A347766 Irregular table read by rows, T(n, k) is the rank of the k-th positive Euler permutation of {1,...,n}, permutations sorted in lexicographical order. If no such permutation exists, then T(n, 0) = 0 by convention.

1, 0, 0, 2, 3, 1, 6, 8, 11, 14, 15, 17, 3, 8, 24, 28, 29, 30, 32, 35, 50, 55, 57, 68, 71, 74, 79, 92, 2, 6, 15, 16, 21, 26, 30, 40, 44, 54, 55, 60, 68, 99, 104, 120, 121, 123, 124, 125, 137, 138, 142, 143, 144, 146, 150, 161, 164, 167, 174, 175, 177, 179, 185

Offset: 0

Author

Peter Luschny, Sep 12 2021

Keywords

• nonn
• tabf

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.

A347767 gives the table of negative Euler permutations. Related sequences are A347599 (Genocchi permutations) and A347600 (Seidel permutations).

Examples

			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].
		

Crossrefs

Cf. A347601, A346831, A347767, A347599, A347600.

Programs

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;