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A347600 Irregular table read by rows, T(n, k) is the rank of the k-th Seidel permutation of {1,...,n}, permutations sorted in lexicographical order.

%I A347600 #7 Sep 09 2021 09:38:24
%S A347600 2,11,17,187,211,307,331,451,452,571,572,6937,7057,7657,7777,8497,
%T A347600 8498,9217,9218,11977,12097,12697,12817,13537,13538,14257,14258,17737,
%U A347600 17739,17857,17859,18577,18578,18579,18580,19297,19298,19299,19300,22777,22779,22897
%N A347600 Irregular table read by rows, T(n, k) is the rank of the k-th Seidel permutation of {1,...,n}, permutations sorted in lexicographical order.
%C A347600 Let M be the 2n X 2n matrix with M(j, k) = floor((2*j - k - 1) / 2*n). A Seidel permutation of order n is a permutation sigma of {1,...,2n} if Product_{k=1..2n} M(k, sigma(k)) does not vanish.
%C A347600 Let P(n) denote the number of Seidel permutations of order n. We conjecture that P(n) = A005439(n). This conjecture was inspired by the conjecture of _Zhi-Wei Sun_ in A036968. The name 'Seidel permutations' follows a comment of _Don Knuth_: "The earliest known reference for these numbers (A005439) is Seidel ...."
%C A347600 The related sequence A347599 lists Genocchi permutations.
%e A347600 Table starts:
%e A347600 [1] 2;
%e A347600 [2] 11, 17;
%e A347600 [3] 187, 211, 307, 331, 451, 452, 571, 572.
%e A347600 .
%e A347600 The 8 permutations corresponding to the ranks are for n = 3:
%e A347600 187 -> [246135]; 211 -> [256134]; 307 -> [346125]; 331 -> [356124];
%e A347600 451 -> [456123]; 452 -> [456132]; 571 -> [546123]; 572 -> [546132].
%o A347600 (Julia)
%o A347600 function SeidelPermutations(n)
%o A347600     f(m) = m >= 2n ? 1 : m < 0 ? -1 : 0
%o A347600     Mat(n) = [[f(2*j - k - 1) for k in 1:2n] for j in 1:2n]
%o A347600     M = Mat(n); P = permutations(1:2n); R = Int64[]
%o A347600     S, rank = 0, 1
%o A347600     for p in P
%o A347600         m = prod(M[k][p[k]] for k in 1:2n)
%o A347600         if m != 0
%o A347600             S += m
%o A347600             push!(R, rank)
%o A347600         end
%o A347600         rank += 1
%o A347600     end
%o A347600     # println(n, " -> ", (-1)^n*S)
%o A347600     return R
%o A347600 end
%o A347600 for n in 1:5 println(SeidelPermutations(n)) end
%Y A347600 Cf. A005439, A036968, A347599.
%K A347600 nonn,tabf
%O A347600 1,1
%A A347600 _Peter Luschny_, Sep 08 2021