A008307 Table T(n,k) giving number of permutations of [1..n] with order dividing k, read by antidiagonals.

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

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

Rows give A000034, A284517, A284518.

Columns give A000012, A000085, A001470, A001472, A052501, A053496, A053497, A053498, A053499, A053500, A053501, A053502, A053503, A053504, A053505.

Main diagonal gives A074759. - Alois P. Heinz, Feb 14 2013

Sequence in context: A294587 A064645 A285425 * A249694 A192005 A333810

Adjacent sequences: A008304 A008305 A008306 * A008308 A008309 A008310

Keyword

nonn,tabl,easy,look,nice

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Apr 13 2001

Status

approved