Narrow Results

We define the generalized vertex operators expressed by charged fermions and state vectors corresponding to the n-partition. The action of the generalized vertex operators on state vectors and the generalized universal character (UC) plane partitions have been investigated by the fermion calculus techniques. Furthermore, we construct the infinite scalar product and the generating function associated with the generalized UC plane partitions.

We construct Fock and MacMahon modules for the quantum toroidal superalgebra ${{ℰ}}_s$ associated with the Lie superalgebra ${𝔤𝔩}_{m|n}$ and parity $s$. The bases of the Fock and MacMahon modules are labeled by super-analogs of partitions and plane partitions with various boundary conditions, while the action of generators of ${{ℰ}}_s$ is given by Pieri-type formulas. We study the corresponding characters.

In 2021 da Silva, Hirschhorn, and Sellers studied a wide variety of congruences for the $k$-elongated plane partition function $d_k(n)$ by various primes. They also conjectured the existence of an infinite congruence family modulo arbitrarily high powers of 2 for the function $d_7(n)$. We prove that such a congruence family exists — indeed, for powers of 8. The proof utilizes only classical methods, i.e. integer polynomial manipulations in a single function, in contrast to all other known infinite congruence families for $d_k(n)$ which require more modern methods to prove.

In this paper, we provide an algorithm to detect linear congruences of $p{l}_k(n)$, the number of MacMahon’s $k$-rowed plane partitions, and give a quantitative result on the nonexistence of Ramanujan-type congruences of the $k$-rowed plane partition functions. We also show $p(n,m)$ that the number of partitions at most $m$ parts always admits linear congruences.