TY - JOUR

T1 - Congruences modulo squares of primes for FU'S k dots bracelet partitions

AU - Radu, Cristian Silviu

AU - Sellers, James A.

N1 - Funding Information:
C.-S. Radu was funded by the Austrian Science Fund (FWF), W1214-N15, project DK6 and by grant P2016-N18. The research was supported by the strategic program “Innovatives OÖ 2010 plus” by the Upper Austrian Government.

PY - 2013/6

Y1 - 2013/6

N2 - In 2007, Andrews and Paule introduced the family of functions Δk(n) which enumerate the number of broken k-diamond partitions for a fixed positive integer k. In that paper, Andrews and Paule proved that, for all n ≥ 0, Δ1(2n+1) ≡ 0 (mod 3) using a standard generating function argument. Soon after, Shishuo Fu provided a combinatorial proof of this same congruence. Fu also utilized this combinatorial approach to naturally define a generalization of broken k-diamond partitions which he called k dots bracelet partitions. He denoted the number of k dots bracelet partitions of n by k(n) and proved various congruence properties for these functions modulo primes and modulo powers of 2. In this note, we extend the set of congruences proven by Fu by proving the following congruences: For all n ≥ 0, $$\begin{array}{r@{}cl}\mathfrak{B}-5(10n+7) &\equiv& 0 \pmod{5 2},\\[4pt]\mathfrak{B}-7(14n+11) &\equiv& 0 \pmod{7 2}, \quad {\rm and}\\[4pt]\mathfrak{B}-{11}(22n+21) &\equiv& 0 \pmod{11 2}\end{array}$$ We also conjecture an infinite family of congruences modulo powers of 7 which are satisfied by the function 7.

AB - In 2007, Andrews and Paule introduced the family of functions Δk(n) which enumerate the number of broken k-diamond partitions for a fixed positive integer k. In that paper, Andrews and Paule proved that, for all n ≥ 0, Δ1(2n+1) ≡ 0 (mod 3) using a standard generating function argument. Soon after, Shishuo Fu provided a combinatorial proof of this same congruence. Fu also utilized this combinatorial approach to naturally define a generalization of broken k-diamond partitions which he called k dots bracelet partitions. He denoted the number of k dots bracelet partitions of n by k(n) and proved various congruence properties for these functions modulo primes and modulo powers of 2. In this note, we extend the set of congruences proven by Fu by proving the following congruences: For all n ≥ 0, $$\begin{array}{r@{}cl}\mathfrak{B}-5(10n+7) &\equiv& 0 \pmod{5 2},\\[4pt]\mathfrak{B}-7(14n+11) &\equiv& 0 \pmod{7 2}, \quad {\rm and}\\[4pt]\mathfrak{B}-{11}(22n+21) &\equiv& 0 \pmod{11 2}\end{array}$$ We also conjecture an infinite family of congruences modulo powers of 7 which are satisfied by the function 7.

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U2 - 10.1142/S1793042113500073

DO - 10.1142/S1793042113500073

M3 - Article

AN - SCOPUS:84877645162

SN - 1793-0421

VL - 9

SP - 939

EP - 943

JO - International Journal of Number Theory

JF - International Journal of Number Theory

IS - 4

ER -