OFFSET
1,1
COMMENTS
Let M be the 2n X 2n matrix with M(j, k) = floor((2*j - k - 1) / 2*n). A Seidel permutation of order n is a permutation sigma of {1,...,2n} if Product_{k=1..2n} M(k, sigma(k)) does not vanish.
Let P(n) denote the number of Seidel permutations of order n. We conjecture that P(n) = A005439(n). This conjecture was inspired by the conjecture of Zhi-Wei Sun in A036968. The name 'Seidel permutations' follows a comment of Don Knuth: "The earliest known reference for these numbers (A005439) is Seidel ...."
The related sequence A347599 lists Genocchi permutations.
EXAMPLE
Table starts:
[1] 2;
[2] 11, 17;
[3] 187, 211, 307, 331, 451, 452, 571, 572.
.
The 8 permutations corresponding to the ranks are for n = 3:
187 -> [246135]; 211 -> [256134]; 307 -> [346125]; 331 -> [356124];
451 -> [456123]; 452 -> [456132]; 571 -> [546123]; 572 -> [546132].
PROG
(Julia)
function SeidelPermutations(n)
f(m) = m >= 2n ? 1 : m < 0 ? -1 : 0
Mat(n) = [[f(2*j - k - 1) for k in 1:2n] for j in 1:2n]
M = Mat(n); P = permutations(1:2n); R = Int64[]
S, rank = 0, 1
for p in P
m = prod(M[k][p[k]] for k in 1:2n)
if m != 0
S += m
push!(R, rank)
end
rank += 1
end
# println(n, " -> ", (-1)^n*S)
return R
end
for n in 1:5 println(SeidelPermutations(n)) end
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Peter Luschny, Sep 08 2021
STATUS
approved