694
Internat. J. Math. & Math. Sci.
VOL. 21 NO. 4 (1998) 694
CORRIGENDUM
A NOTE ON SELF-CONJUGATE n-COLOR PARTITIONS
A.K. AGARWAL
Department of Mathematics
Panjab University
Chandigarh 160014, INDIA
A theorem involving self-conjugate n-color partitions was proved by A K. Agarwal and R.
Balasubrananian in their paper entitled n-Color Partitions with Weighted Differences Equal to Minus Two,
Internat. d. Math. Math. Sci., Vol. 20, No. 4 (1997), 759-768. There are some errors in this theorem which
are corrected as follows:
THEOREM 1. Let A(v) denote the number ofn-color self-conjugate partitions ofv such that each
part is self-conjugate. Let B(v) denote the number of ordinary partitions of v imo odd parts. Then
A(v) B(u), for all v > 0. Hence
II 1
+
,=1
=1
q2-1
(1)
3, since the relevant partitions are 5a, 311111 and 111]111111. Also, B(5) 3,
the
relevant
this
case
since in
partitions are 5, 311, 11111.
THEOREM 2. Let C(v) denote the number of all n-color self-conjugate partitions ofv. LetD(v)
dznote the number of partitions ofv such that each even part2n can come in[n/2], where[ is the greatest
EXAMPLE. A(5)
integer function, different colors denoted by
(2n)1, (2n)2,’’-, (2n)W2 I.
D(v), for all v > 0. Hence,
Then C(v)
+
u=]
II
ELE. C(5) 4, s e relt fiom e 5s, 3211, 2221, 11111. so,
D(5) 4. In is e e relt tiom e 54]1, 3 1 1, 1 1 1 1 1.
in ch p
n-color pfion enat by A(u).
PROOF OF OM 1. L H
m 2i 1. m if we iore e subscrip of all
of i m mint be d. Bemme mi m=_,+]
of u into odd p. Convsely, if we mider
p in H, we get a ique ordi
on
or
d rh ch 2a-1 by (2a-1) we g a uque nition
pfion ofu imo odd
enat by A (u). h bijeon prov eor 1.
PROOF OF OM 2. L a
n-color pfion at byC(). is impli at th
p of a e ei seff-conjugam or ey app pm of mum1y conjugam p. It w obsewed in
seff-conjugam if it is odd. Also, e humor of pairs of mutually
e proof of eor at a p
njugam p pong to y en imer2n In/2]. e gen toger prove eorem 2
sho in [1] atC() of rem 2 ao eqs e nber of seic ple
RE It
pfion of
FENCES
1. A.K. Ag, Prdings ofRCOM (l, 35M6, As Ademy ofe.
.