m a x i m a or minima, so F(4) (x) = (1 — x )_ 1. T h u s , from T h e o r e m 12.2,
k
2 3 fk(1)~û = oi^shax — (1 — M)a~2(chax — 1)
k>0 K\
k
1 — 22 fk 77 = ch ax — cT sh (xx k>0
K-k
^2 fk 77 = oT sh ax — (1 — m)a~ (ch ax — 1)
A;>0 K\
k
]C fk 4 77 = x — (m + M — 2)a~3(sh ax — ax)
A;>0 ^ '
a n d the result follows immediately.
COROLLARY 12.5. The number of permutations on n with i strict maxima is
I M ' - J U - a - ' t a n h a * ) -
1where a = (1 - M)1/2.
Proof. P u t w = 1 in Corollary 12.4.
W e observe t h a t the n u m b e r of p e r m u t a t i o n s on n is given by
lim {1 - (1 - M ) -1 / 2t a n h (1 - M )1'2* } "1 = "1 (1 ~ ^ )_ 1 = nl
M->I LnlJ as required. Also the n u m b e r of p e r m u t a t i o n s on n with no strict m a x i m a is
x_
n\.
[^](l-tanhx)-
1=2-
1,
a result which m a y be obtained trivially by induction on n.
13. C o n c l u d i n g r e m a r k s . T h e e n u m e r a t i v e m e t h o d which has been described above has certain limitations. For example, problems such as the Davenport-Shinzel problem ( D a v e n p o r t and Shinzel [13]) a n d the T e r q u e m problem (Moser and A b r a m s o n [28]), both of which involve positional infor-m a t i o n , c a n n o t be treated by this infor-method. However, a certain class of such problems has already been considered by Stanley [36] by means of binomial posets. In addition, no w a y has been found for treating by this method problems which involve both maximal increasing p a t h s a n d maximal decreasing p a t h s occurring together, since no usable unique maximal decomposition exists in this case. T h e Erdôs-Szekeres problem (Erdos and Szekeres [17]), involving e m b e d d e d increasing and decreasing subsequences of length n + 1 in a per-m u t a t i o n on n2 + 1, is an example of such a problem. However, this problem a l r e a d y a d m i t s an elegant solution by means of plane p a r t i t i o n s (Schenstead
[33]; see also S t a n l e y [35]).
N o a t t e m p t h a s been m a d e to a p p l y this e n u m e r a t i v e m e t h o d in an e x h a u s t i v e fashion. P r o b a b l y , several specialisations a n d applications of t h e t h e o r e m s remain. In particular, t h e " p a r t i t i o n t r i c k " , n a m e l y t h e s u b s t i t u t i o n xt = q\
leads to a v a r i e t y of expressions in t e r m s of E u l e r i a n g e n e r a t i n g functions.
T o a certain e x t e n t t h e t h e o r y presented here m a y be a service in establishing t h e positivity of t h e coefficients in t h e expansion of certain rational functions.
C e r t a i n l y positivity is established provided a c o m b i n a t o r i a l i n t e r p r e t a t i o n to t h e rational function is found. If t h e rational function is s y m m e t r i c it is reasonable to seek, in t h e first instance, a c o m b i n a t o r i a l i n t e r p r e t a t i o n in-volving t h e e n u m e r a t i o n of sequences. Accordingly, b y reversing t h e applica-tion of T h e o r e m 4 . 1 , for example, we m a y c o n s t r u c t an e n u m e r a t o r F(x) from a knowledge of {1 — 2 ^ > O / * 7 À : }_ 1. In t h e cases where t h e coefficients in F(x) are non-negative, t h e problem a d m i t s a c o m b i n a t o r i a l i n t e r p r e t a t i o n . In this context, t h e c o n t r i b u t i o n of t h e t h e o r y presented here lies chiefly in t h e con-s t r u c t i o n of t h e combinatorial problem, r a t h e r t h a n in d e m o n con-s t r a t i n g pocon-sitivity since t h e l a t t e r m a y be more readily proved b y o t h e r m e a n s . T h e following r e m a r k d e m o n s t r a t e s t h e principle.
Remark 13.1. L e t
A(x) = {1 — 2(xi + x2 + x3) + (xix2 + x2x3 + x i x3) i_ 1
= X)
ai^
i
T h e n i) a,- is t h e sum of t h e weights A o u(a) over all sequences a in {1, 2, 3 } * of t y p e l where
AOM(<X) = (|^| + 1)(|H + ! ) • • • (\P„\ + D
a n d u(a) — (pip2 . . . pn), t h e m a x i m a l decomposition of a into strictly in-creasing subsequences (in P4) .
Also ii) oi ^ 0.
Proof. L e t P = PA (see Definition 2.5 ( i v ) ) . T h e n from Section 7 ( h ) we h a v e 70 = 1, 7l = #1 + X2 + #3, 72 = X1X2 + X2X3 + X1X3
T h u s
A(x) = (1 -fai - / 2 7 2 ) -1 where f(x) = 1 - fxx - J2x2 - (1 - x )2
T h e n
F(x) = \f(x)}~1 = (1 - x ) "2 = 1 + 2x + 3x2 + . . . a n d t h e result follows from T h e o r e m 4 . 1 .
ii) I m m e d i a t e , since t h e coefficients in t h e expansion of F(x) are positive.
T h e following example shows how t h e weights are d e t e r m i n e d .
Example 13.2. By direct expansion of A(x) we have au.i.i) = 36. T h e fol-lowing table lists the sequences a of t y p e (1, 1, 1) together with their maximal decompositions pip2 . . . pn a n d their weights A o u(a).
T A B L E 2
a- P\p2. . . pn A o u(<r) 123 (123) 4 132 (13) (2) 3.2 213 (2)(13) 2.3 231 (23) (1) 3.2 312 (3)(12) 2.3 321 (3)(2)(1) 2.2.2
Thus
X Aou(a) = 4 + 6 + 6 + 6 + 6 + 8 = 36 = a(1|1(1) r(<r)=( 1,1,1)
as asserted.
N o such i n t e r p r e t a t i o n exists for the function
{(1 - * i ) ( l - x2) + (1 - x2) ( l -xz) + (1 ~ x3) ( l - X i ) } -1
considered by Friedrichs a n d Lewy since the corresponding e n u m e r a t o r F(x) has some negative coefficients. Positivity has been proved by Szego [37] by an a r g u m e n t relying on special functions. N o combinatorial i n t e r p r e t a t i o n to the coefficients in the expansion of this function has been discovered.
A c k n o w l e d g e m e n t s . T h e a u t h o r s wish to acknowledge the s u b s t a n t i a l assistance offered t h e m by A. C. N o r m a n and J. Fitch ( C o m p u t e r L a b o r a t o r y , University of C a m b r i d g e , E n g l a n d ) in connexion with the expansion, b y c o m p u t e r , of a n u m b e r of the generating functions for checking purposes. T h e a u t h o r s h a v e found the use of algebraic m a n i p u l a t i o n systems invaluable both in the d e v e l o p m e n t of this theory a n d in the checking of expansions. A portion of this work was carried o u t when one of the a u t h o r s (D. M . J.) was visiting the D e p a r t m e n t of P u r e M a t h e m a t i c s a n d M a t h e m a t i c a l Statistics a t the University of C a m b r i d g e , E n g l a n d . One of the a u t h o r s ( D . M. J . ) would like to t h a n k Professor J. W . S. Cassels and Professor M. V. Wilkes for permitting him to m a k e use of the facilities of the D e p a r t m e n t of P u r e M a t h e m a t i c s a n d M a t h e m a t i c a l Statistics a n d the C o m p u t e r L a b o r a t o r y , respectively. T h i s work was s u p p o r t e d by a research g r a n t from the N a t i o n a l Research Council of C a n a d a .
A p p e n d i x . T h e following table lists some of the more common e n u m e r a t i o n problems which m a y be treated by the m e t h o d s described here. T h e generating functions are given in the cited corollaries. T h e list does n o t exhaust the possible applications.
T A B L E 3
permutation (P) Configurations Conditions
Corollary sequence (S) Path Type recognised on ^-paths
6.1 S increasing sequence terminators
6.2 S strictly increasing sequence terminators
7.1 S arbitrary £-path exactly i
7.2
s
arbitrary £-path exactly 07 . 3
s
arbitrary longest path8.5 p increasing run p-path exactly i
8.6 p increasing run £-path exactly 0
8.7 p strictly increasing sequence p-path exactly i
8.9 p increasing run longest path
8.10 p increasing run longest path unique length
8.11 p strictly increasing sequence 2 (alternating) even leng th
11.4 S strictly increasing sequence rise, non-rise, maximum 11.5 P strictly increasing sequence rise, non-rise,
maximum
11.6 S strictly increasing sequence length
2 (alternating)
12.3 p strictly increasing sequence length
2 (alternating) 12.4 p strictly ncreasing sequence j strict maximum
( strict minimum j strict maximum ( strict minimum 12.5 p strictly increasing sequence strict maximum
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