

Making the same replacements in Q(x, y), we get

For y=O this reduces to ca X?n - I

(6.23) tan x= g(n, n) - , n = l ( 2 n - l ) !

in agreement with the second of (6.19).

rent,iate (6.20) with respect to y and then put y = 0, we get

(6.24) sec~x-secx:--xtanxsecx= Cf(n, f b - 1 ) -. This is equivalent to

The relat,ion

(6.25)

is the case k=1z+l of (6.16). For k=n, (6.15) becomes

Additional results of this kind can be obtained. For example, if we diffe-

- 1 2 n = l (2n) !

f (n, n - 1 ) = A (2% + 1 ) - (n + 3 ) A ( 2 1 2 ) = g (n + 1, 12 + 1 ) - (n + 1) f (n, 7%) .

f (n , n- l ) = g ( n + 1,n + 1) - (n+ 1) f (n , n)

(6.26) s(n, n - 1) =fb, 12) - ng(n, n) , while (6.16) gives

(6.27) g(n+ I , n) = 2f(n, - 2) + (n+ 1 ) f ( ~ , 12- 1) +nf(n, n) . Hence, by (6.25), (6.26) and (6.27), we get

(6.28) 2f(n, 9%-2) =f(n+ 1, n+ 1 ) - 2 (n+ I ) ~ ( T L + 1, n+ 1) + (n2+n+ 1) f ( m , 12.).

In terms of A(n) this is (6.29) 2f(n, n-2) = A (212+2) - 2 (n + 1 ) A (2n+ 1) + ( n ' + ~ + 1) A(2n) .

For k = n - I , (6.15) reduces to 2g(n,n-%)=f(n ,n- - l ) - (n- - l )g (n,n-I)

=g(n+ 1, n+ 1) - (n+ 1) f (n , 12) - (n- 1) f (n , 12) -ng(n,n) ,

Carlitz, Permuta.tions with Prescribed Pattern 121

that is,

(6.30) 2g(?%, ?%-2)=g(?2+1, %+1)-2Rf(n, n)+n (n-I)g(n, R )

= A (2n+1)-2nA(2n)+n ( n - 1 ) A ( 3 n - 1 ) .

Similarly we find tha t

(6.31)

md (6.32) 6g(n, 12-3)=A (2n+2)-3nA (2n+l )

Gf(n, n - 3 ) = A ( 2 n + 3 ) - 3 ( ? % f l ) A ( 2 n f 2 )

+ (3n2+ 3% +4) A (2rt + 1 ) - (Iza+2n + 3) A(%)

+(3n2-3n+l )A(2n) -n (n-1) (n-2)A (212-1).

These results suggest that

(0 .33) k ! f ( u , n - k ) = C ( - I ) ~ P ~ , ~ ( ~ ) A (2n-t-k-j) ( O s k < n )

and

(6.34) k ! g ( n , n - k ) = z (-l)'f&&€)A (2n+k-j-l) (Or!%-=%) ,

where Y,,?(n), &k,j(n) are polynomials in n of degreej.

A!

1 =o

k

j = O

In order to prove (6.33) and (6.34) we require the following

Lemma. The enumerant A(n) satisfies no reczcrrence, of order independeqat of n,

Proof. See [3]. Izut

with coefficients that are polynomials in n.

c., $2" - $2" + 1 - (6.35) secr x = s;, ~ - - sec' x tan x= C S!,n+l I_--

N = O (272) ! ' n=O (2n+ I ) !

so that P

(6.36) f (n, n - k ) = 2 (-1)i j = O


I \ j l 0 1 2 3 k\

1 1

2 4 1

3 16 20 1

4 720 736 56 1

_ _ _ _

Similarly, using (6.22), we get

On the other hand, by differentiation of - X 2 n ca X2n f 1

secx= C ~ ( 2 n ) t a n x = C A (2n+i ) - - - ( 2 n + i ) ! n=O (2%) * n =O

we get

where c ~ , ~ , dk , j are independent of n and satisfy the recurrences

%+l, i = c k , j - i + ( 2 k + l ) 2 C k , j , (6.39) { dk+i,j=dk,j-l+ ( 2 k ) 2 d k , j - The above Lemma is used in the derivation.

It can be shown that (see [5 , p. 2211)

A

(6.42) ( k - l ) ! seckx= c-- 2 U ~ , ~ A ( 2 n f j ) . .n=o ('n)! j = o

Carlitz, Permutations with Prescribed Pattern 123

Differentiat,ion gives

(6.43) k! sect% tan x=

Comparing (6.42) and (6.43) wit.h (6.35), we get

- %2n--i k

C a,,iA (2n- 1 +j) . n=l (2n--1)! j=o

Hence, by

Therefore,

(6.44)

(6.45)

k - 1

J =(I

(k-l)! k&= C a , , i A (2n+j)

k

) =o Ic! S & + , = C a , , , A ( 2 n + l + j ) .

(6.36) and (6.37), I i

k! f(n, n - k ) = C A (2n+k- j ) C j = 1 8 =o

k i k! g (n, n - k ) = C A (2n+k-j-1) C

j =(I r=O (:)(:) '! ' k - r , k - j *

by (6.33) and (6.34), we get I

P,,&) = 2 ( - 1 y - g (":')(!) s! a k - s i l . k - i ,

QJn) = 2 ( - 1 y - 8

8 = ( I

(:)(f) a k - s , k - i . 6=11

Clearly PJn) , Qp,?(n) are polynomials in n of degree j.

Theorem 7 . The enunzerants f (n, k), g(n, k ) are ezpressible as linear combinations of A (2n+j ) :

k k! f(n, k) = C ( - l)i Pk,i(n) A (2n+k- j )

3 =(I

k (0 5 k < n) ;

k ! g ( n , k ) = C ( - l ) ' Q k , i ( n ) A ( 2 n + k - j - 1 ) I =o

the coefficients P,,;(n), QI,j(n) satisfy (6.44) a?ad (6.45) and are polynomials in n of degreej.

7. Explicit formulas for f (n, A!), g(n, k) . We have

- 1 - Y ___ ___ ~~

1 -Y 1 - y cosh (z fT--i) 1 - y - y (coshx 1 1 -y - 1)

124 Oerlitz, Permutations with Prescribed Pattern

Hence we get

(7.2) n

k = O f,(y)= C 2-"yk (I - Y ) " - ~ 8(2k, 212) .

The right hand side of (7.2) is equal t o n n-k

C 2 - 2 k 8 =o ( - l ) ' (nsk)ykt .d(2k , 2n) k=O

- - 2 yk C ( - 1)s 2- -2k+ ' l8 ( " - k + s )S(2k-2*, 2%) k

k = O s = o

and therefore k

9-2k.f2S ( n - k + s )8(2k-Zs, 2n) f(n, k)- c (-1) a

8 = O (7.3)

As for g(n, k), we have . -. -

y 1 1 - y sinh x f r y - y 11 - y sin11 x 1- - -- --

1-ycoshx 11-y 1-y-y (coshx ~ G J - 1 )

Carlitz, Permutations with Prescribed PR.tterii 125

C2-2h"+l y h (1 -y)"-k 0(2k- l ,212-1) , =c n = l ( 2 1 2 - 1 ) ! k = ,

where

I t now follows irorn (6.2) and (ti.6) that

(7.5)

This yields

n y ~ y ) = C 2 - 2 k + i y k (1 - Y ) " - ~ d(2k- 1,212- 1) .

E = l

lye may now state

Theorem 8. The enurnerants f(n, k ) , g(n, k ) have the explicit evaluation k

f ( n , k ) = c ( - l )k-s 2-'8 S =o

where 2k

j =o 8(2k, 2%) = c ( - l)i (7) (k- j )2n

6 ( 2 k - 1 , 2 n - I ) = C ( - I )? ( k - j ) 2 f i - 1 . 2 k - i

j =O

126 Carlitz, Permutations with Presaribed Pattern

In particular, for k = n, n

A(2n) =f(n, a) = C ( - I ) ~ - ' 2-2s 6(2s, 2n)

A (%-1)=g(n,n)= C (-1)n-s2-288f18(2s-1, 2%--1) .

s=o

n

s = l

We remark that S(s, n) can be exhibited a8 L central difference of zero (6, p. 13).

References

[l] L. CABLITZ, Generating functions for a special class of permutations, Proceedings of the

[2] - , Permutations with prescribed pattern, Math. Nachr. 58, 31 -53 (1973). 131 - , Recurrences for the Bernoulli and Euler numbers, Math. Nachr. 29, 151 - 160 (1965). [4] L. CABLYTZ and RICHARD SCOVILLE, Generalized Eulerian numbers: combhatorial applications,

[5] N. E. NORLUHD, Vorlesungen iiber Differenzenrechnung, Berlin 1924. [6] J. F. STEFFENSEN, Interpolation, Baltimore 1927.

American math. Society, vol. 47, 251 -256 (1975).

Journal fiir die rehe und angewandte Mathematik 266,110 - 137 (1974).

Duke University Deprtnte9zt of dfathematics Durham, North Carolina 27706 U.S.A.