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.

2, 11, 17, 187, 211, 307, 331, 451, 452, 571, 572, 6937, 7057, 7657, 7777, 8497, 8498, 9217, 9218, 11977, 12097, 12697, 12817, 13537, 13538, 14257, 14258, 17737, 17739, 17857, 17859, 18577, 18578, 18579, 18580, 19297, 19298, 19299, 19300, 22777, 22779, 22897

Offset: 1

Author

Peter Luschny, Sep 08 2021

Keywords

• nonn
• tabf

Comments

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.

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 ...."

The related sequence A347599 lists Genocchi permutations.

Examples

			Table starts:
[1] 2;
[2] 11, 17;
[3] 187, 211, 307, 331, 451, 452, 571, 572.
.
The 8 permutations corresponding to the ranks are for n = 3:
187 -> [246135]; 211 -> [256134]; 307 -> [346125]; 331 -> [356124];
451 -> [456123]; 452 -> [456132]; 571 -> [546123]; 572 -> [546132].
		

Crossrefs

Cf. A005439, A036968, A347599.

Programs

Julia

function SeidelPermutations(n)
    f(m) = m >= 2n ? 1 : m < 0 ? -1 : 0
    Mat(n) = [[f(2*j - k - 1) for k in 1:2n] for j in 1:2n]
    M = Mat(n); P = permutations(1:2n); R = Int64[]
    S, rank = 0, 1
    for p in P
        m = prod(M[k][p[k]] for k in 1:2n)
        if m != 0
            S += m
            push!(R, rank)
        end
        rank += 1
    end
    # println(n, " -> ", (-1)^n*S)
    return R
end
for n in 1:5 println(SeidelPermutations(n)) end