Permutations with prescribed pattern

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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 so-called up-down 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 GP-17031.

  • 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?, * . - > k7n-2, km-1 + km) and generally

    (1.5) 8, = c (81 , 8 2 , . . . > S , ) . s1 = k, +

    where + k,,, 82 = kj l+ l + * * * + kj2,

    . . . , 8" = k,r-l+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 up-down permutations [J], [ 2 , pp. 105-1121. 3 Math. Nachr. 1973, Bd. 58, H. 1-6

  • 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 (real-valued) 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) . . ., 0-fi = 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 r-tuples (sl, . . ., sr) is

    ). Similarly t,he number of R-tuples (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 R-tuple (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 so-called 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 I-inclines and 2-inclines. More precisely if x has 712 inclines then z has exactly 7% - 1 1-inclines. It follows that the number of periiiutatioiis with t 1-inclines 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 up-down 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. 1-6

  • 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...

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