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Permutations with prescribed pattern *)
By L. CARLITZ, Durham (U.S.A.)
(Eingegangen am 17. 3. 1972)
1.
Introduction. LetZ, = (1, 2, . . . , n> and let n = (a l , a 2 , . . ., a,) denote an arbitrary permutation of Z,L. Let k, , L,, . . . , kfib be positive integers such that (1.1) k, + k2 + + k, = n. We shall say that the permutation n has pattern [k17 k , , . . ., k,] if the following conditions are satisfied :
(1.2)
and
UI < a2 < . * < akI; a k l + l < nklt2 < *  . < ak l+kZ; . * .; akl+ ... + E ,  i + l < * * * < a,
(1.3) ukf akl+l akl+k2 > a k l + k a + l , * . We may represent z graphically: Thus for example the graph
represents the pattern [4, 1, 4, 1, 1, 21. The graph
with pattern [2, 2, 2 , 2 , 21 represents a socalled updown permutation. We let A ( k i , k 2 , . . ., k,) denote the number of permutations of 2,
with pattern [k, , k,, . . . , k,]. The first main result of the present paper follows :
Put (k l + b + * * * + k m ) !
ki! kz! * * * k,! (hi, k,,. . ., Ic,) =
1) Supported in part by NSF grant GP17031.
32 Carlitz, Permutations with prescribed pattern
Then 112
A ( k 1 , h , . . ., k,) == C ( l ) " "Sr , r = l
(1.4)
where E m = ('1, h, . . .> km) f 4  l = ( k , + k , , k , , * . . > k,) + (h, 4 + k , , k,, . . ., k,
+ . , . i k l , k?, * .  > k7n2, km1 + km) and generally
(1.5) 8, = c (81 , 8 2 , . . . > S , ) . s1 = k, +
where + k,,, 82 = kj l+ l + * * * + kj2,
. . . , 8" = k,rl+l + * * ' i kjr and the summation in (1.5) is over all j , , j , , . . . , j , satisfying
1 5 j , < j , <  *  < j , = 92%. While t.his result theoretically enables one to compute A (k i , k , , . . . , km)
in all cases, i t is unfortunately rather complicated. The remainder of the paper is concerned with the construction of generating functions in certain special cases. I n the first place, if
f(n, nl) C A ( h , k2, . . ., k , ) , ki+ . . . +BW?i
(1.6)
kt >o we show that
It follows that f(n, m) is equal to the EULERian number A,,,, which enu merates the number of permutations of 2, with m rises [3, Ch. 81. Thus this result can be thought of as a partial check of the general formula (1.4).
I n the next place, if we put
g ( n , 9%) = 2 A ( k 1 , k2, . .  , k m ) , kit ... +kwL=7t
kt>l
(1.8)
then
where M, B are the roots of x2  x + y = 0. Now the EULERian number An,k can also be defined by
Carlitz, Permutations with prescribed pattern 33
where  A(') 8) = A r + s + i , s + i  A r + s + i , r + i == A(', '1.
This suggests that we define the array of numbers A(r, s) by means of
Then
n  2 s n  m  s m g ( n , m) = C ( I),'  A(n  8,s)
s = o n  m  s ( m  s ) (2 m < n),
m
g ( 2 m, m) = c ( 1)"SA(2 m  8, s) . s = o
The numbers A ( r , s) and A(', s) are closely related; see (8.9) and (8.10) below.
Finally we consider the case
Then we show that 00 .mk
J,
(1.10) 2 A,(m k)  = {F,(x))l, ?n=O (m I c ) !
where M
FI,(X) = c ( 1)' r(, j = O ( j k ) ! .
Moreover if we put
where 00 ..ik+t
& Fk, t (x) = C ( I)' ___
j = O ( j k + t ) ! * These results, (1.10) and (1.1 1), evidently generalize the known results
for updown permutations [J], [ 2 , pp. 1051121. 3 Math. Nachr. 1973, Bd. 58, H. 16
34 Carlitz, Permutations with prescribed pattern
2.
M \Te begin with the case ? ) L = 2. Froin the pattern it is clear that
(2. 1)
since the ele~nent 12 is situated at one of the peaks. Moreover it is evident that
A (k1, k.) = (kl 1 1 + A ( k , , k 2  1) (k2 > I ) , (2.3) A ( k , , 1) = k, ( k , 2 l ) , since there are precisely k , elements that may be placed i n the extreme right hand position. For a like reason we have also
Indeed (2 .1) also holds when k , = 1. 12.3) A(1, k2) = k2 (k2 2 1).
In the next place we have, for 7 ) ~ 2 3, k, + k, 
(2.5) A(k1, k , , k3) = (I + k ; + k , l ) k J
+ ( k 1  k 2  k 1  ) A ( k , , k 2  1) + A ( k , , k 2 , k ,  1) ( k , > l , k 3 > 1 ) .
Indeed if the element rL is at the left hand peak, we get the first product on the right of ( 2 . 5 ) ; if it is at the second peak, we get the middle product; if it is at the extreme right, we get the third term.
Then, by (2.4), A ( k , , k2, k,)  A(li1, k ? , kj  1)
k , + k2 + k ,  = ( k2 + k3
h.9 + k3  1) [ ( h + k z  1)  1] k ,
Carlitz, Permutations with prescribed pattern 35
Simplifying we get
(2.6) (k + k2 + b)!
k,! kl! kj! A ( k , , k,, k3) =  ~ ~ 
36 Carlitz, Permutations with prescribed pattern
It will be convenient to use the notation
We have seen above that
 (k l+ kz) = ( k , . k,, 1)  ( k , , x.2 + 1)  (k, + k , , I ) + 1,
so that (2.8) holds for k2 > 1, k,; = 1. For k, = 1, in place of (3.5) we iiare
+ kJ k , + A ( k l , 1, k ,  1) (k, > 1). (B, 1 ) (2.9) A ( k , : 1, k,) = Hence
= (4 , 1, k3)  $ 1 , k3 + 1)  (El + 1, 4) + 1. It follows that (2.8) holds for all positive k l , k 2 , k,.
Carlitz, Permutations with prescribed pattern 37
3.
We now take m = 4. Then, to begin with, we have the recurrence
38 Carlitz, Permutations with prescribed pattern
4.
Carlitz, Permutations with prescribed pattern 39
We show first that
where sl, s2 , . . ., s, are defined by (4.2) and (4.3).
the general case. Thus we shall prove It will suffice to prove (4.5) when m = 3 as the method is the same in
(4.6)
where of course
(kl, k2 > k3) = A (kl k2 > k3) + A (kl 9 k, + k3) + A (kl + k2 9 k,) + 4 k i + k; + JCd. A(kl + kz + k3) = 1.
To prove (4.6) we partition
2, = (1, 2,. . ., n} (n = k1 + k2 + k3) into three sets
Ii, = (al > . * . > u2 = (b, 9 . * . > bk2)' u3 = (%, . *  > ckJ) * This can be done in (k,, k2, k3) ways. We assume that the elements of U 1 , U 2 , U3 are numbered so that
< a2 < ' ' ' < a k l ; b , < b, < ' . ' < b&; c1 < c2 < ' ' < ckJ. If akl > b l , bkz > c l , the partition corresponds to a permutation with pattern [ki . k 2 , k3]. If ak, > bt , bka < cl, we have the pattern [k,, k, + hJ. If ak, < bl , b k Z > c2 we have [kl + k2, k3]. Finally if ak, < b, , bkl < c, we have [k, + k, + k3]. Moreover in each case the correspondence is one to one. This evidently proves (4.6).
It should be noted that (4.5) holds for
(4.7) k, 2 1, k2 2 1,. , . , E m 2 3
It remains to show that (4.5) implies (4.i). To do this we prove the following
gn (x, , . . . , 2,) be arbitrary (realvalued) functions. Define emm ma. Letf1(x,), f%(XI,X2), . . . , f r l ( x , , . . . , x,J, g i ( x l ) , g ~ ( ~ ~ , x p ) , . . . ,
j1,j2,. . .?jr; 8 1 ~ ~ 2 , . . ., 8, by means of (4.2) and (4.3). Then
m
r = l q(k1, . . ., k,) = 2 f (si, . . . , s,) (nz = 1, 2 , . . . ,n) (5.1)
40 Carlitz, Permutations \I itli prescribed pattern
It is easily verified that ( 3 . 6 ) and ( 5 . 6 ) are equivalent. For the general case we require some additional notation. Let
( 5 . 7 ) 1 I  t , < t , < * * < t , = r and put
GI = 8, + . * * , St, , 51 = St,> 1 +  * * + Stz' (5.8) . . ., 0fi = S ( l I  I L 1 +  * * + Stn ,
so that G, , . . . , G~ are related t o ,sl, . . . , s, as s J , . . ., s, are related to k , > * . * , k,,$.
Carlitz, Permutations with prescribed pattern 41
Now it is easily verified that the number of rtuples (sl, . . ., sr) is
). Similarly t,he number of Rtuples (bl, . . . , bR) obtained m  1 r  1
equal to ( from (5.7) and (5.8) is equal t o (L 1 ). In order to show that (5.1) implies (5.2) we substitute from (5.1) into (5.2). Then for a fixed Rtuple (u i , . . . , uB) we get the coefficient
The sum on the right vanishes unless m = R. This completes the proof of the implicat,ion (5.1) + (5.2). The proof of (5.2) r==. (5.1) is exactly the same.
As a variant of the above proof, we define (5.9) 1 > ~ r ) > (5.10) G r ( k , ,  . ., = C g ( s , , . . ., S T ) ,
Fr (ki > . * . > k,) =z CfCsl> .
where the summations are over all (sl, . . ., sr) that satisfy (4.2) and (4.3). It can be verified that
which is the same as
It is familiar that (5.11) is equivalent to
For r = m, we have
Fm = f ( k t , * .  9 km), 2 g(kr,   9 km) and the equivalence of (5.1) and (5.2) follows at once.
6.
We shall now discuss some applications of (4.1). It will be convenient to change the notation slightly. Put
nz A ( k l , kp, . . . ) k,) = C ( I),, 'r7
r = 1 (6.1)
42 Carlitz, Permutations with prescribed pattern
where
( 6 . 2 ) 8, = c (s1,s?_, . . ., s,), SI = kl + . . * + kj . , , = kj ,+i + * * + + kj ,+ ia ,
. . . > S, == kjl+ ... + j ,  l + l + . * + kj ,+ . . .+j , , and the summation is over all j , , . . . ~ j , such that
j , + j ? +  . * + j , = m, j , > 0 , j, > 0, . . . , j, > 0 . As a first application we consider the sum
f(n, nz) = c A(X.1. k ? . . . ., k,, ,) , !I ,I = I 1 kl . . . (6.3)
where the summation is over all positive k, , . . . , k,, such that kt>0
k , + * * f k,,, = 7 1 . We construct the generating function
Then by (6.3)
I\;ow apply (6 .1) . Since the number of positive solutions of x., + . . + k . =
Carlitz, Permutations with prescribed pattern 43
Hence
so that
We recall that the socalled EuLERian numbers may be defined by
Moreover is the number of permutations of 2, with k rises [3, p. 2141. By a rise in the permutation (a l , a 2 , . . . , a,) is meant a pair aj, ul+ with u j < aj+i; also a conventional rise is counted to the left of ai.
Comparison of (6.6) with (6.7) gives
(6.8) f(n, m) = Returning t o (6.3), the function f ( n , m) is equal to the number of per mutations with k , + *  * + k , = n. Clearly the number of rises (plus the conventional one) in a permutation of pattern [ k , , . . ., k,] is equal to
m
I + z ( k j  l ) = n  m + I i=i
Since An,, = A n , n  r n + i 7
we again get (6.8).
the general formula (6.1). Thus the known result concerning A,,m furnishes a, partial check on
7.
As a second application of (6.1) we take
9(% m) = 2 A ( k , : . .  3 k?).J? kl+.. . + k , = n
ki>l
(7.1)
where now each E , > 1. Thus the pattern has the appearance
44 Carlitz, Permutations with prescribed pattern
P u t
,& f. ' . +k, M C c = /JJ y"L r A (K,, . * . , k,,,)   _ _ _ _  ~
( k , + * * * + k,) ! * 2 l l L = I t l , . . ., k , 5 2 Since the number of solutions of
is equal to
me get
c= c . (S) s1,. . . , S r = 2
We rexrite ( 7 . 2 ) as
Clearly
Cnrlitz, Permutations with prescribed pattern 45
If we put
it is easily verified that M i
Since
 1
___  l  z f y x 2
x2  x + y = 0, where u, are the roots of
it follows that
Thus
and ( 7 . 3 ) becomes
Finally therefore
Now it can be shown that the EuLERian numbers An.k defined by (6 .7 ) also satisfy
46 Carlitz, Permutations with presrribed pattern
where
A ( r , s ) A r + s + l , s  i = A r , s c l , r + 1 = A(s, r ) . This suggests that we define an array of numbers B(r, s) by means of
Clearly
d ( r , s) =A(& T ) . By (7.6) and (7.8)
Since
it follows that
Thus (5.9) becomes
Carlitz, Permutations with prescribed pattern 47
S
48 Uarlitz. Permutations with presrrlbed pattern
This can be described l.)y saying that TC has only Iinclines and 2inclines. More precisely if x has 712 inclines then z has exactly 7%  1 1inclines. It follows that the number of periiiutatioiis with t 1inclines and s %inclines is equal to g ( r + 2 s. r  1).
8.
Put
and let H,. H , denote partial derivatives. It can be verified that n y ( . ~  y) ( e r  e)
(.r e  y e r ) 2 T H , ~ f y H , 1
and
so that
(8.2)
(8.3)
It is clear frolii (7.8) thnt
r ( 1  y) H,r 2 y ( l  2 ) H , = .LyH.
d ( r . 8 ) = r q r , s  1) A . s d ( r  1, s ) Compariiig (8.2) with (7.8) we get the ierurrence

+ (1. + s  I ) & r  1, s  1) ( r 2 1, s 2 1).
(8.4) A ( r , 0) =A((); r ) = 0 ( r > O ) , while J ( 0 , 0) = 1. For s = 1, (8 .3) reduces to
so that
(8.5) A(r, 1) = d ( l , r ) = 1 ( r 2 1). The first few values of &$(r. s) are easily computed by means of (8.3).
d ( r , 1) = A ( ,  1 . 1) + r 3 ( r  1, 0 ) ( 7 1 11, 
1 1 1 1 2 1 7 1 9 1 21 21 1 44 1 51 161 51 1 265
The numbers in the right hand column are obtained by summing in the rows. Since
Carlitz, Permutations with prescribed pattern 49
where
is the number of derangements of Z,, i t follows that n
, rA(n  s, s) = Dn. s = o
(8.6)
I n the next place, i t we take y =  x, (8.1) becomes
where En denotes the EULER number. On the other hand, by (7.8), M xr + s
H ( x ,  x) = 2 (  1)8A(r, s) ~ r ,s=O ( r + s)!
r(i
= c  P ( I)sA(n  s, s ) n=O n ! s%
so that n 2 ( i)'B(n  S, S) = E,.
s =o (8.7)
Therefore, by (7.11) and (8.7), we get
(8.8) g(2 m, m) = ( E2m. This is a known result for the number of updown permutations of Zlr,&
The number A(r, s) can be expressed in terms of EuLEItian numbers. PI, PI. Indeed by (7.7) and (7.8) we have
It follows that
and
(8.10) A ( r , s ) = ( l)'j( r + s ) A ( r , j  1 ) . j=i
4 Math. Nachr. 1973, Bd. 58, H. 16
50 Carlitz, Permutations with prescribed pattern
9.
The numbers A(r, s ) defined above were introduced by ROSELLE [4] in an entirely different setting. A succession in a permutation (a,, a2 , . . . , a,) is apair a,, witha,_, = n, + 1. Forexample23145 hastwo successions. Let P(n, r: s) denote the number of permutations of 2, with r rises and s successions. It js proved that
n  I (9.1) P(n , r , s ) = ( ) P(n  s, I'  s, 0) .
Put,
P(n, r ) = P(n, r , 0) .
Clearly r  I
(9.2) 2 P(., r , s) = A,,, , s =o
the EULERIAX number. Coinbining (9.1) with (9.2) we get
which is equivalent to
Also i t is proved that
(9.5) P ( n + 1: r ) = rP(vb: r ) + (n  r + I ) P(n , r  1) + (?L  1) P(n  I, r  I ) .
If we define P* (12, r ) by iiieans of
i t follows from (9.5) that
(9.7) P * ( ~ L + I; Y) = rP*(n, r ) + ('R,  r + I) P*((n, P  i) .+ ?ZP*(?L  1, r  1).
Comparing (9.7) with (8.3), we get
(9.8) B(r, s ) = P*(I' + s, r ) . Finally we may state the following conibinatorial interpretation : P* (n, r ) is the number of permutations of Z,, with r rises, no successions and ai > I.
Carlitz, Permutations with prescribed pattern 51
10.
Returning to (6.1), we now consider the case
(10.1)
It is convenient to put
(10.2)
Then by (6.1) we have
k I   * . . = k, = k.
A, (m k) = A, (k, Ic, . . . ) k) .
A,(m k) = 2 ( m
(10.3) C (jl k) j , k . . . j , 4 . r = l j l + . . . + j , = m
j S > O
It follows from (10.3) that
Therefore
(s k) !
M
Xmk M
(10.4) 1 + 2 A, ( m k ) ~ = {c ( 1)' m=l (m A ) ! s = o
For k = 2 , (10.4) is in agreement with (8.8). For k = 1, (10.4) beconies 1
m=O
Do
m = l
so that
(10.5) Al(m) = 1 (m = 1, 2 , 3) . . .). This is also evident from the definition of A , (1, 1, . . . , I ) . 4*
52 Carlitz, Permutations with prescribed pattern
By (6.1), (10.5), is equivalent to the identity m...