A319564 -id:A319564 - cp's OEIS frontend

A059201 Number of T_0-covers of a labeled n-set.

1, 1, 4, 96, 31692, 2147001636, 9223371991763269704, 170141183460469231473432887375376674952, 57896044618658097711785492504343953920509909728243389682424010192567186540224

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Jan 16 2001

A cover of a set is a T_0-cover if for every two distinct points of the set there exists a member (block) of the cover containing one but not the other point.
From Gus Wiseman, Aug 13 2019: (Start)
A set-system is a finite set of finite nonempty sets. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges). For example, the a(2) = 4 covers are:
  {{1},{2}}
  {{1},{1,2}}
  {{2},{1,2}}
  {{1},{2},{1,2}}
(End)

G. C. Greubel, Table of n, a(n) for n = 0..11
Vladeta Jovovic, T_0-covers of a labeled 3-set

Row sums of A059202.
Cf. A059203, A059084, A059085, A059086, A059087, A059088, A059089.
Covering set-systems are A003465.
The unlabeled version is A319637.
The version with empty edges allowed is A326939.
The non-covering version is A326940.
BII-numbers of T_0 set-systems are A326947.
The same with connected instead of covering is A326948.
The T_1 version is A326961.
Cf. A245567, A316978, A319559, A319564, A323818, A326941, A326946, A326970.

Table[Sum[StirlingS1[n + 1, k]*2^(2^(k - 1) - 1), {k, 0, n + 1}], {n,0,5}] (* G. C. Greubel, Dec 28 2016 *)
dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&UnsameQ@@dual[#]&]],{n,0,3}] (* Gus Wiseman, Aug 13 2019 *)

a(n) = Sum_{i=0..n+1} stirling1(n+1, i)*2^(2^(i-1)-1).
a(n) = Sum_{m=0..2^n-1} A059202(n,m).
Inverse binomial transform of A326940 and exponential transform of A326948. - Gus Wiseman, Aug 13 2019

A001035 Number of partially ordered sets ("posets") with n labeled elements (or labeled acyclic transitive digraphs).

Original entry on oeis.org

1, 1, 3, 19, 219, 4231, 130023, 6129859, 431723379, 44511042511, 6611065248783, 1396281677105899, 414864951055853499, 171850728381587059351, 98484324257128207032183, 77567171020440688353049939, 83480529785490157813844256579, 122152541250295322862941281269151, 241939392597201176602897820148085023

Offset: 0

N. J. A. Sloane

From Altug Alkan, Dec 22 2015: (Start)
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

			R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, Chap. 3, page 98, Fig. 3-1 shows the unlabeled posets with <= 4 points.
From _Gus Wiseman_, Aug 14 2019: (Start)
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)

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.

Christian Bean, Émile Nadeau, Jay Pantone, and Henning Ulfarsson, Permutations avoiding bipartite partially ordered patterns have a regular insertion encoding, The Electronic Journal of Combinatorics, Volume 31, Issue 3 (2024); arXiv preprint, arXiv:2312.07716 [math.CO], 2023.
Juliana Bowles and Marco B. Caminati, A Verified Algorithm Enumerating Event Structures, arXiv:1705.07228 [cs.LO], 2017.
G. Brinkmann and B. D. McKay, Posets on up to 16 Points, Order 19 (2) (2002) 147-179.
J. I. Brown and S. Watson, The number of complements of a topology on n points is at least 2^n (except for some special cases), Discr. Math., 154 (1996), 27-39.
K. K.-H. Butler, The number of partially ordered sets, Journal of Combinatorial Theory, Series B 13.3 (1972): 276-289.
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]
K. K.-H. Butler and G. Markowsky, The number of partially ordered sets. II., J. Korean Math. Soc 11 (1974): 7-17.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
S. D. Chatterji, The number of topologies on n points, Manuscript, 1966. [Annotated scanned copy]
Narendrakumar R. Dasre and Pritam Gujarathi, Approximating the Bounds for Number of Partially Ordered Sets with n Labeled Elements, Computing in Engineering and Technology, Advances in Intelligent Systems and Computing, Vol. 1025, Springer (Singapore 2019), 349-356.
M. Erné, Struktur- und Anzahlformeln für Topologien auf Endlichen Mengen, Manuscripta Math., 11 (1974), 221-259.
M. Erné, Struktur- und Anzahlformeln für Topologien auf Endlichen Mengen, Manuscripta Math., 11 (1974), 221-259. (Annotated scanned copy)
M. Erné and K. Stege, The number of partially ordered (labeled) sets, Preprint, 1989. (Annotated scanned copy)
M. Erné and K. Stege, Counting Finite Posets and Topologies, Order, 8 (1991), 247-265.
J. W. Evans, F. Harary and M. S. Lynn, On the computer enumeration of finite topologies, Commun. ACM, 10 (1967), 295-297, 313.
J. W. Evans, F. Harary and M. S. Lynn, On the computer enumeration of finite topologies, Commun. ACM, 10 (1967), 295-297, 313. [Annotated scanned copy]
S. R. Finch, Transitive relations, topologies and partial orders.
S. R. Finch, Transitive relations, topologies and partialorders, June 5, 2003. [Cached copy, with permission of the author]
Eldar Fischer, Johann A. Makowsky, and Vsevolod Rakita, MC-finiteness of restricted set partition functions, arXiv:2302.08265 [math.CO], 2023.
Didier Garcia, Proof of a(19) formula [in French].
Didier Garcia, Two conjectures concerning a(n) [in French].
Joël Gay and Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.
G. Grekos, Letter to N. J. A. Sloane, Oct 31 1994, with attachments.
J. Heitzig and J. Reinhold, The number of unlabeled orders on fourteen elements, Order 17 (2000) no. 4, 333-341.
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.
Dongseok Kim, Young Soo Kwon, and Jaeun Lee, Enumerations of finite topologies associated with a finite graph, arXiv preprint arXiv:1206.0550 [math.CO], 2012. - From _N. J. A. Sloane_, Nov 09 2012
M. Y. Kizmaz, On The Number Of Topologies On A Finite Set, arXiv preprint arXiv:1503.08359 [math.NT], 2015.
D. J. Kleitman and B. L. Rothschild, Asymptotic enumeration of partial orders on a finite set, Trans. Amer. Math. Soc., 205 (1975) 205-220.
G. Kreweras, Dénombrement des ordres étagés, Discrete Math., 53 (1985), 147-149.
Institut f. Mathematik, Univ. Hanover, Erne/Heitzig/Reinhold papers.
Sami Lazaar, Houssem Sabri, and Randa Tahri, Structural and Numerical Studies of Some Topological Properties for Alexandroff Spaces, Bull. Iran. Math. Soc. (2021).
N. Lygeros and P. Zimmermann, Computation of P(14), the number of posets with 14 elements: 1.338.193.159.771.
G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
Bob Proctor, Chapel Hill Poset Atlas.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Ivo Rosenberg, The number of maximal closed classes in the set of functions over a finite domain, J. Combinatorial Theory Ser. A 14 (1973), 1-7.
Ivo Rosenberg and N. J. A. Sloane, Correspondence, 1971.
D. Rusin, Further information and references. [Broken link]
D. Rusin, Further information and references. [Cached copy]
A. Shafaat, On the number of topologies definable for a finite set, J. Austral. Math. Soc., 8 (1968), 194-198.
N. J. A. Sloane, List of sequences related to partial orders, circa 1972.
N. J. A. Sloane, List of sequences related to partial orders, circa 1972.
N. J. A. Sloane, Classic Sequences.
Gus Wiseman, Hasse diagrams of the a(4) = 219 posets.
J. A. Wright, There are 718 6-point topologies, quasiorderings and transgraphs, Preprint, 1970. [Annotated scanned copy]
J. A. Wright, Letter to N. J. A. Sloane, Apr 06 1972, listing 18 sequences.
Index entries for sequences related to posets

Cf. A000798 (labeled topologies), A001930 (unlabeled topologies), A000112 (unlabeled posets), A006057.
Sequences in the Erné (1974) paper: A000798, A001035, A006056, A006057, A001929, A001927, A006058, A006059, A000110.
Cf. A316978, A319564, A326876, A326906, A326939, A326943, A326944, A326947.

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 *)

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).
From Altug Alkan, Dec 22 2015: (Start)
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

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

A319559 Number of non-isomorphic T_0 set systems of weight n.

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 16, 35, 82, 200, 517, 1373, 3867, 11216, 33910, 105950

Offset: 0

Gus Wiseman, Sep 23 2018

In a set system, two vertices are equivalent if in every block the presence of the first is equivalent to the presence of the second. The T_0 condition means that there are no equivalent vertices.

The weight of a set system is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 7 set systems:
1:        {{1}}
2:      {{1},{2}}
3:     {{2},{1,2}}
      {{1},{2},{3}}
4:    {{1,3},{2,3}}
     {{1},{2},{1,2}}
     {{1},{3},{2,3}}
    {{1},{2},{3},{4}}
5:  {{1},{2,4},{3,4}}
    {{2},{3},{1,2,3}}
    {{2},{1,3},{2,3}}
    {{3},{1,3},{2,3}}
   {{1},{2},{3},{2,3}}
   {{1},{2},{4},{3,4}}
  {{1},{2},{3},{4},{5}}

Cf. A007716, A007718, A049311, A053419, A056156, A059201, A283877, A305854, A306006, A316980, A317757.

Cf. A319557, A319558, A319560, A319564, A319565, A319566, A319567.

a(11)-a(15) from Bert Dobbelaere, May 04 2025

A319558 The squarefree dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted without multiplicity. Then a(n) is the number of non-isomorphic multiset partitions of weight n whose squarefree dual is strict (no repeated blocks).

Original entry on oeis.org

1, 1, 3, 7, 21, 55, 169, 496, 1582, 5080, 17073

Offset: 0

Gus Wiseman, Sep 23 2018

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1, a(2) = 3, and a(3) = 7 multiset partitions:
1:    {{1}}
2:   {{1,1}}
    {{1},{1}}
    {{1},{2}}
3:  {{1,1,1}}
   {{1},{1,1}}
   {{1},{2,2}}
   {{2},{1,2}}
  {{1},{1},{1}}
  {{1},{2},{2}}
  {{1},{2},{3}}

Cf. A007716, A007718, A049311, A053419, A056156, A059201, A283877, A316980.

Cf. A319557, A319559, A319560, A319564, A319565, A319566, A319567.

A319557 Number of non-isomorphic strict connected multiset partitions of weight n.

Original entry on oeis.org

1, 1, 2, 5, 12, 30, 91, 256, 823, 2656, 9103, 31876, 116113, 432824, 1659692, 6508521, 26112327, 106927561, 446654187, 1900858001, 8236367607, 36306790636, 162724173883, 741105774720, 3428164417401, 16099059101049, 76722208278328, 370903316203353, 1818316254655097

Offset: 0

Gus Wiseman, Sep 23 2018

nonn

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Also the number of non-isomorphic connected T_0 multiset partitions of weight n. In a multiset partition, two vertices are equivalent if in every block the multiplicity of the first is equal to the multiplicity of the second. The T_0 condition means that there are no equivalent vertices.

			Non-isomorphic representatives of the a(4) = 12 strict connected multiset partitions:
    {{1,1,1,1}}
    {{1,1,2,2}}
    {{1,2,2,2}}
    {{1,2,3,3}}
    {{1,2,3,4}}
   {{1},{1,1,1}}
   {{1},{1,2,2}}
   {{2},{1,2,2}}
   {{3},{1,2,3}}
   {{1,2},{2,2}}
   {{1,3},{2,3}}
  {{1},{2},{1,2}}
Non-isomorphic representatives of the a(4) = 12 connected T_0 multiset partitions:
     {{1,1,1,1}}
     {{1,2,2,2}}
    {{1},{1,1,1}}
    {{1},{1,2,2}}
    {{2},{1,2,2}}
    {{1,1},{1,1}}
    {{1,2},{2,2}}
    {{1,3},{2,3}}
   {{1},{1},{1,1}}
   {{1},{2},{1,2}}
   {{2},{2},{1,2}}
  {{1},{1},{1},{1}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A007716, A007718, A049311, A056156, A283877, A316980.

Cf. A319558, A319559, A319560, A319564, A319565, A319566, A319567.

Inverse Euler transform of A316980.

Terms a(11) and beyond from Andrew Howroyd, Jan 19 2023

A326947 BII-numbers of T_0 set-systems.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 67, 69, 70, 71, 73, 74, 75, 77, 78

Offset: 1

Gus Wiseman, Aug 08 2019

The dual of a set-system has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

			The sequence of all T_0 set-systems together with their BII numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   3: {{1},{2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   7: {{1},{2},{1,2}}
   8: {{3}}
   9: {{1},{3}}
  10: {{2},{3}}
  11: {{1},{2},{3}}
  13: {{1},{1,2},{3}}
  14: {{2},{1,2},{3}}
  15: {{1},{2},{1,2},{3}}
  17: {{1},{1,3}}
  19: {{1},{2},{1,3}}
  20: {{1,2},{1,3}}
  21: {{1},{1,2},{1,3}}
  22: {{2},{1,2},{1,3}}
  23: {{1},{2},{1,2},{1,3}}

John Tyler Rascoe, Table of n, a(n) for n = 1..10000

T_0 set-systems are counted by A326940, with unlabeled version A326946.

Cf. A059201, A316978, A319559, A319564, A326939, A326941, A326949.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
TZQ[sys_]:=UnsameQ@@dual[sys];
Select[Range[0,100],TZQ[bpe/@bpe[#]]&]

from itertools import count, chain, islice
def bin_i(n): #binary indices
    return([(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1'])
def a_gen():
    for n in count(0):
        a,b,s = [bin_i(k) for k in bin_i(n)],[],set()
        for i in {i for i in chain.from_iterable(a)}:
            b.append([])
            for j in range(len(a)):
                if i in a[j]:
                    b[-1].append(j)
            s.add(tuple(b[-1]))
        if len(s) == len(b):
            yield n
A326947_list = list(islice(a_gen(), 100)) # John Tyler Rascoe, Jul 25 2024

cp's OEIS Frontend

A059201 Number of T_0-covers of a labeled n-set.

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A001035 Number of partially ordered sets ("posets") with n labeled elements (or labeled acyclic transitive digraphs).

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A319559 Number of non-isomorphic T_0 set systems of weight n.

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A319557 Number of non-isomorphic strict connected multiset partitions of weight n.

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A326947 BII-numbers of T_0 set-systems.

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A319565 Number of non-isomorphic connected strict T_0 multiset partitions of weight n.

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A319560 Number of non-isomorphic strict T_0 multiset partitions of weight n.

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A326946 Number of unlabeled T_0 set-systems on n vertices.

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