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Partition theory abounds with bijections between different types ofpartitions. One of the most famous partition bijections maps eachself-conjugate partition of a positive integer $n$ to a partition of $n$ intodistinct odd parts, and vice versa. Here we prove new necessary and sufficientconditions for a self-conjugate partition to be $t$-core, in terms of only theparts of the corresponding partition into distinct odd parts, by proving a newhook length formula. Corollaries of these results include new applications of$t$-core self-conjugate partitions to subsets of the natural numbers, due tothe recent investigation of a new partition statistic called the supernorm bythe first author, Just, and Schneider, as well as many results on $t$-cores byBringmann, Kane, Males, Ono, Raji, and others. We provide several examples ofthese applications, one of which gives a new formula for certain families ofHurwitz class numbers.

Self-conjugate $t$-core partitions and applications