A395658 Number of strict ternary closure operators on a set of n elements.

1, 1, 4, 45, 2271, 1361850

Offset

0,3

Comments

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.

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

Links

Table of n, a(n) for n=0..5.

Aidar Dulliev and Daniil Naumikhin, Binomial Transform of Sequences Counting N-ary Convexities, arXiv:2606.11252 [math.GM], 2026. See p. 6 (Table 1).

Wiesław Kubiś, Separation properties of convexity spaces, J. Geom. 74 (2002), 110-119.

Tian Vlasic, Python Program.

Wikipedia, Closure operator.

Example

The a(3) = 45 set-systems are the following ({} and {1,2,3} not shown).
    {1}   {1}{2}   {1}{2}{3}    {1}{2}{3}{12}   {1}{2}{3}{12}{13}
    {2}   {1}{3}   {1}{2}{12}   {1}{2}{3}{13}   {1}{2}{3}{12}{23}
    {3}   {2}{3}   {1}{2}{13}   {1}{2}{3}{23}   {1}{2}{3}{13}{23}
    {12}  {1}{12}  {1}{2}{23}   {1}{2}{12}{13}
    {13}  {1}{13}  {1}{3}{12}   {1}{2}{12}{23}
    {23}  {1}{23}  {1}{3}{13}   {1}{3}{12}{13}        {1}{2}{3}{12}{13}{23}
          {2}{12}  {1}{3}{23}   {1}{3}{13}{23}
          {2}{13}  {2}{3}{12}   {2}{3}{12}{23}
          {2}{23}  {2}{3}{13}   {2}{3}{13}{23}
          {3}{12}  {2}{3}{23}
          {3}{13}  {1}{12}{13}
          {3}{23}  {2}{12}{23}
                   {3}{13}{23}

Crossrefs

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

Sequence in context: A335014 A396809 A364656 * A102894 A132552 A189273

Adjacent sequences: A395655 A395656 A395657 * A395659 A395660 A395662

Keyword

nonn,hard,more

Author

Tian Vlasic, May 02 2026

Status

approved