Arithmetic properties of $5$-regular partitions into distinct parts

A partition is said to be $\mathrm{\ell }$-regular if none of its parts is a multiple of $\mathrm{\ell }$. Let $b_5^{\mathrm{\prime }}(n)$ denote the number of 5-regular partitions into distinct parts (equivalently, into odd parts) of $n$. This function has also close connections to representation theory and combinatorics. In this paper, we study arithmetic properties of $b_5^{\mathrm{\prime }}(n)$. We provide full characterization of the parity of $b_5^{\mathrm{\prime }}(2n+1)$, present several congruences modulo 4, and prove that the generating function of the sequence $(b_5^{\mathrm{\prime }}(5n+1))$ is lacunary modulo any arbitrary positive powers of 5.