A373760 Number of noncrossing partitions of the n-set including a part containing both 1 and n (with n different from 1), with no pair of singletons {i} and {j} that can be merged into {i,j} and leave the partition a noncrossing partition.

0, 0, 1, 2, 4, 11, 30, 88, 266, 825, 2613, 8408, 27421, 90422, 300987, 1010008, 3413027, 11604237, 39668334, 136258178, 470060495, 1627913941, 5657649569, 19725571728, 68975054956, 241834515725, 849993720642, 2994348927858, 10570741932441, 37390372928207, 132497284947463

Offset: 0

Author

Julien Rouyer, Jun 17 2024

Keywords

• nonn

Examples

			For n=3, the a(3)=2 partitions are {{1,3},{2}} and {{1,2,3}}.
For n=4, the a(4)=4 partitions are {{1,4},{2,3}}, {{1,2,4},{3}}, {{1,3,4},{2}} and {{1,2,3,4}}.
		

Links

• Julien Rouyer, Table of n, a(n) for n = 0..47

Crossrefs

Cf. A363448 (lonely singles partitions), A363449 (marriageable singles partitions), A000108 (noncrossing partitions).

Programs

Sage

t, P, Q = var('t, P, Q')
P = Q / ( 1 - Q ) + t / ( 1 - Q )^2 + 1
solQ=solve([Q == t / (1 - t * P) - t],Q)
q=solQ[1].rhs()
n = 47
DL_Q = (taylor(q, t,0,n)).simplify_full()
Qn = DL_Q.list()
# Julien Rouyer, Wenjie Fang, and Alain Ninet, Jun 17 2024

Formula

With P the generating function of A363448, the generating function Q of (a(n)) is a solution of the system of two equations

P(t)=Q(t)/(1-Q(t))+t/(1-Q(t))^2+1

Q(t)=t/(1-tP(t))-t.