396
MAGAZINE
MATHEMATICS
Partitionsinto Consecutive Parts
M. D. HIRSCHHORN
Universityof New SouthWales
Sydney2052, Australia
[email protected]
P. M. HIRSCHHORN
5/312 FinchleyRoad,Hampstead
LondonNW3 7AG,U.K.
[email protected]
It is known,thoughperhapsnot as well as it shouldbe, thatthe numberof partitions
of n into (one or more)consecutivepartsis equalto the numberof odd divisorsof n.
(Thisis the specialcase k = 1 of a theoremof J. J. Sylvester[1, §46], to the effectthat
the numberof partitionsof n into distinctpartswith k sequencesof consecutiveparts
is equalto the numberof partitionsof n into odd parts(repetitionsallowed)precisely
k of which aredistinct.)
Forinstance,
1+ 2+ 3 +4+5,
15=7+8=4+5+6=
so 15 has four partitionsinto consecutiveparts,and 15 has fourodd divisors,1, 3, 5,
and 15.
We shallprovethe followingresult.
THEOREM.
Thenumberof partitionsof n intoan odd numberof consecutiveparts
is equal to the numberof odd divisorsof n less than /2-n, while the numberof partitions into an even numberof consecutiveparts is equal to the numberof odd divisors
greaterthanA/2-n.
Proof. Suppose n is the sum of an odd numberof consecutiveparts. Then the
middle partis an integerand is the averageof the parts.Supposethe middlepartis
a, andthe numberof partsis 2k + 1. The partitionof n is
n
(a - k)
+a
+ (a + k)
+-.-+-.-andn = (2k + 1)a. So d = 2k + 1 is an odd divisorof n andits codivisoris d' = a.
Note that a - k > 1, thatis, 2a - (2k + 1) > 0, d < 2d', d < 2n/d, and d2 < 2n.
Conversely,supposed is an odd divisorof n with d2 < 2n, and codivisord'. Then
d < 2d', andif we write2k + 1 = d, a = d' then
n = (a - k)
+a
+ +
+.-.
+-.-- (a k)
is a partitionof n into 2k + 1 consecutiveparts.
Next, supposen is the sum of an even number,2k, of consecutiveparts.Thenthe
averagepartis a + 1/2 for some integera, the partitionof n is
n= (a+ 1-k)+...+a+(a+
1) +... + (a +k),
and n = 2k(a + 1/2) = k(2a + 1). Then d = 2a + 1 is an odd divisorof n and its
codivisor is d' = k. Note that a - k > 0, (2a + 1) - 2k > 0, d > 2d', d > 2n/d, and
d2 > 2n.
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397
VOL.78, NO. 5, DECEMBER
2005
Conversely,supposed is an odd divisorof n withd2 > 2n, with codivisord'. Then
d > 2d', andif we write2a + 1 = d, k = d', then
n= (a+ 1-k) + -+
a + (a+ 1)
+ (a +k)
+.
is a partitionof n into an even numberof consecutiveparts.
U
REFERENCE
1. J. J. Sylvester, A constructivetheory of partitions,arrangedin three acts, an interactand an exodion, Amer J.
Math. 5 (1882), 251-330.
MeansGeneratedby an Integral
HONGWEI CHEN
Departmentof Mathematics
Christopher
NewportUniversity
NewportNews,VA23606
[email protected]
Fora pairof distinctpositivenumbers,a andb, a numberof differentexpressionsare
knownas means:
the arithmeticmean:A(a, b) = (a + b)/2
the geometricmean:G(a, b) = -b
the harmonicmean:H(a, b) = 2ab/(a + b)
the logarithmicmean:L(a, b) = (b - a)/(ln b - Ina)
5. the Heronianmean:N(a, b) = (a + Na-b+ b)/3
6. the centroidalmean:T(a, b) = 2(a2 + ab + b2)/3(a + b)
1.
2.
3.
4.
Recently,ProfessorHowardEves [1] showedhow manyof thesemeansoccurin geometricalfigures.The integralin ourtitle is
(1)
fb
whichencompassesall thesemeans:particularvaluesof t in (1) give eachof the means
on ourlist. Indeed,it is easy to verifythat
f
Moreover,uponshowingthatf(t) is strictlyincreasing,we can concludethat
H(a, b) < G(a, b) < L(a, b) <N(a, b) <A(a, b) <T(a, b),
with equalityif andonly if a = b.
(2)