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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. 5
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 6 13:45 EST 2021. Contains 349563 sequences. (Running on oeis4.)