ISSN:
1572-9303
Keywords:
partitions
;
congruences
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Let $$k = p_1^{a_1 } p_2^{a_2 } \cdot \cdot \cdot p_m^{a_m } $$ be the prime factorization of a positive integer k and let b k (n) 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 S k (N; M) be the number of positive integers ≤ N for which b k(n )≡ 0(mod M). If $$p_i^{a_i } \geqslant \sqrt k $$ we prove that, for every positive integer j $$\mathop {\lim }\limits_{N \to \infty } \frac{{S_k (N;p_i^j )}} {N} = 1. $$ In other words for every positive integer j, b k(n) is a multiple of $$p_i^j $$ for almost every non-negative integer n. In the special case when k=p is prime, then in representation-theoretic terms this means that the number ofp -modular irreducible representations of almost every symmetric groupS n is a multiple of p j. We also examine the behavior of b k(n) (mod $$p_i^j $$ ) where the non-negative integers n belong to an arithmetic progression. Although almost every non-negative integer n≡ (mod t) satisfies b k(n) ≡ 0 (mod $$p_i^j $$ ), we show that there are infinitely many non-negative integers n≡ r (mod t) for which b k(n) ≢ 0 (mod $$p_i^j $$ ) provided that there is at least one such n. Moreover the smallest such n (if there are any) is less than 2 $$\cdot 10^8 p_i^{a_i + j - 1} k^2 t^4 $$ .
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1009711020492
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