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A304902 Let (P,<) be the strict partial order on the subsets of {1,2,...,n} ordered by their cardinality. Then a(n) is the number of paths of any length from {} to {1,2,...,n}.

%I A304902 #11 May 29 2018 19:20:54
%S A304902 1,1,3,16,175,4356,263424,40144896,15714084159,15953234222500,
%T A304902 42223789335548788,292262228709213966336,5302397936652484482131200,
%U A304902 252622720869371754406993137664,31660291085217875120800516475520000,10454334647424614439930776175842716286976
%N A304902 Let (P,<) be the strict partial order on the subsets of {1,2,...,n} ordered by their cardinality.  Then a(n) is the number of paths of any length from {} to {1,2,...,n}.
%C A304902 A001142 counts such paths of length n.
%C A304902 A000670 counts such paths under the inclusion relation.
%H A304902 Alois P. Heinz, <a href="/A304902/b304902.txt">Table of n, a(n) for n = 0..69</a>
%p A304902 b:= proc(n, k) option remember; `if`(k=0, 1,
%p A304902       add(b(n, j), j=0..k-1)*binomial(n, k))
%p A304902     end:
%p A304902 a:= n-> b(n$2):
%p A304902 seq(a(n), n=0..17);  # _Alois P. Heinz_, May 20 2018
%t A304902 Table[f[list_] := Apply[Times, Map[Binomial[n, #] &, list]];
%t A304902 Total[Map[f, Map[Accumulate,Level[Map[Permutations, Partitions[n]], {2}]]]], {n, 0, 15}]
%Y A304902 Cf. A000670, A001142.
%K A304902 nonn
%O A304902 0,3
%A A304902 _Geoffrey Critzer_, May 20 2018