A247494 Number of crossing partitions of {1,2,...,n} that contain singletons.

0, 0, 0, 0, 0, 5, 45, 322, 2086, 13092, 82060, 523116, 3429481, 23279555, 164244262, 1206458632, 9228941572, 73471779239, 608000100209, 5222503739340, 46493341311706, 428345495309624, 4078254436854598, 40073317276815681, 405883920183989049, 4232700263388189325

Offset: 0

Author

Peter Luschny, Oct 02 2014

Keywords

• nonn

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

Examples

			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].
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..170
• Peter Luschny, Set partitions

Crossrefs

Cf. A016098, A005043, A000110, A000296, A106640, A247491.

Programs

Maple

A247494 := n -> add((-1)^(n-k+1)*combinat:-bell(k+1), k=0..n-1) + (-1)^n*hypergeom([-n,1/2],[2],4) - binomial(2*n,n)/(n+1):
seq(round(evalf(A247494(n),100)), n=0..25);

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 *)

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

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)]

Formula

a(n) = Sum_{k = 0..n-1} (-1)^(n-k+1)*binomial(n,k)*(Bell(k)-Catalan(k)).

a(n) = A016098(n) - A247491(n).

a(n) = A000296(n+1) - A106640(n-1), for n>0 (i.e., remove the non-crossing partitions from the cyclical adjacencies partitions). - Yuchun Ji, Nov 11 2020