A208655 -id:A208655 - cp's OEIS frontend

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

1, 2, 6, 36, 480, 15000, 1134000, 211768200, 99131719680, 117595223746560, 356467003200000000, 2779532232516963000000, 56049508602150185041920000, 2935889842347365340037522521600

Offset: 1

Clark Kimberling, Mar 01 2012

nonn

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
...
(k,1,1),(k,1,2),...,(k,1,r),(k,2,1),...,(k,2,r),...,(k,n,1),...,(k,n,r),
...
where r=C(n-1,k-1).
Definitions:
(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).
(2) The trace of such a path p is the n-tuple (i(1),i(2),...,i(n)).
(3) The range of p is the set {i(1),i(2),...,i(n)}.
(4) Path p has property P if its trace or range has property P.
...
Guide to sequences which count paths according to selected properties:
property................................sequence
range = {1}.............................A001142(n-1)
constant (range just one element).......A208650
range = {1,2,...,n}.....................A208651
palindromic.............................A208654
palindromic with i(1)=1.................A208655

			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,
2 ways to choose same number from row 2,
1 way to choose same number from row 3.
Total:  a(3) = 1*2*3 = 6 paths.

Cf. A208651.

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 *)

a(n) = n*Product_{k=1..n-1} binomial(n-1,k). - Jason Yuen, Feb 18 2025

A208654 Number of palindromic paths through the subset array of {1,2,...,n}; see Comments.

Original entry on oeis.org

1, 2, 18, 144, 12000, 540000, 388962000, 108425318400, 650403212820480, 1175952237465600000, 57409367332363200000000, 691636564481660937216000000, 270540272566435932512004833280000

Offset: 1

Clark Kimberling, Mar 02 2012

nonn

A palindromic path through the subset array of {1,2,...,n} is essentially a palindrome using numbers i from {1,2,...n}, where the number of times i can be used in position k equals the multiplicity of i in the multiset of numbers in the k-element subsets of {1,2,...,n}. See A208650 for a discussion and guide to related sequences.

			For n=4, write
row 1:  1,2,3,4
row 2:  1,2; 1,3; 1,4; 2,3; 2,4; 3;4
row 3:  1,2,3; 1,2,4; 1,3,4; 2,3,4
row 4:  1,2,3,4
To form a palindromic path of length 4, there are 4 ways to choose 1st term from row 1, then 12 ways to choose 2nd term from row 2, then 3 ways to choose 3rd term, then 1 way to finish, so that a(4)=4*12*3*1=144.

Cf. A208650, A208655.

m[n_] := Floor[(n + 1)/2]; z = 21;
g[n_] := Product[i*Binomial[n, i], {i, 1, m[n]}]
h[n_] := Product[Binomial[n - 1, i], {i, m[n], n - 1}]
Table[g[n], {n, 1, z}]   (* A208652 *)
Table[h[n], {n, 1, z}]   (* A208653 *)
Table[g[n] h[n], {n, 1, 2 z/3}]   (* A208654 *)
Table[g[n] h[n]/n, {n, 1, 2 z/3}] (* A208655 *)

cp's OEIS Frontend

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

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