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

1,5

Comments

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).

Links

Table of n, a(n) for n=1..25.

Example

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.

Prog

(Julia)
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)

Crossrefs

Cf. A000793, A074059, A074064, A383459.

Sequence in context: A299229 A392079 A289496 * A371256 A378056 A248973

Adjacent sequences: A383455 A383456 A383457 * A383459 A383460 A383461

Keyword

nonn,more

Author

Anand Jain, Mar 22 2025

Status

approved