cp's OEIS Frontend

A322442 Number of pairs of set partitions of {1,...,n} where every block of one is a subset or superset of some block of the other.

%I A322442 #10 Jan 20 2024 11:27:29
%S A322442 1,1,4,25,195,1894,22159,303769,4790858,85715595,1720097275,
%T A322442 38355019080,942872934661,25383601383937,744118939661444,
%U A322442 23635548141900445,809893084668253151,29822472337116844174,1175990509568611058299,49504723853840395163221,2218388253903492656783562
%N A322442 Number of pairs of set partitions of {1,...,n} where every block of one is a subset or superset of some block of the other.
%H A322442 Andrew Howroyd, <a href="/A322442/b322442.txt">Table of n, a(n) for n = 0..200</a>
%F A322442 E.g.f.: exp(exp(x)-1) * (2*B(x) - 1) where B(x) is the e.g.f. of A319884. - _Andrew Howroyd_, Jan 19 2024
%e A322442 The a(3) = 25 pairs of set partitions (these are actually all pairs of set partitions of {1,2,3}):
%e A322442   (1)(2)(3)|(1)(2)(3)
%e A322442   (1)(2)(3)|(1)(23)
%e A322442   (1)(2)(3)|(12)(3)
%e A322442   (1)(2)(3)|(13)(2)
%e A322442   (1)(2)(3)|(123)
%e A322442     (1)(23)|(1)(2)(3)
%e A322442     (1)(23)|(1)(23)
%e A322442     (1)(23)|(12)(3)
%e A322442     (1)(23)|(13)(2)
%e A322442     (1)(23)|(123)
%e A322442     (12)(3)|(1)(2)(3)
%e A322442     (12)(3)|(1)(23)
%e A322442     (12)(3)|(12)(3)
%e A322442     (12)(3)|(13)(2)
%e A322442     (12)(3)|(123)
%e A322442     (13)(2)|(1)(2)(3)
%e A322442     (13)(2)|(1)(23)
%e A322442     (13)(2)|(12)(3)
%e A322442     (13)(2)|(13)(2)
%e A322442     (13)(2)|(123)
%e A322442       (123)|(1)(2)(3)
%e A322442       (123)|(1)(23)
%e A322442       (123)|(12)(3)
%e A322442       (123)|(13)(2)
%e A322442       (123)|(123)
%e A322442 Non-isomorphic representatives of the pairs of set partitions of {1,2,3,4} for which the condition fails:
%e A322442     (12)(34)|(13)(24)
%e A322442     (12)(34)|(1)(3)(24)
%e A322442   (1)(2)(34)|(13)(24)
%t A322442 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A322442 costabQ[s_,t_]:=And@@Cases[s,x_:>Select[t,SubsetQ[x,#]||SubsetQ[#,x]&]!={}];
%t A322442 Table[Length[Select[Tuples[sps[Range[n]],2],And[costabQ@@#,costabQ@@Reverse[#]]&]],{n,5}]
%o A322442 (PARI)
%o A322442 F(x)={my(bell=(exp(y*(exp(x) - 1))  )); subst(serlaplace( serconvol(bell, bell)), y, exp(exp(x) - 1)-1)}
%o A322442 seq(n) = {my(x=x + O(x*x^n)); Vec(serlaplace( exp( 2*exp(exp(x) - 1) - exp(x) - 1) * F(x) ))} \\ _Andrew Howroyd_, Jan 19 2024
%Y A322442 Cf. A000110, A000258, A001247, A008277, A059849, A060639, A181939, A318393, A322435, A322437, A322439, A322440, A322441.
%K A322442 nonn
%O A322442 0,3
%A A322442 _Gus Wiseman_, Dec 08 2018
%E A322442 a(8) onwards from _Andrew Howroyd_, Jan 19 2024