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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.
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%I #12 Jun 17 2024 08:26:45

%S 0,0,1,2,4,11,30,88,266,825,2613,8408,27421,90422,300987,1010008,

%T 3413027,11604237,39668334,136258178,470060495,1627913941,5657649569,

%U 19725571728,68975054956,241834515725,849993720642,2994348927858,10570741932441,37390372928207,132497284947463

%N 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.

%H Julien Rouyer, <a href="/A373760/b373760.txt">Table of n, a(n) for n = 0..47</a>

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

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

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

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

%e 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}}.

%o (Sage) t, P, Q = var('t, P, Q')

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

%o solQ=solve([Q == t / (1 - t * P) - t],Q)

%o q=solQ[1].rhs()

%o n = 47

%o DL_Q = (taylor(q, t,0,n)).simplify_full()

%o Qn = DL_Q.list()

%o # _Julien Rouyer_, _Wenjie Fang_, and Alain Ninet, Jun 17 2024

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

%K nonn

%O 0,4

%A _Julien Rouyer_, Jun 17 2024