A395658 - OEIS

%I #23 Jun 15 2026 10:12:36

%S 1,1,4,45,2271,1361850

%N Number of strict ternary closure operators on a set of n elements.

%C A closure operator cl on a set X is strict if the empty set is closed; it is n-ary if for all subsets S of X, cl(A) being a subset of S for all subsets A of X with cardinality no more than n implies that S is closed.

%C a(n) is also the number of ternary convexities on a set of n elements (see Kubiś).

%H Aidar Dulliev and Daniil Naumikhin, <a href="https://arxiv.org/abs/2606.11252">Binomial Transform of Sequences Counting N-ary Convexities</a>, arXiv:2606.11252 [math.GM], 2026. See p. 6 (Table 1).

%H Wiesław Kubiś, <a href="https://doi.org/10.1007/PL00012529">Separation properties of convexity spaces</a>, J. Geom. 74 (2002), 110-119.

%H Tian Vlasic, <a href="/A395658/a395658.py.txt">Python Program</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Closure_operator">Closure operator</a>.

%e The a(3) = 45 set-systems are the following ({} and {1,2,3} not shown).

%e     {1}   {1}{2}   {1}{2}{3}    {1}{2}{3}{12}   {1}{2}{3}{12}{13}

%e     {2}   {1}{3}   {1}{2}{12}   {1}{2}{3}{13}   {1}{2}{3}{12}{23}

%e     {3}   {2}{3}   {1}{2}{13}   {1}{2}{3}{23}   {1}{2}{3}{13}{23}

%e     {12}  {1}{12}  {1}{2}{23}   {1}{2}{12}{13}

%e     {13}  {1}{13}  {1}{3}{12}   {1}{2}{12}{23}

%e     {23}  {1}{23}  {1}{3}{13}   {1}{3}{12}{13}        {1}{2}{3}{12}{13}{23}

%e           {2}{12}  {1}{3}{23}   {1}{3}{13}{23}

%e           {2}{13}  {2}{3}{12}   {2}{3}{12}{23}

%e           {2}{23}  {2}{3}{13}   {2}{3}{13}{23}

%e           {3}{12}  {2}{3}{23}

%e           {3}{13}  {1}{12}{13}

%e           {3}{23}  {2}{12}{23}

%e                    {3}{13}{23}

%Y Cf. A364656, A334255, A358144, A358152, A356544.

%K nonn,hard,more

%O 0,3

%A _Tian Vlasic_, May 02 2026