A conjectural basis for the $(1,2)$-bosonic-fermionic coinvariant ring

We give the first conjectural construction of a monomial basis for the coinvariant ring Rₙ(1,2), for the symmetric group Sₙ acting on one set of bosonic (commuting) and two sets of fermionic (anticommuting) variables. Our construction interpolates between the modified Motzkin path basis for Rₙ(0,2) of Kim-Rhoades (2022) and the super-Artin basis for Rₙ(1,1) conjectured by Sagan-Swanson (2024) and proven by Angarone et al. (2024). We prove that our proposed basis has cardinality 2ⁿ⁻¹n!, aligning with a conjecture of Zabrocki (2020) on the dimension of Rₙ(1,2), and show how it gives a combinatorial expression for the Hilbert series. We also conjecture a Frobenius series for Rₙ(1,2). We show that these proposed Hilbert and Frobenius series are equivalent to conjectures of Iraci, Nadeau, and Vanden Wyngaerd (2023) on Rₙ(1,2) in terms of segmented Smirnov words, by exhibiting a weight-preserving bijection between our proposed basis and their segmented permutations. We extend some of their results on the sign character to hook characters, and give a formula for the m_μ coefficients of the conjectural Frobenius series. Finally, we conjecture a monomial basis for the analogous ring in type Bₙ, and show that it has cardinality 4ⁿn!.