A350225 Number of ordered pairs (a,g) with a in IS_n the symmetric inverse semigroup on [n] and g in symmetric group on [n] such that ag=ga.

1, 2, 10, 60, 480, 4320, 46800, 554400, 7459200, 108864000, 1745452800, 30017433600, 558036864000, 11021826816000, 232330146048000, 5173159799808000, 121812482727936000, 3012672515973120000, 78301030421053440000, 2127572806150471680000, 60438151687124090880000

Offset: 0

Author

Geoffrey Critzer, Dec 20 2021

Keywords

• nonn

Crossrefs

Cf. A000142, A000712, A053529, A349739.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, 2*(n-1)!*
      add(a(j)/j!*numtheory[sigma](n-j), j=0..n-1))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Dec 20 2021

Mathematica

nn = 16; Table[Sum[PartitionsP[k] PartitionsP[n - k], {k, 0, n}], {n, 0, nn}] Table[n!, {n, 0, nn}]

Formula

a(n) = A000712(n)*n!