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A series expansion involving the harmonic numbers
INI-‘ORMATiON PROCESSING LETTERS
Volume 5, number 3
August 1971;
A SERIES EXPANSION INVOLVING THE HARMONIC NUMbERS
tanford thiversity,
Received 13 Apd 1976
faMons,
In the analysis of
left to right m ximum, tight to left maimurn, random permutation
orithms -” particularly algo-
which involve permutations - generating functions with terms of the form
rithms
n
1 (I-
zy””
the polynomials are pIGI)= S] 9 pzcspsz)=s::-
, (1ni-t; - )
9,
f~(Sl.S2,sj)=s:-3sls~+2S~.
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
P4(~~,~2,~~,~4)=~:-6~t~2+‘r.‘1~3+3~t_6~q.
RoojI Let y be a second variable. We may then write
HU’=lt;+-L*...+f. n
3k Such a representation is particularly useful for asymptotic analysis since the asymptotic properties of the generalized harmonic numbers are well tinderstood 12,exercise 6.1-8).
-m-y-l
= Oandn;;bO,
we
(-*f
k
have
=
l+-”
rn4.J )I(
where the polynomial P&l. .... s,,) is defined by PO=1 and = fi
m+k m
)
,k
[ ii P (H;;:-H,$e.., kg) n=O n .-
(-St,
-32)
-
2sp .... -(tt
where Yn is the familiar
- I)! $1 ,
b
p-L
August 1970
INFORMATION PROCESSING LETTERS
.
have made use of th$ relationship po~ynotvrfaleand symme t tic functions 4’0sf i3j which assurer us that P,(H$& the &xxabient ofy” in
00 n-l
+ II)) .**(I +v/(@l +R)) *
=W’ c z (A#, n=l k=O
completed by comparing the coefficients
=xy A@, l,z)A,~,y,z)
q ur expansiot- is just the binomial expoilent. The expansion for the IIin the solution of Exercise 5.2.2-29
an application, we will compute the covanumber ou”left-to-right maxima and the -Weft maxima of a random permuta, ... . n. A left-to-right maximuin is !iat m=l orpk
~)~k)(~n_~_l(~,y)Z”-k-l)
.
This differential equation is to be solved subject to the condition that A@, y, 0) = 1. By setting .X*y = 1, we may solve the equation for A(1) 1, t). Next we set onl one of x and y equal to one and solve to get A( ?BJ and A@, 1, z). Using these results, we solve the ation one more time to obtain.
1 A@, y, z) = ,-$‘. *--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
*
pmb(L,(P) * i
and
R,(P)
= k) g
permutations are considered to be equally he entries to the left of the occurrenceof ribute to t,(p) and the entries to the ly contribute to R,(p), we obtain, by considering ous positions for rr,
E(L, R,)*
jk
x
j,k30
jka,$),
we may obtain
n
.&&~)=~~a,&
l)A,&l,y),
i
re we assume that A&,
y) = 1. If we define
A(R,,.Y,z) z n4 An(x,y)Z” 9
Carrying out the c
n=O
E(L&P =-&
In &,
awehe t
A&y, z) = ii nA,(x,y)? n=l
*1
ii E(L, n=O
R#l
F Gz + cz
From our theorem, we obtain
INFORMATION PROCESSING LETTERS
August 1976
References 11) D.E. Kmri , .I :Qrrleutd Al~witlms, nw 4rt of computer Rogramming 1 (liJdism-Weslay, Read&z, Mass., 1968) pp. 634. [2] D.E. Knuth, Sortiq ad Se~rdu’ng, The Art of Computer hgmmming 3 (AddismWeslq Wedding, Mass., 1973) pp. 722. (3 1 J. Rio&n, An Introduction to Combbutorial Analysis Wiley, 1958) pp. 244. )
77
Volume 5, number 3
August 1971;
A SERIES EXPANSION INVOLVING THE HARMONIC NUMbERS
tanford thiversity,
Received 13 Apd 1976
faMons,
In the analysis of
left to right m ximum, tight to left maimurn, random permutation
orithms -” particularly algo-
which involve permutations - generating functions with terms of the form
rithms
n
1 (I-
zy””
the polynomials are pIGI)= S] 9 pzcspsz)=s::-
, (1ni-t; - )
9,
f~(Sl.S2,sj)=s:-3sls~+2S~.
frequently appear. In this note we will show that the c&‘fkients of the powers of r in the =I ies expansion of the above function can be expressed in terms of the generalized harmonic numbers [ 11
P4(~~,~2,~~,~4)=~:-6~t~2+‘r.‘1~3+3~t_6~q.
RoojI Let y be a second variable. We may then write
HU’=lt;+-L*...+f. n
3k Such a representation is particularly useful for asymptotic analysis since the asymptotic properties of the generalized harmonic numbers are well tinderstood 12,exercise 6.1-8).
-m-y-l
= Oandn;;bO,
we
(-*f
k
have
=
l+-”
rn4.J )I(
where the polynomial P&l. .... s,,) is defined by PO=1 and = fi
m+k m
)
,k
[ ii P (H;;:-H,$e.., kg) n=O n .-
(-St,
-32)
-
2sp .... -(tt
where Yn is the familiar
- I)! $1 ,
b
p-L
August 1970
INFORMATION PROCESSING LETTERS
.
have made use of th$ relationship po~ynotvrfaleand symme t tic functions 4’0sf i3j which assurer us that P,(H$& the &xxabient ofy” in
00 n-l
+ II)) .**(I +v/(@l +R)) *
=W’ c z (A#, n=l k=O
completed by comparing the coefficients
=xy A@, l,z)A,~,y,z)
q ur expansiot- is just the binomial expoilent. The expansion for the IIin the solution of Exercise 5.2.2-29
an application, we will compute the covanumber ou”left-to-right maxima and the -Weft maxima of a random permuta, ... . n. A left-to-right maximuin is !iat m=l orpk
~)~k)(~n_~_l(~,y)Z”-k-l)
.
This differential equation is to be solved subject to the condition that A@, y, 0) = 1. By setting .X*y = 1, we may solve the equation for A(1) 1, t). Next we set onl one of x and y equal to one and solve to get A( ?BJ and A@, 1, z). Using these results, we solve the ation one more time to obtain.
1 A@, y, z) = ,-$‘. *--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
*
pmb(L,(P) * i
and
R,(P)
= k) g
permutations are considered to be equally he entries to the left of the occurrenceof ribute to t,(p) and the entries to the ly contribute to R,(p), we obtain, by considering ous positions for rr,
E(L, R,)*
jk
x
j,k30
jka,$),
we may obtain
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
Carrying out the c
n=O
E(L&P =-&
In &,
awehe t
A&y, z) = ii nA,(x,y)? n=l
*1
ii E(L, n=O
R#l
F Gz + cz
From our theorem, we obtain
INFORMATION PROCESSING LETTERS
August 1976
References 11) D.E. Kmri , .I :Qrrleutd Al~witlms, nw 4rt of computer 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. )
77