A347767 Irregular table read by rows, T(n, k) is the rank of the k-th negative Euler permutation of {1,...,n}, permutations sorted in lexicographical order. If no such permutation exists, then T(n, 0) = 0 by convention.

0, 0, 1, 0, 2, 7, 4, 5, 6, 7, 10, 12, 19, 20, 27, 31, 33, 43, 44, 47, 49, 52, 54, 56, 59, 69, 73, 78, 80, 83, 86, 87, 89, 93, 1, 3, 4, 5, 17, 18, 22, 23, 24, 25, 27, 28, 29, 37, 38, 42, 43, 46, 48, 51, 52, 53, 56, 58, 61, 62, 66, 67, 72, 100, 101, 102, 103, 106

Offset

0,5

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 negative Euler permutation of order n if P = -1. See A347601 for further details.

A347766 gives the table of positive Euler permutations. Related sequences are A347599 (Genocchi permutations) and A347600 (Seidel permutations).

Links

Table of n, a(n) for n=0..67.

Example

Table of negative Euler permutations, length of rows is A347602:
[0] 0;
[1] 0;
[2] 1;
[3] 0;
[4] 2, 7;
[5] 4, 5, 6, 7, 10, 12, 19, 20, 27, 31, 33, 43, 44, 47, 49, ...
.
The first 8 permutations corresponding to the ranks are for n = 5:
    4 -> [12453],  5 -> [12534],  6 -> [12543],  7 -> [13245],
   10 -> [13452], 12 -> [13542], 19 -> [15234], 20 -> [15243].

Maple

# Uses function EulerPermutationsRank from A347766.
A347767Row := n -> `if`(n < 4, [[0, 0, 1, 0][n+1]], EulerPermutationsRank(n, 'neg')): for n from 0 to 6 do A347767Row(n) od;

Crossrefs

Cf. A347601, A347602, A346831, A347766, A347599, A347600.

Sequence in context: A049249 A336000 A222237 * A325644 A198036 A316249

Adjacent sequences: A347764 A347765 A347766 * A347768 A347769 A347770

Keyword

nonn,tabf

Author

Peter Luschny, Sep 12 2021

Status

approved