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 2^n-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_ 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!.