Rule based lexicographical permutation sequences

In a permutation sequence built by means of sub permutations, the transitions between successive permutations are subject to a set of n(n – 1)/2 rules that naturally group into n – 1 matrices with a high degree of regularity. By means of these rules, the sequence can be produced in O(3n!) time and O(n) space. To generate all permutations of a given set one can go for (among other things) a minimum of changes from one permutation to the next [1], [2], [4], or lexicographic order, at the cost of having to perform more changes [3]. Here, we go for the latter. A lexicographic order can be established by recursively producing successively larger sub-permutations, starting from the right, as in the following example on six elements. This makes the permuted elements move back and forth according to a set of rules, from one permutation to the next. 1 ( a b c d e f ) 705 ( f e b c a d ) 2 ( a b c d f e ) 706 ( f e b c d a ) 3 ( a b c e d f ) 707 ( f e b d a c ) 4 ( a b c e f d ) 708 ( f e b d c a ) 5 ( a b c f d e ) 709 ( f e c a b d ) 6 ( a b c f e d ) 710 ( f e c a d b ) 7 ( a b d c e f ) 711 ( f e c b a d ) 8 ( a b d c f e ) 712 ( f e c b d a ) 9 ( a b d e c f ) ... 713 ( f e c d a b ) 10 ( a b d e f c ) 714 ( f e c d b a ) 11 ( a b d f c e ) 715 ( f e d a b c ) 12 ( a b d f e c ) 716 ( f e d a c b ) 13 ( a b e c d f ) 717 ( f e d b a c ) 14 ( a b e c f d ) 718 ( f e d b c a ) 15 ( a b e d c f ) 719 ( f e d c a b ) 16 ( a b e d f c ) 720 ( f e d c b a ) Here is one way of producing the sequence: Make a 6  6 matrix M with identical rows = [a b c d e f], take the downwards diagonal D and the 5  6 matrix N = M \ D, make the permutations of each row in N recursively and prepend D[i] to each permutation of N[i]. The principle is illustrated in Figure 1, with four instead of six elements.