A347599 Irregular table read by rows, T(n, k) is the rank of the k-th Genocchi permutation of {1,...,n}, permutations sorted in lexicographical order. If no Genocchi permutation of {1,...,n} exists, then T(n, 1) = 0 by convention.

1, 0, 5, 0, 67, 91, 92, 0, 1897, 2017, 2018, 2617, 2619, 2737, 2738, 2739, 2740, 3457, 3458, 3459, 3460, 4177, 4178, 4179, 4180, 0, 99241, 99961, 99962, 104281, 104283, 105001, 105002, 105003, 105004, 110041, 110042, 110043, 110044, 115081, 115082, 115083

Offset

1,3

Comments

Let M be the n X n matrix with M(j, k) = floor((2*j - k ) / n). A Genocchi permutation of order n is a permutation sigma of {1,...,n} if Product_{k=1..n} M(k, sigma(k)) does not vanish.

Let P(n) denote the number of Genocchi permutations of order n. Zhi-Wei Sun conjectured, using permanents, that P(n - 1) = G(n), where G(n) are the Genocchi numbers A036968. From the well-known relation between Genocchi and Bernoulli numbers this implies, assuming the conjecture:

Bernoulli(n) = P(n - 1) / ((-1)^floor(n/2)*(2^(n + 2) - 2)) for n >= 2.

The related sequence A347600 lists Seidel permutations.

Links

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

Zhi-Wei Sun, A novel identity connecting permanents to Bernoulli numbers, MathOverflow 2021-09-07.

Example

Table starts:
[1] 1;
[2] 0;
[3] 5;
[4] 0;
[5] 67, 91, 92;
[6] 0;
[7] 1897, 2017, 2018, 2617, 2619, 2737, 2738, 2739, 2740, 3457, 3458, 3459, 3460, 4177, 4178, 4179, 4180;
.
The 17 permutations corresponding to the ranks are for n = 7:
1897 -> [3571246]; 2017 -> [3671245]; 2018 -> [3671254]; 2617 -> [4571236];
2619 -> [4571326]; 2737 -> [4671235]; 2738 -> [4671253]; 2739 -> [4671325];
2740 -> [4671352]; 3457 -> [5671234]; 3458 -> [5671243]; 3459 -> [5671324];
3460 -> [5671342]; 4177 -> [6571234]; 4178 -> [6571243]; 4179 -> [6571324];
4180 -> [6571342].
.
17 / (-510) = -1/30 = Bernoulli(8).

Prog

(Julia)
using Combinatorics
function GenocchiPermutations(n)
    f(m) = m >= n ? 1 : m < 0 ? -1 : 0
    Mat(n) = [[f(2*j - k) for k in 1:n] for j in 1:n]
    M = Mat(n); P = permutations(1:n); R = Int64[]
    S, rank = 0, 1
    for p in P
        m = prod(M[k][p[k]] for k in 1:n)
        if m != 0
            S += m
            push!(R, rank)
        end
        rank += 1
    end
    # println(n, "  ", S, "  ", S // (2^(n + 2) - 2)) # Bernoulli number
    return R
end
for n in 1:11 println(GenocchiPermutations(n)) end

Crossrefs

Cf. A036968, A226158, A027641/A027642, A347600.

Sequence in context: A103709 A122045 A294314 * A266324 A073911 A157302

Adjacent sequences: A347596 A347597 A347598 * A347600 A347601 A347602

Keyword

nonn,tabf

Author

Peter Luschny, Sep 08 2021

Status

approved