A365554 Number of increasing paths from the bottom to the top of the n-hypercube (as a graded poset) which first encounter a vector of isolated zeros at stage k, weighted by k.

2, 10, 60, 396, 2976, 25056, 234720, 2423520, 27371520, 335819520, 4449150720, 63318931200, 963548006400, 15614378035200, 268480048435200, 4882321001779200, 93627018326016000, 1888394741194752000, 39963486306078720000, 885457095215616000000

Offset: 2

Author

Brian Darrow, Jr. and Joe Fields, Feb 20 2024

Keywords

• nonn

Comments

These are the numerators in calculating an expected value. The expectation of the number of steps one takes in marking the elements of a predetermined list before reaching a state where only isolated unmarked entries remain.

Examples

			For n=5, an example vector of isolated 0's is 01011, which has k=3 1's.
For n=3, the following paths (from 000 to 111) reach isolated 0's at k=1 many 1's (010):
  000,010,011,111
  000,010,110,111
The following paths reach isolated 0's only at k=2 1's:
  000,100,110,111
  000,100,101,111
  000,001,101,111
  000,001,011,111
So 2 paths of k=1 and 4 paths of k=2 are weighted total a(3) = 2*1 + 4*2 = 10.
		

Crossrefs

Cf. A067331.

Programs

PARI

a(n) = sum(k=n\2, n-1, k*(binomial(k+1,n-k)-binomial(k-1,n-k))*k!*(n-k)!) \\ Andrew Howroyd, Feb 23 2024

SageMath

k, n = var('k,n')
sum((binomial(k+1,n-k)-binomial(k-1,n-k))*factorial(k)*factorial(n-k), k, floor(n/2),n-1)

Formula

a(n) = Sum_{k=floor(n/2)..n-1} k*(binomial(k+1,n-k)-binomial(k-1,n-k))*k!*(n-k)!.