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

1,1

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.

Links

Table of n, a(n) for n=1..42.

Example

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].

Prog

(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

Crossrefs

Cf. A005439, A036968, A347599.

Sequence in context: A176985 A089592 A106982 * A043461 A030551 A121848

Adjacent sequences: A347597 A347598 A347599 * A347601 A347602 A347603

Keyword

nonn,tabf

Author

Peter Luschny, Sep 08 2021

Status

approved