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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 Zhi-Wei Sun, A novel identity connecting permanents to Bernoulli numbers, MathOverflow 2021-09-07. EXAMPLE Table starts:  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; . The 17 permutations corresponding to the ranks are for n = 7: 1897 -> ; 2017 -> ; 2018 -> ; 2617 -> ; 2619 -> ; 2737 -> ; 2738 -> ; 2739 -> ; 2740 -> ; 3457 -> ; 3458 -> ; 3459 -> ; 3460 -> ; 4177 -> ; 4178 -> ; 4179 -> ; 4180 -> . . 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.)