OFFSET
0,6
COMMENTS
A partition p of the set {1,2,...,n} whose elements are arranged in their natural order, is crossing if there exist four numbers 1 <= i < k < j < l <= n such that i and j are in the same block, k and l are in the same block, but i,j and k,l belong to two different blocks.
Also number of crossing partitions of {1,2,...,n} that contain cyclical adjacencies. a(5) = 5, [124|35, 134|25, 135|24, 13|245, 14|235]. - Yuchun Ji, Nov 13 2020
LINKS
Indranil Ghosh, Table of n, a(n) for n = 0..170
Peter Luschny, Set partitions
FORMULA
EXAMPLE
The crossing partitions of {1,2,3,4,5} that contain singletons are: [1|24|35], [2|14|35], [3|14|25], [4|13|25], [5|13|24].
MAPLE
MATHEMATICA
Table[Sum[(-1)^(n-k+1)*Binomial[n, k]*(BellB[k]-CatalanNumber[k]), {k, 0, n-1}], {n, 0, 25}] (* Indranil Ghosh, Mar 04 2017 *)
PROG
(Sage)
A247494 = lambda n: sum((-1)^(n-k+1)*binomial(n, k)*(bell_number(k)-catalan_number(k)) for k in (0..n-1))
[A247494(n) for n in range(26)]
(PARI)
B(n) = sum(k=0, n, stirling(n, k, 2));
a(n) = sum(k=0, n-1, (-1)^(n-k+1)*binomial(n, k)*(B(k) - binomial(2*k, k)/(k+1))); \\ Indranil Ghosh, Mar 04 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Luschny, Oct 02 2014
STATUS
approved