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A208650 Number of constant paths through the subset array of {1,2,...,n}; see Comments. 5
1, 2, 6, 36, 480, 15000, 1134000, 211768200, 99131719680, 117595223746560, 356467003200000000, 2779532232516963000000, 56049508602150185041920000, 2935889842347365340037522521600 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..14.

FORMULA

(See the Mathematica section.)

EXAMPLE

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.

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 *)

CROSSREFS

Cf. A208651.

Sequence in context: A262234 A055512 A078973 * A152480 A001660 A275904

Adjacent sequences:  A208647 A208648 A208649 * A208651 A208652 A208653

KEYWORD

nonn

AUTHOR

Clark Kimberling, Mar 01 2012

STATUS

approved

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Last modified July 30 02:04 EDT 2021. Contains 346346 sequences. (Running on oeis4.)