A281774 -id:A281774 - cp's OEIS frontend

A326882 Irregular triangle read by rows where T(n,k) is the number of finite topologies with n points and k nonempty open sets, 0 <= k <= 2^n - 1.

1, 0, 1, 0, 1, 2, 1, 0, 1, 6, 9, 6, 6, 0, 1, 0, 1, 14, 43, 60, 72, 54, 54, 20, 24, 0, 12, 0, 0, 0, 1, 0, 1, 30, 165, 390, 630, 780, 955, 800, 900, 500, 660, 240, 390, 120, 190, 10, 100, 0, 60, 0, 0, 0, 20, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Gus Wiseman, Aug 01 2019

			Triangle begins:
  1
  0  1
  0  1  2  1
  0  1  6  9  6  6  0  1
  0  1 14 43 60 72 54 54 20 24  0 12  0  0  0  1
Row n = 3 counts the following topologies:
{}{123} {}{1}{123}  {}{1}{12}{123} {}{1}{2}{12}{123}  {}{1}{2}{12}{13}{123}
        {}{2}{123}  {}{1}{13}{123} {}{1}{3}{13}{123}  {}{1}{2}{12}{23}{123}
        {}{3}{123}  {}{1}{23}{123} {}{2}{3}{23}{123}  {}{1}{3}{12}{13}{123}
        {}{12}{123} {}{2}{12}{123} {}{1}{12}{13}{123} {}{1}{3}{13}{23}{123}
        {}{13}{123} {}{2}{13}{123} {}{2}{12}{23}{123} {}{2}{3}{12}{23}{123}
        {}{23}{123} {}{2}{23}{123} {}{3}{13}{23}{123} {}{2}{3}{13}{23}{123}
                    {}{3}{12}{123}
                    {}{3}{13}{123}      {}{1}{2}{3}{12}{13}{23}{123}
                    {}{3}{23}{123}

Andrew Howroyd, Table of n, a(n) for n = 0..254
Wikipedia, Topological space

Row lengths are A000079.
Row sums are A000798.
Cf. A001930, A014466, A102894, A102895, A102896, A102897, A306445, A326876, A326878, A326881.
Columns: A281774 and refs therein.

Table[Length[Select[Subsets[Subsets[Range[n]],{k}],MemberQ[#,{}]&&MemberQ[#,Range[n]]&&SubsetQ[#,Union[Union@@@Tuples[#,2],Intersection@@@Tuples[#,2]]]&]],{n,0,4},{k,2^n}]

Terms a(31) and beyond from Andrew Howroyd, Aug 10 2019

A281773 Number of distinct topologies on an n-set that have exactly 4 open sets.

Original entry on oeis.org

0, 0, 1, 9, 43, 165, 571, 1869, 5923, 18405, 56491, 172029, 521203, 1573845, 4742011, 14266989, 42882883, 128812485, 386765131, 1160950749, 3484162963, 10455110325, 31370573851, 94122207309, 282387593443, 847204723365, 2541698056171, 7625261940669

Offset: 0

Submitted on behalf of Moussa Benoumhani by Geoffrey Critzer, Jan 29 2017

			a(3) = 9 because we have: {{}, {c}, {a,b}, {a,b,c}} with 3 labelings and {{}, {c}, {b,c}, {a,b,c}} with 6 labelings.

Colin Barker, Table of n, a(n) for n = 0..1000
Moussa Benoumhani, The Number of Topologies on a Finite Set, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.6.
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

The number of distinct topologies on an n-set with exactly k open sets for k=2..12 is given by A000012, A000918, A281773, A028244, A281774, A281775, A281776, A281777, A281778, A281779, A281780.

Partial sums are given in A298564.

CoefficientList[Series[x^2*(1 + 3 x)/((1 - x) (1 - 2 x) (1 - 3 x)), {x, 0, 27}], x] (* Michael De Vlieger, Jan 21 2018 *)

a(n) = stirling(n,2,2) + 3!*stirling(n,3,2) \\ Colin Barker, Jan 30 2017

concat(vector(2), Vec(x^2*(1 + 3*x) / ((1 - x)*(1 - 2*x)*(1 - 3*x)) + O(x^30))) \\ Colin Barker, Jan 30 2017

a(n) = A000392(n+1) + 3*A000392(n).
E.g.f.: (exp(x)-1)^3 + (exp(x)-1)^2/2!.
From Colin Barker, Jan 30 2017: (Start)
G.f.: x^2*(1 + 3*x)/((1 - x)*(1 - 2*x)*(1 - 3*x)).
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n>3.
a(n) = 2 - 5*2^(n-1) + 3^n for n>0. (End)

A281775 Number of distinct topologies on an n-set that have exactly 7 open sets.

Original entry on oeis.org

0, 0, 0, 0, 54, 780, 7830, 67620, 535374, 3992940, 28483110, 196316340, 1317106494, 8650141500, 55853351190, 355770438660, 2241509994414, 13998294536460, 86795899256070, 535048203626580, 3282628800655134, 20061393719417820, 122212221633141750

Offset: 0

Submitted on behalf of Moussa Benoumhani by Geoffrey Critzer, Jan 29 2017

Colin Barker, Table of n, a(n) for n = 0..1000
Moussa Benoumhani, The Number of Topologies on a Finite Set, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.6.
Index entries for linear recurrences with constant coefficients, signature (21,-175,735,-1624,1764,-720).

The number of distinct topologies on an n-set with exactly k open sets for k=2..12 is given by A000012, A000918, A281773, A281774, A028244, A281775, A281776, A281777, A281778, A281779, A281780.

a(n) = 9*4!*stirling(n, 4, 2)/4 + 2*5!*stirling(n, 5, 2) + 6!*stirling(n, 6, 2) \\ Colin Barker, Jan 30 2017

concat(vector(4), Vec(6*x^4*(9 - 59*x + 150*x^2) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)) + O(x^30))) \\ Colin Barker, Jan 30 2017

a(n) = 9/4*4! Stirling2(n, 4) + 2*5! Stirling2(n, 5) + 6! Stirling2(n, 6).
From Colin Barker, Jan 30 2017: (Start)
G.f.: 6*x^4*(9 - 59*x + 150*x^2)/((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)).
a(n) = 21*a(n-1) - 175*a(n-2) + 735*a(n-3) - 1624*a(n-4) + 1764*a(n-5) - 720*a(n-6) for n>6.
a(n) = -5 + 17*2^(n-1) - 3^(2+n) + 29*4^(n-1) - 4*5^n + 6^n for n>0. (End)

A281776 Number of distinct topologies on an n-set that have exactly 8 open sets.

Original entry on oeis.org

0, 0, 0, 1, 54, 955, 11760, 122941, 1175034, 10595215, 91506420, 763624081, 6194818014, 49084747075, 381338401080, 2914184784421, 21965095364994, 163656285828535, 1207613518375740, 8838842878371961, 64253768864671974, 464416229729871595, 3340518964319750400

Offset: 0

Geoffrey Critzer, Jan 29 2017

Colin Barker, Table of n, a(n) for n = 0..1000
Moussa Benoumhani, The Number of Topologies on a Finite Set, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.6.
Index entries for linear recurrences with constant coefficients, signature (28,-322,1960,-6769,13132,-13068,5040).

The number of distinct topologies on an n-set with exactly k open sets for k=2..12 is given by A000012, A000918, A281773, A028244, A281774, A281775, A281776, A281777, A281778, A281779, A281780.

concat(vector(3), Vec(x^3*(1 + 26*x - 235*x^2 + 448*x^3 + 2100*x^4) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)) + O(x^30))) \\ Colin Barker, Jan 30 2017

a(n) = Stirling2(n, 3) + 2*4! Stirling2(n, 4) + 15/4*5! Stirling2(n, 5) + 5/2*6! Stirling2(n, 6) + 7! Stirling2(n, 7).
From Colin Barker, Jan 30 2017: (Start)
a(n) = 13/4 - 19*2^(n-1) + 44*3^(n-1) - 2^(n-1)*3^(2+n) - 57*4^(n-1) + (39*5^n)/4 + 7^n for n>0.
G.f.: x^3*(1 + 26*x - 235*x^2 + 448*x^3 + 2100*x^4) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)).
(End)

A281777 Number of distinct topologies on an n-set that have exactly 9 open sets.

Original entry on oeis.org

0, 0, 0, 0, 20, 800, 14260, 189280, 2181060, 23241120, 235737620, 2308206560, 21979728100, 204477713440, 1864504348980, 16707856095840, 147469451067140, 1284607771225760, 11063319237792340, 94343562846289120, 797685042851814180, 6694943490279586080

Offset: 0

Submitted on behalf of Moussa Benoumhani by Geoffrey Critzer, Jan 29 2017

Colin Barker, Table of n, a(n) for n = 0..1000
Moussa Benoumhani, The Number of Topologies on a Finite Set, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.6.
Index entries for linear recurrences with constant coefficients, signature (36,-546,4536,-22449,67284,-118124,109584,-40320).

The number of distinct topologies on an n-set with exactly k open sets for k=2..12 is given by A000012, A000918, A281773, A028244, A281774, A281775, A281776, A281777,A281778, A281779, A281780.

LinearRecurrence[{36,-546,4536,-22449,67284,-118124,109584,-40320},{0,0,0,0,20,800,14260,189280,2181060},30] (* Harvey P. Dale, Aug 19 2020 *)

concat(vector(4), Vec(20*x^4*(1 + 4*x - 181*x^2 + 1100*x^3 - 1344*x^4) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)*(1 - 8*x)) + O(x^30))) \\ Colin Barker, Jan 30 2017

a(n) = 5/6*4! Stirling2(n, 4) + 5*5! Stirling2(n, 5) + 11/2*6! Stirling2(n, 6) + 3*7! Stirling2(n, 7) + 8! Stirling2(n, 8).

G.f.: 20*x^4*(1 + 4*x - 181*x^2 + 1100*x^3 - 1344*x^4) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)*(1 - 8*x)). - Colin Barker, Jan 30 2017

A281778 Number of distinct topologies on an n-set that have exactly 10 open sets.

Original entry on oeis.org

0, 0, 0, 0, 24, 900, 18030, 276570, 3680964, 45065160, 523292010, 5859909990, 63862084704, 680829769620, 7122705252390, 73284607133010, 742843170653244, 7429450873589280, 73416173732059170, 717721593866613630, 6949589106333898584, 66721599431782204140

Offset: 0

Submitted on behalf of Moussa Benoumhani by Geoffrey Critzer, Jan 29 2017

Colin Barker, Table of n, a(n) for n = 0..1000
Moussa Benoumhani, The Number of Topologies on a Finite Set, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.6.
Index entries for linear recurrences with constant coefficients, signature (45,-870,9450,-63273,269325,-723680,1172700,-1026576,362880).

The number of distinct topologies on an n-set with exactly k open sets for k=2..12 is given by A000012, A000918, A281773, A028244, A281774, A281775, A281776, A281777,A281778, A281779, A281780.

concat(vector(4), Vec(6*x^4*(4 - 30*x - 265*x^2 + 3570*x^3 - 10839*x^4 + 22680*x^5) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)*(1 - 8*x)*(1 - 9*x)) + O(x^30))) \\ Colin Barker, Jan 30 2017

a(n) = 4! Stirling2(n, 4) + 11/2*5! Stirling2(n, 5) + 73/8*6! Stirling2(n, 6) + 15/2*7! Stirling2(n, 7) + 7/2*8! Stirling2(n, 8) + 9! Stirling2(n, 9).

G.f.: (6*(4 - 30*x - 265*x^2 + 3570*x^3 - 10839*x^4 + 22680*x^5))*x^4/Product_{j=1..9} (1-j*x). - Robert Israel, Jan 29 2017

A281779 Number of distinct topologies on an n-set that have exactly 11 open sets.

Original entry on oeis.org

0, 0, 0, 0, 0, 500, 16980, 342160, 5486040, 77926380, 1031160060, 13047426920, 160124426880, 1921105846660, 22632779709540, 262513678889280, 3002768326532520, 33914184260797340, 378596540805849420, 4181330954328313240, 45727913513193402960, 495618273676457274420

Offset: 0

Geoffrey Critzer, Jan 29 2017

Colin Barker, Table of n, a(n) for n = 0..950
Moussa Benoumhani, The Number of Topologies on a Finite Set, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.6.
Index entries for linear recurrences with constant coefficients, signature (55,-1320,18150,-157773,902055,-3416930,8409500,-12753576,10628640,-3628800).

The number of distinct topologies on an n-set with exactly k open sets for k=2..12 is given by A000012, A000918, A281773, A028244, A281774, A281775, A281776, A281777,A281778, A281779, A281780.

a(n) = 25*5!*stirling(n, 5, 2)/6 + 79*6!*stirling(n, 6, 2)/6 + 29*7!*stirling(n, 7, 2)/2 + 39*8!*stirling(n, 8, 2)/4 + 4*9!*stirling(n, 9, 2) + 10!*stirling(n, 10, 2) \\ Colin Barker, Jan 30 2017

concat(vector(4), Vec(20*x^5*(25 - 526*x + 3413*x^2 + 292*x^3 - 72756*x^4 + 226800*x^5) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)*(1 - 8*x)*(1 - 9*x)*(1 - 10*x)) + O(x^30))) \\ Colin Barker, Jan 30 2017

a(n) = 25/6*5! Stirling2(n, 5) + 79/6*6! Stirling2(n, 6) + 29/2*7! Stirling2(n, 7) + 39/4*8! Stirling2(n, 8) + 4*9! Stirling2(n, 9) + 10! Stirling2(n, 10).

G.f.: 20*x^5*(25 - 526*x + 3413*x^2 + 292*x^3 - 72756*x^4 + 226800*x^5) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)*(1 - 8*x)*(1 - 9*x)*(1 - 10*x)). - Colin Barker, Jan 30 2017

A281780 Number of distinct topologies on an n-set that have exactly 12 open sets.

Original entry on oeis.org

0, 0, 0, 0, 12, 660, 20400, 445620, 7977732, 126860580, 1873839000, 26381789940, 359484471852, 4784481401700, 62538498859200, 805447464281460, 10241415118476372, 128722997969290020, 1600670708273985000, 19705915838479512180, 240330009637668935292

Offset: 0

Geoffrey Critzer, Jan 29 2017

nonn

Ray Chandler, Table of n, a(n) for n = 0..960
Moussa Benoumhani, The Number of Topologies on a Finite Set, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.6.
Index entries for linear recurrences with constant coefficients, signature (66, -1925, 32670, -357423, 2637558, -13339535, 45995730, -105258076, 150917976, -120543840, 39916800).

The number of distinct topologies on an n-set with exactly k open sets for k=2..12 is given by A000012, A000918, A281773, A028244, A281774, A281775, A281776, A281777, A281778, A281779, A281780.

a(n) = 1/2*4! Stirling2(n, 4) + 9/2*5! Stirling2(n, 5) + 16*6! Stirling2(n, 6) + 295/12*7! Stirling2(n, 7) + 85/4*8! Stirling2(n, 8) + 49/4*9! Stirling2(n, 9) + 9/2*10! Stirling2(n, 10) + 11!*Stirling2(n, 11).

cp's OEIS Frontend

A326882 Irregular triangle read by rows where T(n,k) is the number of finite topologies with n points and k nonempty open sets, 0 <= k <= 2^n - 1.

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A281773 Number of distinct topologies on an n-set that have exactly 4 open sets.

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A281775 Number of distinct topologies on an n-set that have exactly 7 open sets.

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A281776 Number of distinct topologies on an n-set that have exactly 8 open sets.

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A281777 Number of distinct topologies on an n-set that have exactly 9 open sets.

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A281778 Number of distinct topologies on an n-set that have exactly 10 open sets.

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A281779 Number of distinct topologies on an n-set that have exactly 11 open sets.

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A281780 Number of distinct topologies on an n-set that have exactly 12 open sets.

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