%I
%S 1,2,6,36,480,15000,1134000,211768200,99131719680,117595223746560,
%T 356467003200000000,2779532232516963000000,56049508602150185041920000,
%U 2935889842347365340037522521600
%N Number of constant paths through the subset array of {1,2,...,n}; see Comments.
%C Let I(n)={1,2,...,n}. Arrange the subsets of I(n) in an
%C array S(n) of n rows, where row k consists of all the
%C numbers in all the kelement subsets, including
%C repetitions. Each i in I(n) occurs C(n1,k1) times in
%C row k of S(n); index these occurrences as
%C ...
%C (k,1,1),(k,1,2),...,(k,1,r),(k,2,1),...,(k,2,r),...,(k,n,1),...,(k,n,r),
%C ...
%C where r=C(n1,k1). Definitions:
%C (1) A path through I(n) is an ntuple of triples,
%C ((1,i(1),j(1)), (2,i(2),j(2)), ..., (n,i(n),j(n)),
%C formed from the above indexing of the numbers in S(n).
%C (2) The trace of such a path p is the ntuple
%C (i(1),i(2),...,i(n)).
%C (3) The range of p is the set {i(1),i(2),...,i(n)}.
%C (4) Path p has property P if its trace or range has
%C property P.
%C ...
%C Guide to sequences which count paths according to
%C selected properties:
%C property................................sequence
%C range = {1}.............................A001142(n1)
%C constant (range just one element).......A208650
%C range = {1,2,...,n}.....................A208651
%C palindromic.............................A208654
%C palindromic with i(1)=1.................A208655
%F (See the Mathematica section.)
%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 2 ways to choose same number from row 2,
%e 1 way to choose same number from row 3.
%e Total: a(3) = 1*2*3 = 6 paths.
%t p[n_]:=Product[Binomial[n1,k],{k,1,n1}]
%t Table[p[n],{n,1,20}] (* A001142(n1) *)
%t Table[p[n]*n,{n,1,20}] (* A208650 *)
%t Table[p[n]*n!,{n,1,20}] (* A208651 *)
%Y Cf. A208651.
%K nonn
%O 1,2
%A _Clark Kimberling_, Mar 01 2012
