Congruences modulo powers of 3 for k-colored partitions

Xin-Qi Wen ()
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Xin-Qi Wen: Taiyuan University

Indian Journal of Pure and Applied Mathematics, 2025, vol. 56, issue 1, 324-338

Abstract: Abstract We study congruence properties of k-colored partition $$p_{-k}(n)$$ p - k ( n ) when $$k=9t+3$$ k = 9 t + 3 and $$k=9t+6$$ k = 9 t + 6 for some integers $$t\ge 0$$ t ≥ 0 . In particular, we establish some dissection formulas for $$p_{-(9t+3)}(n)$$ p - ( 9 t + 3 ) ( n ) when $$t\in \{3r, 3r+1, 3r+2\}$$ t ∈ { 3 r , 3 r + 1 , 3 r + 2 } , $$r=0,1$$ r = 0 , 1 , and for $$p_{-(9t+6)}(n)$$ p - ( 9 t + 6 ) ( n ) where $$t\in \{3r, 3r+1, 3r+2\}$$ t ∈ { 3 r , 3 r + 1 , 3 r + 2 } , $$r=0$$ r = 0 . Furthermore, we prove some infinite families of congruences modulo powers of 3 for them. For example, for partition $$p_{-(9t+3)}(n)$$ p - ( 9 t + 3 ) ( n ) when $$t=1$$ t = 1 , we prove that for any integer $$n\ge 0$$ n ≥ 0 and $$\alpha \ge 1$$ α ≥ 1 , $$\begin{aligned} p_{-12}\left( 3^{\alpha }n+\frac{3^{\alpha }+1}{2}\right) \equiv 0 \pmod {3^{\alpha +1}}, \end{aligned}$$ p - 12 3 α n + 3 α + 1 2 ≡ 0 ( mod 3 α + 1 ) , and $$\begin{aligned} p_{-12}\left( 3^{\alpha +1}n+\frac{5\times 3^{\alpha }+1}{2}\right) \equiv 0 \pmod {3^{\alpha +2}}. \end{aligned}$$ p - 12 3 α + 1 n + 5 × 3 α + 1 2 ≡ 0 ( mod 3 α + 2 ) .

Keywords: Congruences; k-Colored partitions; Dissection formulas; 05A17; 11P83 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s13226-023-00483-2

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