cp's OEIS Frontend

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.

%I A347767 #7 Sep 16 2021 03:47:28
%S A347767 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,
%T A347767 73,78,80,83,86,87,89,93,1,3,4,5,17,18,22,23,24,25,27,28,29,37,38,42,
%U A347767 43,46,48,51,52,53,56,58,61,62,66,67,72,100,101,102,103,106
%N 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.
%C A347767 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.
%C A347767 A347766 gives the table of positive Euler permutations. Related sequences are A347599 (Genocchi permutations) and A347600 (Seidel permutations).
%e A347767 Table of negative Euler permutations, length of rows is A347602:
%e A347767 [0] 0;
%e A347767 [1] 0;
%e A347767 [2] 1;
%e A347767 [3] 0;
%e A347767 [4] 2, 7;
%e A347767 [5] 4, 5, 6, 7, 10, 12, 19, 20, 27, 31, 33, 43, 44, 47, 49, ...
%e A347767 .
%e A347767 The first 8 permutations corresponding to the ranks are for n = 5:
%e A347767     4 -> [12453],  5 -> [12534],  6 -> [12543],  7 -> [13245],
%e A347767    10 -> [13452], 12 -> [13542], 19 -> [15234], 20 -> [15243].
%p A347767 # Uses function EulerPermutationsRank from A347766.
%p A347767 A347767Row := n -> `if`(n < 4, [[0,0,1,0][n+1]], EulerPermutationsRank(n, 'neg')): for n from 0 to 6 do A347767Row(n) od;
%Y A347767 Cf. A347601, A347602, A346831, A347766, A347599, A347600.
%K A347767 nonn,tabf
%O A347767 0,5
%A A347767 _Peter Luschny_, Sep 12 2021