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%I #8 Feb 25 2018 22:52:46
%S 1,2,12,216,11520,1800000,816480000,1067311728000,3996990937497600,
%T 42672954793151692800,1293547461212160000000000,
%U 110950032218933108678400000000,26847804299643702075375747072000000
%N Number of paths through the subset array whose trace is a permutation of (1,2,...,n); see Comments.
%C 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}.
%H Nick Early, <a href="https://arxiv.org/abs/1709.03686">Generalized Permutohedra, Scattering Amplitudes, and a Cubic Three-Fold</a>, arXiv:1709.03686 [math.CO], 2017.
%e Taking n=3:
%e row 1: {1},{2},{3} ---------> 1,2,3
%e row 2: {1,2},{1,3},{2,3} ---> 1,1,2,2,3,3
%e row 3: {1,2,3} -------------> 1,2,3
%e 3 ways to choose a number from row 1,
%e 4 ways to choose a different number from row 2,
%e 1 way to choose remaining number from row 3.
%e Total: a(3) = 1*4*3 = 12 paths.
%t p[n_]:=Product[Binomial[n-1,k],{k,1,n-1}]
%t Table[p[n],{n,1,20}] (* A001142(n-1) *)
%t Table[p[n]*n,{n,1,20}] (* A208650 *)
%t Table[p[n]*n!,{n,1,20}] (* A208651 *)
%Y Cf. A208650.
%K nonn
%O 1,2
%A _Clark Kimberling_, Mar 01 2012
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