BASIL GORDON
Department of Mathematics, The University of California
at Los Angeles, Los Angeles, California 90024
bg@ sonia.math.ucla.edu
KEN ONO
School of Mathematics, Institute for Advanced Study,
Princeton, New Jersey 08540; and Department of Mathematics,
Penn State University, University Park, Pennsylvannia 16802
ono@math.ias.edu
Dedicated to the memory of Nathan Fine
Received June 21, 1995; Accepted January 25, 1996
Abstract. Let 
be the
prime factorization of a positive integer k and
let 
denote the number of partitions of a non-negative integer n
into parts none of
which are multiples of k.
If M is a positive integer, let 
be the number
of positive integers 
for which 
If 
we prove that, for every positive integer
j
In other words for every positive integer j, 
is a multiple of
for almost every non-negative integer 
In the special case when k=p is prime, then
in representation-theoretic terms this means that the number of
p-modular irreducible representations of almost every symmetric group
is a multiple of 
We also examine the behavior
of 
where
the non-negative integers n belong to an arithmetic
progression. Although almost every non-negative integer 
satisfies 
,
we show that there are infinitely many non-negative integers
for which 
provided that
there is at least one such 
Moreover the smallest such n (if there
are any) is less than 
Keywords: partitions, congruences
1991 Mathematics Classification: Primary 11P83. Secondary 05A17