A327547 Triangular array read by rows: T(n,k) is the number of ordered pairs of n-permutations that generate a group with exactly k orbits, 0 <= k <= n, n >= 0.

1, 0, 1, 0, 3, 1, 0, 26, 9, 1, 0, 426, 131, 18, 1, 0, 11064, 2910, 395, 30, 1, 0, 413640, 92314, 11475, 925, 45, 1, 0, 20946960, 3980172, 438424, 34125, 1855, 63, 1, 0, 1377648720, 224782284, 21632436, 1550689, 84840, 3346, 84, 1, 0, 114078384000, 16158371184, 1353378284, 87036012, 4533249, 185976, 5586, 108, 1

Offset

0,5

Links

Table of n, a(n) for n=0..54.

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; page 139.

Formula

E.g.f.: exp(y*log(Sum_{n>=0} n! * x^n)).

Example

Triangle T(n,k) begins:
  1;
  0,      1;
  0,      3,     1;
  0,     26,     9,    1;
  0,    426,   131,   18,    1;
  0,  11064,  2910,  395,   30,  1;
  0, 413640, 92314, 11475, 925, 45, 1;
T(3,2) = 9 because we have 3 ordered pairs (e,<(1,2)>), (<(1,2)>,e), (<(1,2)>,<(1,2)>) for each of the 3 transpositions in S_3.

Mathematica

nn = 7; Range[0, nn]! CoefficientList[Series[Exp[u Log[Sum[n!^2 z^n/n!, {n, 0, nn}]]], {z, 0, nn}], {z, u}] // Grid

Crossrefs

Cf. A122949 (column 1), A001044 (row sums), A220754.

Sequence in context: A006837 A158782 A187558 * A233293 A361579 A066746

Adjacent sequences: A327544 A327545 A327546 * A327548 A327549 A327550

Keyword

nonn,tabl

Author

Geoffrey Critzer, Sep 16 2019

Status

approved