Inequivalent Z₂ⁿ-graded brackets, n-bit parastatistics and statistical transmutations of supersymmetric quantum mechanics

Given an associative ring of Z₂ⁿ-graded operators, the number of inequivalent brackets of Lie-type which are compatible with the grading and satisfy graded Jacobi identities is b_n= n+lfloor n/2rfloor+1. This follows from the Rittenberg-Wyler and Scheunert analysis of "color" Lie (super)algebras which is revisited here in terms of Boolean logic gates. The inequivalent brackets, recovered from Z₂^ntimes Z₂ⁿ→ Z₂ mappings, are defined by consistent sets of commutators/anticommutators describing particles accommodated into an n-bit parastatistics (ordinary bosons/fermions correspond to 1 bit). Depending on the given graded Lie (super)algebra, its graded sectors can fall into different classes of equivalence expressing different types of (para)bosons and/or (para)fermions. As a first application we construct Z₂² and Z₂³-graded quantum Hamiltonians which respectively admit b₂=4 and b₃=5 inequivalent multiparticle quantizations (the inequivalent parastatistics are discriminated by measuring the eigenvalues of certain observables in some given states). As a main physical application we prove that the N-extended, 1D supersymmetric and superconformal quantum mechanics, for N=1,2,4,8, are respectively described by s_N=2,6,10,14 alternative formulations based on the inequivalent graded Lie (super)algebras. These numbers correspond to all possible "statistical transmutations" of a given set of supercharges which, for {N}=1,2,4,8, are accommodated into a Z₂ⁿ-grading with n=1,2,3,4 the identification is $N= 2^n-1$. In the simplest {N}=2 setting (the 2-particle sector of the de DFF deformed oscillator with sl(2|1) spectrum-generating superalgebra), the Z₂²-graded parastatistics imply a degeneration of the energy levels which cannot be reproduced by ordinary bosons/fermions statistics.