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IDENTITIES RELATING THE NUMBER OF PARTITIONS
INTO AN EVEN AND ODD NUMBER OF PARTS, II
DEAW R. HICKERSON
University of California, Davis, California 95616
Definition.
If / > 0 and n > 1, let q?(n) be the number of partitions of/? into an even number of parts,
where each part occurs at most/times. Let q°(n) be the number of partitions of A? into an odd numberof parts,
where each part occurs at most/times. If/ > 0, let^f(O) = 1 and ^ ( 0 ) = 0.
Definition.
If / > 0 and n > 0, let A{(n) = qf(n) -
qf(n).
The purpose of this paper is to determine Ai(n) when / is any odd positive integer. The only cases previously
known were / = 1, proved by Euler (see [1]), / = 3, proved by this writer (see [2]), and/= 5 and 7, proved by
Alder and Muwafi (see [3]).
Definition.
If s, t, u are positive integers with s odd and 1 < s < t, and n is an integer, let fs> t}U (n) be the
number of partitions of n in which each odd part occurs at most once and is ^ ±s (mod It) and in which each
even part is divisible by It and occurs < u times.
Theorem.
If s, t, u are positive integers with s odd and 1 < s < t, and n is an integer, then
&2tu-i(n)
= (~VnZ
fs,t,u(n-tj2-(t-s)j).
Proof.
z A2tu~i(n)xn = n lzj^1 = n n-*1)- n r / - * ^ w ^ v . . ^ - ^
2\j
= JJ
(1
2t\j
_ x2tj+s)(1 _ x2tj+2t-s)(1 _ x2tj+2t)mYY(J_xJh
j
i
j>0
J
j>l
2t\j
where the last equality follows from Jacobi's identity with k = tant\z = t-s.
tj
+ (t-s)j
Hence, when we substitute -x forx, we obtain
,
iia+xj+x2j+-+x(u~1)j),
d-xh-
JJ
j>l
2\)
2\)
_
j£±s (mod 2t)
2
j
j>l
j>l
2kj
2t\j
j^+s (mod 2t)
= E (-Djxtil+(t-s)^ n
j
Y\(l+xJ+x2j+...+x(u-l)j)
=/
(mod 2).
Since s is odd,
IDENTITIES RELATING THE NUMBER OF PARTITIONS
(-1)nA2tu„l(n)xn
£
=£ x^
n
+
^ - EI
j
d + x')- El
j
(1 +
FEB. 1978
x3+x2i+..-+x(»-l)i)
j
j>l
2\j
f£±s (mod 2t)
j>l
2t\j
= 2 > ^ ^ ' . £ fs,t,Jm)xm,
j
m
from which the theorem follows immediately.
Corollary
1.
\is and t are positive integers with s odd and 1 < s < t, and n is an integer, then
*2t-i(n)
= (~1)n E
fs,t,l(n-tf2-(t-s)/).
j
Note that fS)t,i(n) is the number of partitions of n into distinct odd parts ^ ± s (mod It),
Proof.
Ls\u=
1 in the theorem.
Letting^ = 1 and t = 3 yields Theorem 1 of [ 3 ] .
2. If i> 2 and/? is an integer, then (-J)nAi(n)>
Corollary
0.
Proof.
For even i, this follows from Theorem 3 of [ 2 ] ; for odd i, it follows by letting s = 1 and
t = (i + D/2 in Corollary 1.
Corollary
3. If s and t are positive integers with s odd and 1 <s <t, and/? is an integer, then
A4t-lM
= <-Dn £
j
fs,t,2(n-tj2-(t-s)j).
Note that fSyt,2(n) ' s t n e number of partitions of/? into distinct parts which are either odd but ^±s (mod 2f)
or which are divisible by It
Proof
Corollary
Let £/ = 2 in the theorem.
4. If u is a positive integer and n is an integer, then
±4u-lM = (-Dn L
h,2,Jn-2j2-j).
j
Note that fi}2,u(n)
Proof.
is the number of partitions of/7 into parts divisible by 4, where each part occurs <u times.
Lets= 7, f = ^ in the theorem.
Letting u = 1 yields Theorem 2 of [2] and u = 2, Theorem 2 of [ 3 ] .
REFERENCES
1. Ivan Niven and Herbert S. Zuckerman, An Introduction to the Theory of Numbers, 3rd ed., pp. 221-222,
Wiley, New York, 1972.
2. Dean R. Hickerson, "Identities Relating the Number of Partitions into an Even and Odd Number of Parts,"
J. Combinatorial Theory, Section A (1973), pp. 351-353.
3. Henry L. Alder and Amin A. Muwafi, "Identities Relating the Number of Partitions into an Even and Odd
Number of Parts," The Fibonacci Quarterly, Vol. 13, No. 2 (1975), pp. 337-339.