login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A008307 Table T(n,k) giving number of permutations of [1..n] with order dividing k, read by antidiagonals. 18
1, 1, 1, 1, 2, 1, 1, 4, 1, 1, 1, 10, 3, 2, 1, 1, 26, 9, 4, 1, 1, 1, 76, 21, 16, 1, 2, 1, 1, 232, 81, 56, 1, 6, 1, 1, 1, 764, 351, 256, 25, 18, 1, 2, 1, 1, 2620, 1233, 1072, 145, 66, 1, 4, 1, 1, 1, 9496, 5769, 6224, 505, 396, 1, 16, 3, 2, 1, 1, 35696, 31041, 33616, 1345, 2052, 1, 56, 9, 4, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
Solutions to x^k = 1 in Symm_n (the symmetric group of degree n).
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 257.
J. D. Dixon, B. Mortimer, Permutation Groups, Springer (1996), Exercise 1.2.13.
LINKS
M. B. Kutler, C. R. Vinroot, On q-Analogs of Recursions for the Number of Involutions and Prime Order Elements in Symmetric Groups, JIS 13 (2010) #10.3.6, eq (5) for primes k.
FORMULA
T(n+1,k) = Sum_{d|k} (n)_(d-1)*T(n-d+1,k), where (n)_i = n!/(n - i)! = n*(n - 1)*(n - 2)*...*(n - i + 1) is the falling factorial.
E.g.f. for n-th row: Sum_{n>=0} T(n,k)*t^n/n! = exp(Sum_{d|k} t^d/d).
EXAMPLE
Array begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 1, 2, 1, 2, 1, 2, ...
1, 4, 3, 4, 1, 6, 1, 4, ...
1, 10, 9, 16, 1, 18, 1, 16, ...
1, 26, 21, 56, 25, 66, 1, 56, ...
1, 76, 81, 256, 145, 396, 1, 256, ...
1, 232, 351, 1072, 505, 2052, 721, 1072, ...
1, 764, 1233, 6224, 1345, 12636, 5761, 11264, ...
MAPLE
A:= proc(n, k) option remember; `if`(n<0, 0, `if`(n=0, 1,
add(mul(n-i, i=1..j-1)*A(n-j, k), j=numtheory[divisors](k))))
end:
seq(seq(A(1+d-k, k), k=1..d), d=1..12); # Alois P. Heinz, Feb 14 2013
# alternative
A008307 := proc(n, m)
local x, d ;
add(x^d/d, d=numtheory[divisors](m)) ;
exp(%) ;
coeftayl(%, x=0, n) ;
%*n! ;
end proc:
seq(seq(A008307(1+d-k, k), k=1..d), d=1..12) ; # R. J. Mathar, Apr 30 2017
MATHEMATICA
t[n_ /; n >= 0, k_ /; k >= 0] := t[n, k] = Sum[(n!/(n - d + 1)!)*t[n - d, k], {d, Divisors[k]}]; t[_, _] = 1; Flatten[ Table[ t[n - k, k], {n, 0, 12}, {k, 1, n}]] (* Jean-François Alcover, Dec 12 2011, after given formula *)
CROSSREFS
Rows give A000034, A284517, A284518.
Main diagonal gives A074759. - Alois P. Heinz, Feb 14 2013
Sequence in context: A294587 A064645 A285425 * A249694 A192005 A333810
KEYWORD
nonn,tabl,easy,look,nice
AUTHOR
EXTENSIONS
More terms from Vladeta Jovovic, Apr 13 2001
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 22:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)