A074064 -id:A074064 - cp's OEIS frontend

A074859 Number of elements of S_n having the maximum possible order g(n), where g(n) is Landau's function (A000793).

1, 1, 1, 2, 6, 20, 240, 420, 2688, 18144, 120960, 2661120, 7983360, 103783680, 1037836800, 12454041600, 149448499200, 1693749657600, 60974987673600, 289631191449600, 5792623828992000, 121645100408832000, 3568256278659072000, 30776210403434496000, 738629049682427904000, 12310484161373798400000

Offset: 0

Christopher J. Smyth, Sep 11 2002

J.-L. Nicolas, On Landau's function g(n), pp. 228-240 of R. L. Graham et al., eds., Mathematics of Paul Erdős I.

Alois P. Heinz, Table of n, a(n) for n = 0..175
J. Kuzmanovich and A. Pavlichenkov, Finite groups of matrices whose entries are integers, Amer. Math. Monthly, 109 (2002), 173-186.
W. Miller, The Maximum Order of an Element of Finite Symmetric Group, Am. Math. Monthly, Jun-Jul 1987, pp. 497-506.
J.-L. Nicolas, Sur l'ordre maximum d'un élément dans le groupe S_n des permutations, Acta Arith., 14 (1968), 315-332.
J.-L. Nicolas, Ordre maximal d'un élément du groupe S_n de permutations et 'highly composite numbers', Bull. Math. Soc. France 97 (1969) 129-191.

Cf. A000793 (Landau's function g(n)).
Cf. A074064, A074103, A074115, A162682.
Last row element of A057731. - Alois P. Heinz, Feb 14 2013

g[n_] := Max[ Apply[ LCM, IntegerPartitions[n], 1]]; f[x_, n_] := Total[ (MoebiusMu[g[n]/#]*Exp[ Total[ (x^#/# & ) /@ Divisors[#]]] & ) /@ Divisors[g[n]]]; a[n_] := n!*Coefficient[ Series[f[x, n], {x, 0, n}], x^n]; Table[a[n], {n, 1, 25}] (* Jean-François Alcover, Nov 03 2011, after Vladeta Jovovic *)

a(n) = n!*coefficient of x^n in expansion of Sum_{i divides A000793(n)} mu(A000793(n)/i)*exp(Sum_{j divides i} x^j/j). - Vladeta Jovovic, Sep 29 2002

Corrected and extended by Vladeta Jovovic, Sep 20 2002

A256067 Irregular table T(n,k): the number of partitions of n where the least common multiple of all parts equals k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 2, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 0, 0, 1, 0, 1, 1, 4, 2, 4, 1, 5, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 4, 3, 4, 1, 7, 1, 1, 1, 2, 0, 2, 0, 1, 1, 0, 0, 0, 0, 1, 1, 5, 3, 6, 2, 9, 1, 2, 1, 3, 0, 4, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 5, 3, 6, 2

Offset: 0

R. J. Mathar, Mar 18 2015

			The 5 partitions of n=4 are 1+1+1+1 (lcm=1), 1+1+2 (lcm=2), 2+2 (lcm=2), 1+3 (lcm=3) and 4 (lcm=4). So k=1, 3 and 4 appear once, k=2 appears twice.
The triangle starts:
  1 ;
  1 ;
  1  1;
  1  1  1;
  1  2  1  1;
  1  2  1  1  1  1;
  1  3  2  2  1  2;
  1  3  2  2  1  3  1  0  0  1  0  1;
  ...

Alois P. Heinz, Rows n = 0..30, flattened

Cf. A000041 (row sums), A000793 (row lengths), A213952, A074761 (diagonal), A074752 (6th column), A008642 (4th column), A002266 (5th column), A002264 (3rd column), A132270 (7th column), A008643 (8th column), A008649 (9th column), A258470 (10th column).

Cf. A009490 (number of nonzero terms of rows), A074064 (last elements of rows), A168532 (the same for gcd), A181844 (Sum k*T(n,k)).

A256067 := proc(n,k)
        local a,p ;
        a := 0 ;
        for p in combinat[partition](n) do
                ilcm(op(p)) ;
                if % = k then
                        a := a+1 ;
                end if;
        end do:
        a;
end proc:
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0 or i=1, x,
      b(n, i-1)+(p-> add(coeff(p, x, t)*x^ilcm(t, i),
      t=1..degree(p)))(add(b(n-i*j, i-1), j=1..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2)):
seq(T(n), n=0..12);  # Alois P. Heinz, Mar 27 2015

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x, b[n, i-1] + Function[{p}, Sum[ Coefficient[p, x, t]*x^LCM[t, i], {t, 1, Exponent[p, x]}]][Sum[b[n-i*j, i-1], {j, 1, n/i}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jun 22 2015, after Alois P. Heinz *)

T(0,1)=1 prepended by Alois P. Heinz, Mar 27 2015

A256554 Number T(n,k) of cycle types of degree-n permutations having the k-th smallest possible order; triangle T(n,k), n>=0, 1<=k<=A009490(n), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 2, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 1, 1, 1, 4, 2, 4, 1, 5, 1, 1, 1, 1, 1, 1, 4, 3, 4, 1, 7, 1, 1, 1, 2, 2, 1, 1, 1, 1, 5, 3, 6, 2, 9, 1, 2, 1, 3, 4, 1, 1, 1, 1, 1, 1, 5, 3, 6, 2, 12, 1, 2, 1, 4, 1, 6, 2, 2, 1, 2, 1, 1, 1, 2

Offset: 0

Alois P. Heinz, Apr 01 2015

Sum_{k>=0} A256553(n,k)*T(n,k) = A181844(n).

			Triangle T(n,k) begins:
  1;
  1;
  1, 1;
  1, 1, 1;
  1, 2, 1, 1;
  1, 2, 1, 1, 1, 1;
  1, 3, 2, 2, 1, 2;
  1, 3, 2, 2, 1, 3, 1, 1, 1;
  1, 4, 2, 4, 1, 5, 1, 1, 1, 1, 1;
  1, 4, 3, 4, 1, 7, 1, 1, 1, 2, 2, 1, 1, 1;
  1, 5, 3, 6, 2, 9, 1, 2, 1, 3, 4, 1, 1, 1, 1, 1;

Alois P. Heinz, Rows n = 0..60, flattened

Row sums give A000041.
Row lengths give A009490.
Columns k=1-9 give: A000012, A004526, A002264, A008642(n-4), A002266, A074752, A132270, A008643(n-8) for n>7, A008649(n-9) for n>8.
Last elements of rows give A074064.
Main diagonal gives A074761.
Cf. A181844, A256067, A256553.

b:= proc(n, i) option remember; `if`(n=0 or i=1, x,
      b(n, i-1)+(p-> add(coeff(p, x, t)*x^ilcm(t, i),
      t=1..degree(p)))(add(b(n-i*j, i-1), j=1..n/i)))
    end:
T:= n->(p->seq((h->`if`(h=0, [][], h))(coeff(p, x, i))
     , i=1..degree(p)))(b(n$2)):
seq(T(n), n=0..12);

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x, b[n, i - 1] + Function[p, Sum[Coefficient[p, x, t]*x^LCM[t, i], {t, 1, Exponent[p, x]}]][Sum[b[n - i*j, i - 1], {j, 1, n/i}]]]; T[n_] := Function[p, Table[Function[h, If[h == 0, {{}, {}}, h]][Coefficient[p, x, i]], {i, 1, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 23 2017, translated from Maple *)

A074260 Number of labeled cyclic subgroups of S_n having the maximum possible order.

Original entry on oeis.org

1, 1, 1, 1, 3, 10, 120, 105, 336, 2268, 15120, 332640, 498960, 6486480, 43243200, 259459200, 3113510400, 35286451200, 1270312243200, 3016991577600, 60339831552000, 1267136462592000, 37169336236032000, 160292762517888000

Offset: 0

Vladeta Jovovic, Sep 20 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..100

Cf. A051625, A074064.

a(n) = A074859(n)/A000010(A000793(n)).

A383459 Minimum number of cycles in any permutation in S_n of the highest order (A000793(n)).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 2, 2, 3, 2, 3, 4, 3, 3, 3, 4, 3, 4, 5, 5, 4, 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 a(6) = 1 and maximum number is A383458(6) = 3.

Cf. A000793, A074059, A074064, A383458.

A256443 Irregular triangle T(n,k) read by rows: row n gives a smallest partition of n with maximal order (see Comments for precise definition).

Original entry on oeis.org

1, 2, 3, 4, 2, 3, 6, 3, 4, 3, 5, 4, 5, 2, 3, 5, 5, 6, 3, 4, 5, 1, 3, 4, 5, 3, 4, 7, 3, 5, 7, 4, 5, 7, 2, 3, 5, 7, 5, 6, 7, 3, 4, 5, 7, 1, 3, 4, 5, 7, 2, 3, 4, 5, 7, 4, 5, 6, 7, 3, 5, 7, 8, 1, 3, 5, 7, 8, 4, 5, 7, 9, 1, 4, 5, 7, 9, 4, 5, 7, 11, 2, 3, 5, 7, 11

Offset: 1

Bob Selcoe, Mar 29 2015

Consider all partitions of n for which the LCM of the parts is A000793(n) (A000793 is Landau's function g(n), the largest order of a permutation of n elements). Minimize the number of parts. Then take the lexicographically earliest solution. This is row n of the triangle. See A256445 for a partition with the most elements.

			Triangle starts T(1,1) = 1:
1:  1
2:  2
3:  3
4:  4
5:  2,3
6:  6
7:  3,4
8:  3,5
9:  4,5
10: 2,3,5
11: 5,6
12: 3,4,5
13: 1,3,4,5
14: 3,4,7
15: 3,5,7
16: 4,5,7
17: 2,3,5,7
18: 5,6,7
19: 3,4,5,7
20: 1,3,4,5,7
21: 2,3,4,5,7
22: 4,5,6,7
23: 3,5,7,8
T(11,k) = [5,6] rather than [1,2,3,5] because [5,6] has fewer elements.

Cf. A000793, A074064, A256445.

More terms from Alois P. Heinz, Apr 01 2015

A256445 Irregular triangle T(n,k) read by rows: row n gives a largest partition of n with maximal order (see Comments for precise definition).

Original entry on oeis.org

1, 2, 3, 4, 2, 3, 1, 2, 3, 3, 4, 3, 5, 4, 5, 2, 3, 5, 1, 2, 3, 5, 3, 4, 5, 1, 3, 4, 5, 3, 4, 7, 3, 5, 7, 4, 5, 7, 2, 3, 5, 7, 1, 2, 3, 5, 7, 3, 4, 5, 7, 1, 3, 4, 5, 7, 1, 1, 3, 4, 5, 7, 1, 1, 1, 3, 4, 5, 7, 3, 5, 7, 8, 1, 3, 5, 7, 8, 4, 5, 7, 9, 1, 4, 5, 7, 9

Offset: 1

Bob Selcoe, Mar 29 2015

Consider all partitions of n for which the LCM of the parts is A000793(n) (A000793 is Landau's function g(n), the largest order of a permutation of n elements). Maximize the number of parts. Then take the lexicographically earliest solution. This is row n of the triangle. See A256443 for a partition with the fewest elements.

			Triangle starts T(1,1) = 1:
1:  1
2:  2
3:  3
4:  4
5:  2,3
6:  1,2,3
7:  3,4
8;  3,5
9:  4,5
10: 2,3,5
11: 1,2,3,5
12: 3,4,5
13: 1,3,4,5
14: 3,4,7
15: 3,5,7
16: 4,5,7
17: 2,3,5,7
18: 1,2,3,5,7
19: 3,4,5,7
20: 1,3,4,5,7
21: 1,1,3,4,5,7
22: 1,1,1,3,4,5,7
23: 3,5,7,8
T(11,k) = [1,2,3,5] rather than [5,6] because [1,2,3,5] has more elements.

Cf. A000793, A074064, A256443.

More terms from Alois P. Heinz, Apr 01 2015

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

Original entry on oeis.org

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

A074859 Number of elements of S_n having the maximum possible order g(n), where g(n) is Landau's function (A000793).

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A256067 Irregular table T(n,k): the number of partitions of n where the least common multiple of all parts equals k.

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A256554 Number T(n,k) of cycle types of degree-n permutations having the k-th smallest possible order; triangle T(n,k), n>=0, 1<=k<=A009490(n), read by rows.

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A074260 Number of labeled cyclic subgroups of S_n having the maximum possible order.

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A383459 Minimum number of cycles in any permutation in S_n of the highest order (A000793(n)).

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A256443 Irregular triangle T(n,k) read by rows: row n gives a smallest partition of n with maximal order (see Comments for precise definition).

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A256445 Irregular triangle T(n,k) read by rows: row n gives a largest partition of n with maximal order (see Comments for precise definition).

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

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Julia