Partitions with designated summands: $k$-tuples and their congruences

arXiv:2611.05623v1 Announce Type: new Abstract: Let $A_k(n)$ and $B_k(n)$ be the numbers of ordered $k$-tuples of partitions with designated summands, respectively with designated summands and all parts odd, of total size $n$. Using an exact description of the generating function of designated partitions after a $3$-dissection, we classify for every $e$ the values of $k$ for which $A_k(3n+2)$ is divisible by $3^e$: this happens exactly when $k\equiv0$ or $-5\pmod{3^e}$ if $e\le4$, and when $3^e\mid k$ if $e\ge5$. We obtain the corresponding exact $3$-adic and several $2$-adic results for $B_k$ and for $A_k$ along the progressions $4n+3$, a recursion for the classes $3^b(3n+s)$, and for $k=2$ the congruences $A_2(12n+10)\equiv 27f_1^4f_6^3/f_2\pmod{81}$ in generating-function form. We relate the numbers $A_k(n)$ modulo $3$ and $9$ to overpartitions, sums of squares and triangular numbers, give group-action proofs of some of the congruences, and close with several conjectures supported by computation.

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