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A208650 Number of constant paths through the subset array of {1,2,...,n}; see Comments.

%I A208650 #13 Feb 18 2025 08:58:35
%S A208650 1,2,6,36,480,15000,1134000,211768200,99131719680,117595223746560,
%T A208650 356467003200000000,2779532232516963000000,56049508602150185041920000,
%U A208650 2935889842347365340037522521600
%N A208650 Number of constant paths through the subset array of {1,2,...,n}; see Comments.
%C A208650 Let I(n)={1,2,...,n}. Arrange the subsets of I(n) in an array S(n) of n rows, where row k consists of all the numbers in all the k-element subsets, including repetitions. Each i in I(n) occurs C(n-1,k-1) times in row k of S(n); index these occurrences as
%C A208650 ...
%C A208650 (k,1,1),(k,1,2),...,(k,1,r),(k,2,1),...,(k,2,r),...,(k,n,1),...,(k,n,r),
%C A208650 ...
%C A208650 where r=C(n-1,k-1).
%C A208650 Definitions:
%C A208650 (1) A path through I(n) is an n-tuple of triples, ((1,i(1),j(1)), (2,i(2),j(2)), ..., (n,i(n),j(n))), formed from the above indexing of the numbers in S(n).
%C A208650 (2) The trace of such a path p is the n-tuple (i(1),i(2),...,i(n)).
%C A208650 (3) The range of p is the set {i(1),i(2),...,i(n)}.
%C A208650 (4) Path p has property P if its trace or range has property P.
%C A208650 ...
%C A208650 Guide to sequences which count paths according to selected properties:
%C A208650 property................................sequence
%C A208650 range = {1}.............................A001142(n-1)
%C A208650 constant (range just one element).......A208650
%C A208650 range = {1,2,...,n}.....................A208651
%C A208650 palindromic.............................A208654
%C A208650 palindromic with i(1)=1.................A208655
%F A208650 a(n) = n*Product_{k=1..n-1} binomial(n-1,k). - _Jason Yuen_, Feb 18 2025
%e A208650 Taking n=3:
%e A208650 row 1:  {1},{2},{3} ---------> 1,2,3
%e A208650 row 2:  {1,2},{1,3},{2,3} ---> 1,1,2,2,3,3
%e A208650 row 3:  {1,2,3} -------------> 1,2,3
%e A208650 3 ways to choose a number from row 1,
%e A208650 2 ways to choose same number from row 2,
%e A208650 1 way to choose same number from row 3.
%e A208650 Total:  a(3) = 1*2*3 = 6 paths.
%t A208650 p[n_]:=Product[Binomial[n-1,k],{k,1,n-1}]
%t A208650 Table[p[n],{n,1,20}]    (* A001142(n-1) *)
%t A208650 Table[p[n]*n,{n,1,20}]  (* A208650 *)
%t A208650 Table[p[n]*n!,{n,1,20}] (* A208651 *)
%Y A208650 Cf. A208651.
%K A208650 nonn
%O A208650 1,2
%A A208650 _Clark Kimberling_, Mar 01 2012