OFFSET
0,4
COMMENTS
Let M be the tangent matrix of dimension n X n. The definition of a tangent matrix is given in A346831. An Euler permutation of order n is a permutation sigma of {1,...,n} if P = Product_{k=1..n} M(k, sigma(k)) does not vanish. We say sigma is a positive Euler permutation of order n if P = 1. See A347601 for further details.
EXAMPLE
Table of positive Euler permutations, length of rows is A347601:
[0] 1;
[1] 0;
[2] 0;
[3] 2, 3;
[4] 1, 6, 8, 11, 14, 15, 17;
[5] 3, 8, 24, 28, 29, 30, 32, 35, 50, 55, 57, 68, 71, 74, 79, 92.
.
The 16 permutations corresponding to the ranks are for n = 5:
3 -> [12435], 8 -> [13254], 24 -> [15432], 28 -> [21453],
29 -> [21534], 30 -> [21543], 32 -> [23154], 35 -> [23514],
50 -> [31254], 55 -> [32145], 57 -> [32415], 68 -> [35142],
71 -> [35412], 74 -> [41253], 79 -> [42135], 92 -> [45132].
MAPLE
# Uses function TangentMatrix from A346831.
EulerPermutationsRank := proc(n, sgn) local M, P, N, s, p, m, rank;
M := TangentMatrix(n); P := []; N := []; rank := 0;
for p in Iterator:-Permute(n) do
rank := rank + 1;
m := mul(M[k, p(k)], k = 1..n);
if m = 0 then next fi;
if m = 1 then P := [op(P), rank] fi;
if m = -1 then N := [op(N), rank] fi; od;
if sgn = 'pos' then P else N fi end:
A347766Row := n -> `if`(n < 3, [[1, 0, 0][n+1]], EulerPermutationsRank(n, 'pos')):
for n from 0 to 5 do A347766Row(n) od;
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Peter Luschny, Sep 12 2021
STATUS
approved