A383459 -id:A383459 - cp's OEIS frontend

A383458 Maximum number of cycles in any permutation in S_n of the highest order (A000793(n)).

1, 1, 1, 1, 2, 3, 2, 2, 2, 3, 4, 3, 4, 3, 3, 3, 4, 5, 4, 5, 6, 7, 4, 5, 4

Offset: 1

Anand Jain, Mar 22 2025

Landau's function g(n) = A000793(n) gives the maximum order of any permutation on n elements.

The number of permutations of order g(n) is A074059, and the number of different cycle types of permutations of order g(n) is A074064. a(n) is the maximum number of cycles in any permutation of order g(n), and A383459(n) is the minimum number of cycles in any permutation of order g(n).

			There are two different cycle types of permutations in S_6 of the maximum order g(6) = 6, for example (123456) and (12)(345)(6). The minimum number of cycles is A383459(6) = 1 and maximum number is a(6) = 3.

Cf. A000793, A074059, A074064, A383459.

using Combinatorics
arrs = []
for n in 1:25
    ps = integer_partitions(n)
    lcms = lcm.(ps)
    the_max, imax, = findmax(lcms)
    max_order_cyc_idxs = []
    for (i, l) in enumerate(lcms)
        if the_max == l
            push!(max_order_cyc_idxs, i)
        end
    end
    push!(arrs, ps[max_order_cyc_idxs])
end
map(x->maximum(length.(x)), arrs)

cp's OEIS Frontend

A383458 Maximum number of cycles in any permutation in S_n of the highest order (A000793(n)).

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