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Number of inequivalent maximal antichains of the Boolean lattice on a set of n elements.
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%I #45 Sep 06 2023 01:40:23

%S 1,2,3,5,10,30,233,35925

%N Number of inequivalent maximal antichains of the Boolean lattice on a set of n elements.

%C a(n) is the number of orbits for the corresponding families of maximal antichains (see also A326358) of the powerset of {1,2,...,n} under the action of the symmetry group S_n.

%H Dmitry I. Ignatov, <a href="https://doi.org/10.1134/S1995080223010158">On the Number of Maximal Antichains in Boolean Lattices for n up to 7</a>. Lobachevskii J. Math., 44 (2023), 137-146.

%H Dmitry I. Ignatov, <a href="https://github.com/dimachine/NonEquivMACs/">Supporting iPython code and input files for generating and counting inequivalent maximal antichains for n up to 7 and related sequences</a>, Github repository, sections 0,1,4

%H Dmitry I. Ignatov, <a href="/A348260/a348260.ipynb.txt">Supporting iPython notebook: A348260.ipynb</a>

%H Dmitry I. Ignatov, <a href="/A348260/a348260.pdf">PDF of the supporting iPython notebook</a>

%e The a(0)=1 maximal antichains is {}.

%e The a(1)=2 maximal antichains are {}, {1}.

%e The a(2)=3 maximal antichains {}, {1}{2}, {12}.

%e Representatives of the a(3)=5 maximal antichains are: {}, {1}{2}{3}, {12}{3}, {12}{13}{23}, {123}.

%e Representatives of the a(4)=10 maximal antichains are:

%e {}, {1}{2}{3}{4},

%e {12}{3}{4}, {12}{13}{23}{4},

%e {123}{4}, {12}{13}{24}{14}{24}{34},

%e {123}{14}{24}{34}, {123}{124}{34},

%e {123}{124}{134}{234}, {1234}.

%Y Cf. A003182 (not necessarily maximal), A326358 (labeled case).

%Y Cf. A326359, A326360.

%K nonn,more

%O 0,2

%A _Dmitry I. Ignatov_, Oct 13 2021