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Published online by Cambridge University Press: 23 December 2022
A partition of a positive integer n is called $\ell $-regular if none of its parts is divisible by
$\ell $. Denote by
$b_{\ell }(n)$ the number of
$\ell $-regular partitions of n. We give a complete characterisation of the arithmetic of
$b_{23}(n)$ modulo
$11$ for all n not divisible by
$11$ in terms of binary quadratic forms. Our result is obtained by establishing a relation between the generating function for these values of
$b_{23}(n)$ and certain modular forms having complex multiplication by
${\mathbb Q}(\sqrt {-69})$.