A208651 Number of paths through the subset array whose trace is a permutation of (1,2,...,n); see Comments.

1, 2, 12, 216, 11520, 1800000, 816480000, 1067311728000, 3996990937497600, 42672954793151692800, 1293547461212160000000000, 110950032218933108678400000000, 26847804299643702075375747072000000

Offset: 1

Author

Clark Kimberling, Mar 01 2012

Keywords

• nonn

Comments

See A208650. The trace of a path is a permutation of (1,2,...,n) if and only if the range of the path is {1,2,...,n}.

Examples

			Taking n=3:
row 1:  {1},{2},{3} ---------> 1,2,3
row 2:  {1,2},{1,3},{2,3} ---> 1,1,2,2,3,3
row 3:  {1,2,3} -------------> 1,2,3
3 ways to choose a number from row 1,
4 ways to choose a different number from row 2,
1 way to choose remaining number from row 3.
Total:  a(3) = 1*4*3 = 12 paths.
		

Links

• Nick Early, Generalized Permutohedra, Scattering Amplitudes, and a Cubic Three-Fold, arXiv:1709.03686 [math.CO], 2017.

Crossrefs

Cf. A208650.

Programs

Mathematica

p[n_]:=Product[Binomial[n-1,k],{k,1,n-1}]
Table[p[n],{n,1,20}]    (* A001142(n-1) *)
Table[p[n]*n,{n,1,20}]  (* A208650 *)
Table[p[n]*n!,{n,1,20}] (* A208651 *)