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 the number of crossing partitions of {1,2,...,n} that contain no cyclical adjacencies. e.g., a(5) = 5, [13|24|5, 13|25|4, 14|25|3, 14|2|35, 1|24|35]. - Yuchun Ji, Nov 13 2020
LINKS
Indranil Ghosh, Table of n, a(n) for n = 0..163
Peter Luschny, Set partitions
FORMULA
EXAMPLE
The crossing partitions of {1,2,3,4,5} that contain no singletons are: [13|245], [14|235], [24,135], [25|134], [35|124].
MAPLE
MATHEMATICA
Table[Sum[(-1)^(n-k)*Binomial[n, k]*(BellB[k]-CatalanNumber[k]), {k, 0, n}], {n, 0, 26}] (* Indranil Ghosh, Mar 04 2017 *)
PROG
(Sage)
A247491 = lambda n: sum((-1)^(n-k)*binomial(n, k)*(bell_number(k) - catalan_number(k)) for k in (0..n))
[A247491(n) for n in range(27)]
(PARI)
B(n) = sum(k=0, n, stirling(n, k, 2));
a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*(B(k)-binomial(2*k, k)/(k+1))); \\ Indranil Ghosh, Mar 04 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Luschny, Sep 25 2014
STATUS
approved