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
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
KEYWORD
nonn,tabf
AUTHOR
Peter Luschny, Sep 08 2021
STATUS
approved