Congruences for Andrews ’ Spt - Function modulo Powers of 5 , 7 and 13
نویسندگان
چکیده
Abstract. Congruences are found modulo powers of 5, 7 and 13 for Andrews’ smallest parts partition function spt(n). These congruences are reminiscent of Ramanujan’s partition congruences modulo powers of 5, 7 and 11. Recently, Ono proved explicit Ramanujan-type congruences for spt(n) modulo for all primes ≥ 5 which were conjectured earlier by the author. We extend Ono’s method to handle the powers of 5, 7 and 13 congruences. We need the theory of weak Maass forms as well as certain classical modular equations for the Dedekind eta-function.
منابع مشابه
Congruences for Andrews’ Spt-function
Congruences are found modulo powers of 5, 7 and 13 for Andrews’ smallest parts partition function spt(n). These congruences are reminiscent of Ramanujan’s partition congruences modulo powers of 5, 7 and 11. Recently, Ono proved explicit Ramanujan-type congruences for spt(n) modulo ` for all primes ` ≥ 5 which were conjectured earlier by the author. We extend Ono’s method to handle the powers of...
متن کاملCongruences for Andrews' spt-Function Modulo 32760 and Extension of Atkin's Hecke-Type Partition Congruences
New congruences are found for Andrews’ smallest parts partition function spt(n). The generating function for spt(n) is related to the holomorphic part α(24z) of a certain weak Maass form M(z) of weight 3 2 . We show that a normalized form of the generating function for spt(n) is an eigenform modulo 72 for the Hecke operators T (`2) for primes ` > 3, and an eigenform modulo p for p = 5, 7 or 13 ...
متن کاملCombinatorial Interpretations of Congruences for the Spt-function
Let spt(n) denote the total number of appearances of the smallest parts in all the partitions of n. In 1988, the second author gave new combinatorial interpretations of Ramanujan’s partition congruences mod 5, 7 and 11 in terms of a crank for weighted vector partitions. In 2008, the first author found Ramanujantype congruences for the spt-function mod 5, 7 and 13. We give new combinatorial inte...
متن کاملThe Andrews–Sellers family of partition congruences
In 1994, James Sellers conjectured an infinite family of Ramanujan type congruences for 2-colored Frobenius partitions introduced by George E. Andrews. These congruences arise modulo powers of 5. In 2002 Dennis Eichhorn and Sellers were able to settle the conjecture for powers up to 4. In this article, we prove Sellers' conjecture for all powers of 5. In addition, we discuss why the Andrews-Sel...
متن کاملCongruences for the Andrews spt function.
Ramanujan-type congruences for the Andrews spt(n) partition function have been found for prime moduli 5 ≤ ℓ ≤ 37 in the work of Andrews [Andrews GE, (2008) J Reine Angew Math 624:133-142] and Garvan [Garvan F, (2010) Int J Number Theory 6:1-29]. We exhibit unexpectedly simple congruences for all ℓ≥5. Confirming a conjecture of Garvan, we show that if ℓ≥5 is prime and (-δ/ℓ) = 1, then spt[(ℓ2(ℓn...
متن کامل