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A368213
Triangular array read by rows: Number of permutations of [n] that factor into exactly k-cycles, ordered by n (rows) and divisors k of n (columns).
1
1, 1, 1, 1, 0, 2, 1, 3, 0, 6, 1, 0, 0, 0, 24, 1, 15, 40, 0, 0, 120, 1, 0, 0, 0, 0, 0, 720, 1, 105, 0, 1260, 0, 0, 0, 5040, 1, 0, 2240, 0, 0, 0, 0, 0, 40320, 1, 945, 0, 0, 72576, 0, 0, 0, 0, 362880, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3628800, 1, 10395, 246400, 1247400, 0, 6652800, 0, 0, 0, 0, 0, 39916800
OFFSET
1,6
REFERENCES
P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, 2009, pages 120-122.
FORMULA
T(n, k) = n! / ( k^(n/k) * (n/k)! ) if k divides n otherwise 0.
EXAMPLE
Row n=6 is 1, 15, 40, 120 because there is one permutation of [6] consisting of six fixed points, there are 15 permutations consisting of three transpositions, there are forty permutations consisting of two three-cycles and there are one hundred and twenty permutations consisting of just one six-cycle (6!/6).
Triangular array starts:
[ 1] 1;
[ 2] 1, 1;
[ 3] 1, 0, 2;
[ 4] 1, 3, 0, 6;
[ 5] 1, 0, 0, 0, 24;
[ 6] 1, 15, 40, 0, 0, 120;
[ 7] 1, 0, 0, 0, 0, 0, 720;
[ 8] 1, 105, 0, 1260, 0, 0, 0, 5040;
[ 9] 1, 0, 2240, 0, 0, 0, 0, 0, 40320;
[10] 1, 945, 0, 0, 72576, 0, 0, 0, 0, 362880;
MAPLE
T:= (n, m)-> `if`(irem(n, m)=0, n!/m^(n/m)/(n/m)!, 0):
seq(seq(T(n, m), m = 1..n), n=1..15);
MATHEMATICA
A368213[n_, k_]:=If[Divisible[n, k], n!/(k^(n/k)(n/k)!), 0];
Table[A368213[n, k], {n, 15}, {k, n}] (* Paolo Xausa, Dec 18 2023 *)
PROG
(SageMath)
def T(n, d): return factorial(n) // (d ** (n//d) * factorial(n//d))
for n in range(1, 19):
print([T(n, d) if n % d == 0 else 0 for d in range(1, n+1)])
# Peter Luschny, Dec 17 2023
CROSSREFS
Cf. A005225 (row sums), A008290.
Cf. A123023 (column 2), A052502 (column 3), A060706 (column 4).
Sequence in context: A078024 A112469 A330502 * A249455 A248967 A331094
KEYWORD
nonn,tabl
AUTHOR
Marko Riedel, Dec 17 2023
STATUS
approved