A001035 - OEIS

a(p^k) == 1 (mod p) and a(n + p) == a(n + 1) (mod p) for all primes p.

a(0+19) == a(0+1) (mod 19) or a(19^1) == 1 (mod 19), that is, a(19) mod 19 = 1.

a(2+17) == a(2+1) (mod 17). So a(19) == 19 (mod 17), that is, a(19) mod 17 = 2.

a(6+13) == a(6+1) (mod 13). So a(19) == 6129859 (mod 13), that is, a(19) mod 13 = 8.

a(8+11) == a(8+1) (mod 11). So a(19) == 44511042511 (mod 11), that is, a(19) mod 11 = 1.

a(12+7) == a(12+1) (mod 7). So a(19) == 171850728381587059351 (mod 7), that is, a(19) mod 7 = 1.

a(14+5) == a(14+1) (mod 5). So a(19) == 77567171020440688353049939 (mod 5), that is, a(19) mod 5 = 4.

a(16+3) == a(16+1) (mod 3). So a(19) == 122152541250295322862941281269151 (mod 3), that is, a(19) mod 3 = 1.

a(17+2) == a(17+1) (mod 2). So a(19) mod 2 = 1.

In conclusion, a(19) is a number of the form 2*3*5*7*11*13*17*19*n - 1615151, that is, 9699690*n - 1615151.

Additionally, for n > 0, note that the last digit of a(n) has the simple periodic pattern: 1,3,9,9,1,3,9,9,1,3,9,9,...

(End)

Number of rank n sublattices of the Boolean algebra B_n. - Kevin Long, Nov 20 2018

a(n) is the number of n X n idempotent Boolean relation matrices (A121337) that have rank n. - Geoffrey Critzer, Aug 16 2023

a(19) == 163279579 (mod 232792560). - Didier Garcia, Feb 06 2025

G. Birkhoff, Lattice Theory, Amer. Math. Soc., 1961, p. 4.

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 427.

K. K.-H. Butler, A Moore-Penrose inverse for Boolean relation matrices, pp. 18-28 of Combinatorial Mathematics (Proceedings 2nd Australian Conf.), Lect. Notes Math. 403, 1974.

K. K.-H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184.

K. K. H. Butler and G. Markowsky. "The number of partially ordered sets. I." Journal of Korean Mathematical Society 11.1 (1974).

S. D. Chatterji, The number of topologies on n points, Manuscript, 1966.

L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 60, 229.

M. Erné, Struktur- und Anzahlformeln für Topologien auf endlichen Mengen, PhD dissertation, Westfälische Wilhelms-Universität zu Münster, 1972.

M. Erné and K. Stege, The number of labeled orders on fifteen elements, personal communication.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, Chap. 3, pages 96ff; Vol. 2, Problem 5.39, p. 88.

K. K.-H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184.

K. K.-H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184. [Annotated scan of pages 180 and 183 only]

Richard Kenyon, Maxim Kontsevich, Oleg Ogievetsky, Cosmin Pohoata, Will Sawin, and Senya Shlosman, The miracle of integer eigenvalues, arXiv:2401.05291 [math.CO], 2024. See p. 4.

A000798(n) = Sum_{k=0..n} Stirling2(n,k)*a(k).

Related to A000112 by Erné's formulas: a(n+1) = -s(n, 1), a(n+2) = n*a(n+1) + s(n, 2), a(n+3) = binomial(n+4, 2)*a(n+2) - s(n, 3), where s(n, k) = sum(binomial(n+k-1-m, k-1)*binomial(n+k, m)*sum((m!)/(number of automorphisms of P)*(-(number of antichains of P))^k, P an unlabeled poset with m elements), m=0..n).

a(p^k) == 1 (mod p) for all primes p and for all nonnegative integers k.

a(n + p) == a(n + 1) (mod p) for all primes p and for all nonnegative integers n.

If n = 1, then a(1 + p) == a(2) (mod p), that is, a(p + 1) == 3 (mod p).

If n = p, then a(p + p) == a(p + 1) (mod p), that is, a(2*p) == a(p + 1) (mod p).

In conclusion, a(2*p) == 3 (mod p) for all primes p.

(End)

a(n) = Sum_{k=0..n} Stirling1(n,k)*A000798(k). - Tian Vlasic, Feb 25 2022

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, Chap. 3, page 98, Fig. 3-1 shows the unlabeled posets with <= 4 points.

Also the number of T_0 topologies with n points. For example, the a(0) = 1 through a(3) = 19 topologies are:

{} {}{1} {}{1}{12} {}{1}{12}{123}

{}{2}{12} {}{1}{13}{123}

{}{1}{2}{12} {}{2}{12}{123}

{}{2}{23}{123}

{}{3}{13}{123}

{}{3}{23}{123}

{}{1}{2}{12}{123}

{}{1}{3}{13}{123}

{}{2}{3}{23}{123}

{}{1}{12}{13}{123}

{}{2}{12}{23}{123}

{}{3}{13}{23}{123}

{}{1}{2}{12}{13}{123}

{}{1}{2}{12}{23}{123}

{}{1}{3}{12}{13}{123}

{}{1}{3}{13}{23}{123}

{}{2}{3}{12}{23}{123}

{}{2}{3}{13}{23}{123}

{}{1}{2}{3}{12}{13}{23}{123}

(End)

dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}];

Table[Length[Select[Subsets[Subsets[Range[n]]], MemberQ[#, {}]&&MemberQ[#, Range[n]]&&UnsameQ@@dual[#]&&SubsetQ[#, Union@@@Tuples[#, 2]]&&SubsetQ[#, Intersection@@@Tuples[#, 2]]&]], {n, 0, 3}] (* Gus Wiseman, Aug 14 2019 *)

a(15)-a(16) from Jobst Heitzig (heitzig(AT)math.uni-hannover.de), Jul 03 2000

a(17)-a(18) from Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 02 2008