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

0,4

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.

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

Links

Table of n, a(n) for n=0..62.

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

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

Sequence in context: A092271 A054115 A100822 * A198427 A289905 A086211

Adjacent sequences: A347763 A347764 A347765 * A347767 A347768 A347769

Keyword

nonn,tabf

Author

Peter Luschny, Sep 12 2021

Status

approved