A074859
Number of elements of S_n having the maximum possible order g(n), where g(n) is Landau's function (A000793).
Original entry on oeis.org
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
- 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)).
-
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 *)
Original entry on oeis.org
1, 1, 1, 2, 3, 10, 10, 35, 56, 126, 84, 924, 462, 6006, 4290, 3432, 5148, 58344, 58344, 554268, 554268, 554268, 554268, 6374082, 6374082, 21246940, 21246940, 52151580, 34767720, 924241890, 504131940, 15628090140, 26447537160, 26447537160, 15628090140
Offset: 0
-
g:= proc(n) g(n):= `if`(n=0, 1, ilcm(n, g(n-1))) end:
b:= proc(n, i) option remember; local p;
p:= `if`(i<1, 1, ithprime(i));
`if`(n=0 or i<1, 1, max(b(n, i-1),
seq(p^j*b(n-p^j, i-1), j=1..ilog[p](n))))
end:
a:= n->g(n)/b(n, `if`(n<8, 3,
numtheory[pi](ceil(1.328*isqrt(n*ilog(n)))))):
seq(a(n), n=0..40); # Alois P. Heinz, May 22 2013
-
b[n_, i_] := b[n, i] = Module[{p}, p = If[i<1, 1, Prime[i]]; If[n==0 || i<1, 1, Max[b[n, i-1], Table[p^j*b[n-p^j, i-1], {j, 1, Log[p, n] // Floor }]]]]; a[0]=1; a[n_] := LCM @@ Range[n] / b[n, If[n<8, 3, PrimePi[ Ceiling[ 1.328*Sqrt[n*Log[n] // Floor]]]]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 27 2016, after Alois P. Heinz *)
A162682
If S is countable finite set, we can define n as number of elements in S. There are n^n distinct functions f(S)->S. Each function has a fixed point, or an orbit in S. This sequence is a number of distinct functions g(S)->S, with largest orbit.
Original entry on oeis.org
1, 1, 1, 2, 6, 20, 840, 420, 2688, 18144, 120960, 15966720, 7983360, 1349187840, 1037836800, 12454041600, 149448499200, 1693749657600, 579262382899200, 289631191449600, 115852476579840000, 26822744640147456000, 4750241170964889600000, 30776210403434496000
Offset: 0
Dmitriy Samsonov (dmitriy.samsonov(AT)gmail.com), Jul 10 2009
For S={a}, n=1 and only one operation possible {a->a}. For S={a,b}, n=2 and possible operations are {a->a,b->a}, {a->a,b->b}, {a->b,b->a},{a->b,b->b}. Longest orbit generated by applying operation {a->b,b->a}: initial set (a,b), applying function gives orbit - (b,a), (a,b). All other possible functions are generating fixed points.
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