Volume 5, number 3 INI-‘ORMATiON PROCESSING LETTERS August 1971;

A SERIES EXPANSION INVOLVING THE HARMONIC NUMbERS

tanford thiversity,

Received 13 Apd 1976

faMons, left to right m ximum, tight to left maimurn, random permutation

In the analysis of orithms -” particularly algo-

rithms which involve permutations - generating functions with terms of the form

1 ( ) n

1ni-t; , (I- zy”” - frequently appear. In this note we will show that the c&‘fkients of the powers of r in the =I ies expansion of the above function can be expressed in terms of the generalized harmonic numbers [ 11

HU’=lt;+-L*...+f. n

3k

Such a representation is particularly useful for asymp- totic analysis since the asymptotic properties of the generalized harmonic numbers are well tinderstood 12, exercise 6.1-8).

Oandn;;bO, we have

where the polynomial P&l. . . . . s,,) is defined by PO=1 and

(-St, -32) - 2sp . . . . -(tt - I)! $1 ,

where Yn is the familiar

the polynomials are

pIGI)= S] 9

pzcspsz)=s::- 9,

f~(Sl.S2,sj)=s:-3sls~+2S~.

P4(~~,~2,~~,~4)=~:-6~t~2+‘r.‘1~3+3~t_6~q.

RoojI Let y be a second variable. We may then write

= -m-y-l k

(-*f

= l+-” )I( ) m+k ,k

rn4.J m

= fi [ ii P (H;;:-H,$ e.., kg) n=O n .-

b p-L . INFORMATION PROCESSING LETTERS August 1970

have made use of th$ relationship po~ynotvrfale and symme t tic functions 4’0 sf i3j which assurer us that P,(H$&

the &xxabient ofy” in

+ II )) .** (I +v/(@l + R)) *

completed by comparing the coefficients

q

ur expansiot- is just the binomial expoilent. The expansion for the

II in the solution of Exercise 5.2.2-29

an application, we will compute the cova- number ou” left-to-right maxima and the

-Weft maxima of a random permuta- , . . . . n. A left-to-right maximuin is !iat m=l orpk<p,( for all k<m

mum is de&red amkgously. enote the numbers of left-to-

o-left maxima of a permuta-

* pmb (L,(P) * i and R,(P) = k) g

permutations are considered to be equally he entries to the left of the occurrence of

ribute to t,(p) and the entries to the ly contribute to R,(p), we obtain, by considering ous positions for rr,

n

.&&~)=~~a,& l)A,&l,y), i

re we assume that A&, y) = 1. If we define

A(R,,.Y, z) z n4 An(x,y)Z” 9

awehe

t A&y, z) = ii nA,(x,y)? *1 n=l

00 n-l

=W’ c z (A#, ~)~k)(~n_~_l(~,y)Z”-k-l) n=l k=O

=xy A@, l,z)A,~,y,z) .

This differential equation is to be solved subject to the condition that A@, y, 0) = 1. By setting .X *y = 1, we may solve the equation for A (1) 1, t). Next we set onl one of x and y equal to one and solve to get A( ?BJ and A@, 1, z). Using these results, we solve the ation one more time to obtain.

A@, y, z) = ,-$‘- 1 . *--I il_nr+Y-l’

Since L, and R, have the same distribution, w s~z that

cov (L, , R,) = IF&, - E(L,)) (R, - E(Rm ))

= ,E(L,R,) - Ect,,? ,

where E denotes the expectation. S&X we have

E(L,) = c ja@) , j,kaO jk

E(L, R,)* x jka,$), j,k30

we may obtain

Carrying out the

c E(L&P =-& In &, n=O

ii E(L, R#l F Gz + cz n=O

From our theorem, we obtain

INFORMATION PROCESSING LETTERS August 1976

References

11) D.E. Kmri , .I :Qrrleutd Al~witlms, nw 4rt of com- puter Rogramming 1 (liJdism-Weslay, Read&z, Mass., 1968) pp. 634.

[2] D.E. Knuth, Sortiq ad Se~rdu’ng, The Art of Computer hgmmming 3 (AddismWeslq ) Wedding, Mass., 1973) pp. 722.

(3 1 J. Rio&n, An Introduction to Combbutorial Analysis Wiley, 1958) pp. 244.

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