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
Alois P. Heinz, Antidiagonals n = 1..141, flattened
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
AUTHOR
EXTENSIONS
More terms from Vladeta Jovovic, Apr 13 2001
STATUS
approved