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A347600 Irregular table read by rows, T(n, k) is the rank of the k-th Seidel permutation of {1,...,n}, permutations sorted in lexicographical order. 5
2, 11, 17, 187, 211, 307, 331, 451, 452, 571, 572, 6937, 7057, 7657, 7777, 8497, 8498, 9217, 9218, 11977, 12097, 12697, 12817, 13537, 13538, 14257, 14258, 17737, 17739, 17857, 17859, 18577, 18578, 18579, 18580, 19297, 19298, 19299, 19300, 22777, 22779, 22897 (list; graph; refs; listen; history; text; internal format)
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.
LINKS
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
Sequence in context: A176985 A089592 A106982 * A043461 A030551 A121848
KEYWORD
nonn,tabf
AUTHOR
Peter Luschny, Sep 08 2021
STATUS
approved

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Last modified February 22 13:38 EST 2024. Contains 370256 sequences. (Running on oeis4.)