A347599 - OEIS

%I #14 Sep 10 2021 01:36:14

%S 1,0,5,0,67,91,92,0,1897,2017,2018,2617,2619,2737,2738,2739,2740,3457,

%T 3458,3459,3460,4177,4178,4179,4180,0,99241,99961,99962,104281,104283,

%U 105001,105002,105003,105004,110041,110042,110043,110044,115081,115082,115083

%N 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.

%C 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.

%C 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:

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

%C The related sequence A347600 lists Seidel permutations.

%H Zhi-Wei Sun, <a href="https://mathoverflow.net/q/403386">A novel identity connecting permanents to Bernoulli numbers</a>, MathOverflow 2021-09-07.

%e Table starts:

%e [1] 1;

%e [2] 0;

%e [3] 5;

%e [4] 0;

%e [5] 67, 91, 92;

%e [6] 0;

%e [7] 1897, 2017, 2018, 2617, 2619, 2737, 2738, 2739, 2740, 3457, 3458, 3459, 3460, 4177, 4178, 4179, 4180;

%e .

%e The 17 permutations corresponding to the ranks are for n = 7:

%e 1897 -> [3571246]; 2017 -> [3671245]; 2018 -> [3671254]; 2617 -> [4571236];

%e 2619 -> [4571326]; 2737 -> [4671235]; 2738 -> [4671253]; 2739 -> [4671325];

%e 2740 -> [4671352]; 3457 -> [5671234]; 3458 -> [5671243]; 3459 -> [5671324];

%e 3460 -> [5671342]; 4177 -> [6571234]; 4178 -> [6571243]; 4179 -> [6571324];

%e 4180 -> [6571342].

%e .

%e 17 / (-510) = -1/30 = Bernoulli(8).

%o (Julia)

%o using Combinatorics

%o function GenocchiPermutations(n)

%o     f(m) = m >= n ? 1 : m < 0 ? -1 : 0

%o     Mat(n) = [[f(2*j - k) for k in 1:n] for j in 1:n]

%o     M = Mat(n); P = permutations(1:n); R = Int64[]

%o     S, rank = 0, 1

%o     for p in P

%o         m = prod(M[k][p[k]] for k in 1:n)

%o         if m != 0

%o             S += m

%o             push!(R, rank)

%o         end

%o         rank += 1

%o     end

%o     # println(n, "  ", S, "  ", S // (2^(n + 2) - 2)) # Bernoulli number

%o     return R

%o end

%o for n in 1:11 println(GenocchiPermutations(n)) end

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

%K nonn,tabf

%O 1,3

%A _Peter Luschny_, Sep 08 2021