According to the definition given on page 449 of the book Discriminants, Resultants, and Multidimensional Determinants, the output of this method is $s'(M)$, where $s'$ is the inverse permutation of $s$.
i1 : M = genericMultidimensionalMatrix {4,3,2}
o1 = {{{a , a }, {a , a }, {a , a }}, {{a , a }, {a , a }, {a , a }}, {{a ,
0,0,0 0,0,1 0,1,0 0,1,1 0,2,0 0,2,1 1,0,0 1,0,1 1,1,0 1,1,1 1,2,0 1,2,1 2,0,0
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a }, {a , a }, {a , a }}, {{a , a }, {a , a }, {a , a }}}
2,0,1 2,1,0 2,1,1 2,2,0 2,2,1 3,0,0 3,0,1 3,1,0 3,1,1 3,2,0 3,2,1
o1 : 3-dimensional matrix of shape 4 x 3 x 2 over ZZ[a ..a ]
0,0,0 3,2,1
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i2 : permute(M,{1,0,2})
o2 = {{{a , a }, {a , a }, {a , a }, {a , a }}, {{a , a }, {a , a }, {a ,
0,0,0 0,0,1 1,0,0 1,0,1 2,0,0 2,0,1 3,0,0 3,0,1 0,1,0 0,1,1 1,1,0 1,1,1 2,1,0
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a }, {a , a }}, {{a , a }, {a , a }, {a , a }, {a , a }}}
2,1,1 3,1,0 3,1,1 0,2,0 0,2,1 1,2,0 1,2,1 2,2,0 2,2,1 3,2,0 3,2,1
o2 : 3-dimensional matrix of shape 3 x 4 x 2 over ZZ[a ..a ]
0,0,0 3,2,1
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i3 : permute(M,{2,0,1})
o3 = {{{a , a , a }, {a , a , a }, {a , a , a }, {a , a , a }}, {{a , a ,
0,0,0 0,1,0 0,2,0 1,0,0 1,1,0 1,2,0 2,0,0 2,1,0 2,2,0 3,0,0 3,1,0 3,2,0 0,0,1 0,1,1
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a }, {a , a , a }, {a , a , a }, {a , a , a }}}
0,2,1 1,0,1 1,1,1 1,2,1 2,0,1 2,1,1 2,2,1 3,0,1 3,1,1 3,2,1
o3 : 3-dimensional matrix of shape 2 x 4 x 3 over ZZ[a ..a ]
0,0,0 3,2,1
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The object permute is a method function.