  The magnetic correlations in the austenite phase and the consequent
martensitic transition in inverse magnetocaloric alloys,
Ni$_{2}$Mn$_{1+x}$Sn$_{1-x}$, have been a matter of debate for decades. We
conclusively establish using {\it ab initio} phonon calculations that the spin
alignment of excess Mn at the Sn site (Mn$_{Sn}$) with the existing Mn in the
unit cell in the high temperature cubic phase of Ni-Mn-Sn alloy is
ferromagnetic (FM), and not ferrimagnetic (FI), resolving a long lasting
controversy. Using first principles density functional perturbation theory
(DFPT), we observe an instability of the TA$_{2}$ mode along the $\Gamma$-M
direction in the FM phase, very similar to that observed in the prototypical
ferromagnetic shape memory alloy (FSMA) Ni$_{2}$MnGa. This specific instability
is not observed in the FI phase. Further finite temperature first principles
lattice dynamics calculations reveal that at 300 K the FM phase becomes
mechanically stable, while the FI phase continue to remain unstable providing
credence to the fact that the high-temperature phase has FM order. These
results will be primordial to understand the magneto-structural properties of
this class of compounds.
