A056005 -id:A056005 - cp's OEIS frontend

A056069 Number of 4-element ordered antichains on an unlabeled n-element set; T_1-hypergraphs with 4 labeled nodes and n hyperedges.

25, 454, 3818, 21420, 92805, 335152, 1055944, 2990020, 7767357, 18789070, 42797602, 92588216, 191542842, 381000192, 731941256, 1363109096, 2468549141, 4358716470, 7520830306, 12706161124, 21054530855, 34269633840, 54863015040, 86489873580, 134406530985

Offset: 4

Vladeta Jovovic, Goran Kilibarda, Jul 26 2000

nonn

T_1-hypergraph is a hypergraph (not necessarily without empty hyperedges or multiple hyperedges) which for every ordered pair of distinct nodes has a hyperedge containing one but not the other node.

V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6)
V. Jovovic, G. Kilibarda, On enumeration of the class of all monotone Boolean functions, in preparation.

T. D. Noe, Table of n, a(n) for n = 4..1000
K. S. Brown, Dedekind's problem

Cf. A051112 for 4-element (unordered) antichains on a labeled n-element set, A056005.

a(n) = C(n + 15, 15) - 12*C(n + 11, 11) + 24*C(n + 9, 9) + 4*C(n + 8, 8) - 18*C(n + 7, 7) + 6*C(n + 6, 6) - 36*C(n + 5, 5) + 36*C(n + 4, 4) + 11*C(n + 3, 3) - 22*C(n + 2, 2) + 6*C(n + 1, 1).

Empirical G.f.: x^4*(6*x^10 -62*x^9 +271*x^8 -636*x^7 +800*x^6 -328*x^5 -495*x^4 +812*x^3 -446*x^2 +54*x +25)/(x-1)^16. [Colin Barker, May 29 2012]

A056071 Number of 6-element ordered antichains on an unlabeled n-element set; T_1-hypergraphs with 6 labeled nodes and n hyperedges.

Original entry on oeis.org

30, 8340, 780242, 29813578, 657271645, 10037038800, 117733967666, 1130702091428, 9273992351046, 66900184307860, 433616524985590, 2566055594813118, 14037125952339998, 71676448315103924, 344320192201127730, 1566076395413987110, 6779944255517707576

Offset: 4

Vladeta Jovovic, Goran Kilibarda, Zoran Maksimovic, Jul 26 2000

T_1-hypergraph is a hypergraph (not necessarily without empty hyperedges or multiple hyperedges) which for every ordered pair of distinct nodes has a hyperedge containing one but not the other node.

V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6)
V. Jovovic, G. Kilibarda, On enumeration of the class of all monotone Boolean functions, in preparation.

K. S. Brown, Dedekind's problem
Eric Weisstein's World of Mathematics, Antichain covers
Index entries for linear recurrences with constant coefficients, signature (64, -2016, 41664, -635376, 7624512, -74974368, 621216192, -4426165368, 27540584512, -151473214816, 743595781824, -3284214703056, 13136858812224, -47855699958816, 159518999862720, -488526937079580, 1379370175283520, -3601688791018080, 8719878125622720, -19619725782651120, 41107996877935680, -80347448443237920, 146721427591999680, -250649105469666120, 401038568751465792, -601557853127198688, 846636978475316672, -1118770292985239888, 1388818294740297792, -1620288010530347424, 1777090076065542336, -1832624140942590534, 1777090076065542336, -1620288010530347424, 1388818294740297792, -1118770292985239888, 846636978475316672, -601557853127198688, 401038568751465792, -250649105469666120, 146721427591999680, -80347448443237920, 41107996877935680, -19619725782651120, 8719878125622720, -3601688791018080, 1379370175283520, -488526937079580, 159518999862720, -47855699958816, 13136858812224, -3284214703056, 743595781824, -151473214816, 27540584512, -4426165368, 621216192, -74974368, 7624512, -635376, 41664, -2016, 64, -1).

Cf. A051114 for 6-element (unordered) antichains on a labeled n-element set, A056005.

a(n)=C(n + 63, 63) - 30*C(n + 47, 47) + 120*C(n + 39, 39) + 60*C(n + 35, 35) + 60*C(n + 33, 33) - 12*C(n + 32, 32) - 345*C(n + 31, 31) - 720*C(n + 29, 29) + 810*C(n + 27, 27) + 120*C(n + 26, 26) + 480*C(n + 25, 25) + 360*C(n + 24, 24) - 480*C(n + 23, 23) - 720*C(n + 22, 22) - 240*C(n + 21, 21) - 540*C(n + 20, 20) + 1380*C(n + 19, 19) + 750*C(n + 18, 18) + 60*C(n + 17, 17) - 210*C(n + 16, 16) - 1535*C(n + 15, 15) - 1820*C(n + 14, 14) + 2250*C(n + 13, 13) + 1800*C(n + 12, 12) - 2820*C(n + 11, 11) + 300*C(n + 10, 10) + 2040*C(n + 9, 9) + 340*C(n + 8, 8) - 1815*C(n + 7, 7) + 510*C(n + 6, 6) - 1350*C(n + 5, 5) + 1350*C(n + 4, 4) + 274*C(n + 3, 3) - 548*C(n + 2, 2) + 120*C(n + 1, 1).

More terms from Sean A. Irvine, Apr 14 2022

A056073 Number of 7-element ordered antichains on an unlabeled n-element set; T_1-hypergraphs with 7 labeled nodes and n hyperedges.

Original entry on oeis.org

20580, 9209340, 1113220168, 64271300556, 2302652531436, 59028678965286, 1179552813324360, 19421453010531722, 273692092058502488, 3392151018511583748, 37729265705269684476

Offset: 5

Vladeta Jovovic, Goran Kilibarda, Zoran Maksimovic, Jul 26 2000

nonn

T_1-hypergraph is a hypergraph (not necessarily without empty hyperedges or multiple hyperedges) which for every ordered pair of distinct nodes has a hyperedge containing one but not the other node.

V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6)
V. Jovovic and G. Kilibarda, On enumeration of the class of all monotone Boolean functions, in preparation.

K. S. Brown, Dedekind's problem

A051115 for 7-element (unordered) antichains on a labeled n-element set, A056005.

a(n)=C(n + 127, 127) - 42*C(n + 95, 95) + 210*C(n + 79, 79) + 140*C(n + 71, 71) + 210*C(n + 67, 67) - 84*C(n + 65, 65) + 14*C(n + 64, 64) - 819*C(n + 63, 63) - 2520*C(n + 59, 59) + 2730*C(n + 55, 55) + 840*C(n + 53, 53) + 840*C(n + 51, 51) - 420*C(n + 50, 50) + 2940*C(n + 49, 49) + 630*C(n + 47, 47) - 5040*C(n + 45, 45) + 840*C(n + 44, 44) - 1260*C(n + 43, 43) +
1680*C(n + 42, 42) - 9660*C(n + 41, 41) + 1260*C(n + 40, 40) + 3360*C(n + 39, 39) - 7560*C(n + 38, 38) + 11130*C(n + 37, 37) + 5880*C(n + 36, 36) + 9240*C(n + 35, 35) + 2982*C(n + 34, 34) - 6300*C(n + 33, 33) - 8652*C(n + 32, 32) - 9905*C(n + 31, 31) - 8400*C(n + 30, 30) - 8540*C(n + 29, 29) + 13860*C(n + 28, 28) + 14490*C(n + 27, 27) - 5040*C(n + 26, 26) + 10500*C(n + 25, 25) +
10080*C(n + 24, 24) - 8120*C(n + 23, 23) - 15050*C(n + 22, 22) - 5040*C(n + 21, 21) - 11340*C(n + 20, 20) + 20580*C(n + 19, 19) + 15750*C(n + 18, 18) - 1540*C(n + 17, 17) - 5810*C(n + 16, 16) - 16485*C(n + 15, 15) - 21420*C(n + 14, 14) + 26250*C(n + 13, 13) + 21000*C(n + 12, 12) - 29820*C(n + 11, 11) + 3500*C(n + 10, 10) + 17640*C(n + 9, 9) + 2940*C(n + 8, 8) - 16016*C(n + 7, 7) + 4410*C(n + 6, 6) - 9744*C(n + 5, 5) + 9744*C(n + 4, 4) + 1764*C(n + 3, 3) - 3528*C(n + 2, 2) + 720*C(n + 1, 1).

A056078 Number of proper T_1-hypergraphs with 3 labeled nodes and n hyperedges.

Original entry on oeis.org

0, 0, 2, 15, 54, 141, 306, 588, 1036, 1710, 2682, 4037, 5874, 8307, 11466, 15498, 20568, 26860, 34578, 43947, 55214, 68649, 84546, 103224, 125028, 150330, 179530, 213057, 251370, 294959, 344346, 400086, 462768, 533016, 611490, 698887, 795942, 903429, 1022162

Offset: 1

Vladeta Jovovic, Goran Kilibarda, Jul 26 2000

Also number of 3 X 3 matrices with nonnegative integer entries with zero main diagonal and without zero rows or columns, such that sum of all entries is n. - Vladeta Jovovic, Sep 06 2006

A T_1-hypergraph is a hypergraph (not necessarily without empty hyperedges or multiple hyperedges) which for every ordered pair (u,v) of distinct nodes has a hyperedge containing u but not v. A proper hypergraph is a hypergraph without empty hyperedges or hyperedges containing all nodes. - Vladeta Jovovic, Sep 06 2006

			There are 15 proper T_1-hypergraphs with 3 nodes and 4 hyperedges: {{3},{3},{2},{1}}, {{3},{2},{2},{1}}, {{3},{2},{2,3},{1}}, {{3},{2},{1},{1}}, {{3},{2},{1},{1,3}}, {{3},{2},{1},{1,2}}, {{3},{2},{1,3},{1,2}}, {{3},{2,3},{1},{1,2}}, {{3},{2,3},{1,3},{1,2}}, {{2},{2,3},{1},{1,3}}, {{2},{2,3},{1,3},{1,2}}, {{2,3},{2,3},{1,3},{1,2}}, {{2,3},{1},{1,3},{1,2}}, {{2,3},{1,3},{1,3},{1,2}}, {{2,3},{1,3},{1,2},{1,2}}.

V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6)
V. Jovovic, G. Kilibarda, On enumeration of the class of all monotone Boolean functions, in preparation.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A056005, A047707.

[(n^4 + 20*n^3 + 35*n^2 - 140*n + 84)*n/120: n in [0..25]]; // G. C. Greubel, Oct 07 2017

Table[(n^4 + 20*n^3 + 35*n^2 - 140*n + 84)*n/120, {n, 0, 50}] (* G. C. Greubel, Oct 07 2017 *)

for(n=0,25, print1((n^4 + 20*n^3 + 35*n^2 - 140*n + 84)*n/120, ", ")) \\ G. C. Greubel, Oct 07 2017

a(n) = C(n+5,5) -6*C(n+3,3) +6*C(n+2,2) +3*C(n+1,1) -6*C(n,0).
a(n+1) = ( n^4 +20*n^3 +35*n^2 -140*n +84 )*n/120.
From Colin Barker, Jul 11 2013: (Start)
a(n) = (-240+394*n-135*n^2-35*n^3+15*n^4+n^5)/120.
G.f.: x^3 *(x-2) *(2*x^2-2*x-1) / (x-1)^6. (End)

More terms from Colin Barker, Jul 11 2013

A056070 Number of 5-element ordered antichains on an unlabeled n-element set; T_1-hypergraphs with 5 labeled nodes and n hyperedges.

Original entry on oeis.org

30, 2206, 56242, 766198, 7056249, 49662920, 286860862, 1422695104, 6246302316, 24810260818, 90593318410, 307833736038, 982717917851, 2969842897554, 8548862507642, 23559234462890, 62421788882924, 159585012804848, 394875247007432, 948171537489016

Offset: 4

Vladeta Jovovic, Goran Kilibarda, Zoran Maksimovic, Jul 26 2000

T_1-hypergraph is a hypergraph (not necessarily without empty hyperedges or multiple hyperedges) which for every ordered pair of distinct nodes has a hyperedge containing one but not the other node.

V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6)
V. Jovovic and G. Kilibarda, On enumeration of the class of all monotone Boolean functions, in preparation.

K. S. Brown, Dedekind's problem
Index entries for linear recurrences with constant coefficients, signature (32, -496, 4960, -35960, 201376, -906192, 3365856, -10518300, 28048800, -64512240, 129024480, -225792840, 347373600, -471435600, 565722720, -601080390, 565722720, -471435600, 347373600, -225792840, 129024480, -64512240, 28048800, -10518300, 3365856, -906192, 201376, -35960, 4960, -496, 32, -1).

Cf. A051113 for 5-element (unordered) antichains on a labeled n-element set, A056005.

a(n)=C(n + 31, 31) - 20*C(n + 23, 23) + 60*C(n + 19, 19) + 20*C(n + 17, 17) + 10*C(n + 16, 16) - 110*C(n + 15, 15) - 120*C(n + 14, 14) + 150*C(n + 13, 13) + 120*C(n + 12, 12) - 240*C(n + 11, 11) + 20*C(n + 10, 10) + 240*C(n + 9, 9) + 40*C(n + 8, 8) - 205*C(n + 7, 7) + 60*C(n + 6, 6) - 210*C(n + 5, 5) + 210*C(n + 4, 4) + 50*C(n + 3, 3) - 100*C(n + 2, 2) + 24*C(n + 1, 1).

More terms from Sean A. Irvine, Apr 14 2022

A056163 Number of ordered antichains on an unlabeled n-set; labeled T_1-hypergraphs with n hyperedges.

Original entry on oeis.org

2, 3, 5, 11, 120, 191297

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Jul 31 2000

A T_1-hypergraph is a hypergraph (not necessarily without empty hyperedges or multiple hyperedges) which for every ordered pair of distinct nodes has a hyperedge containing one but not the other node.

			a(1)=1+2=3; a(2)=1+3+1=5; a(3)=1+4+4+2=11; a(4)=1+5+10+19+25+30+30=120; a(5)=1+6+20+90+454+2206+8340+20580+38640+60480+60480=191297.
There are 11 ordered antichains on an unlabeled 3-set: 0, (0), ({1}), ({1,2}), ({1,2,3}), ({1},{2}), ({1},{2,3}), ({2,3},{1}), ({1,2},{1,3}), ({1},{2},{3}), ({1,2},{1,3},{2,3}).

V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6)
V. Jovovic, G. Kilibarda, On enumeration of the class of all monotone Boolean functions, in preparation.

K. S. Brown, Dedekind's problem
Eric Weisstein's World of Mathematics, Antichain covers

Cf. A000372 for (unordered) antichains on a labeled n-set, A056005, A056069-A056071, A056073, A056046-A056049, A056052, A056101, A056104, A051112-A051118.

a(n)=Sum_{k=0..C(n, floor(n/2))}b(k, n) where b(k, n) is the number of k-element ordered antichains on an unlabeled n-set.

A056778 Number of 3-element antichains on an unlabeled n-element set; equivalence classes of monotone Boolean functions of n variables with 3 mincuts under action of symmetric group S_n.

Original entry on oeis.org

0, 0, 0, 2, 9, 30, 84, 202, 437, 872, 1627, 2874, 4853, 7882, 12383, 18902, 28130, 40934, 58391, 81812, 112790, 153238, 205430, 272054, 356270, 461754, 592774, 754252, 951831, 1191956, 1481962, 1830144, 2245867, 2739658, 3323305, 4009972, 4814323, 5752624, 6842893

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Aug 17 2000

			There are 30 3-element antichains on an unlabeled 5-element set: {{5},{4},{3}}, {{5},{4},{2,3}}, {{5},{4},{1,2,3}}, {{5},{3,4},{2,4}}, {{5},{3,4},{1,2}}, {{5},{3,4},{1,2,4}}, {{5},{2,3,4},{1,3,4}}, {{4,5},{3,5},{3,4}}, {{4,5},{3,5},{2,5}}, {{4,5},{3,5},{2,4}},{{4,5},{3,5},{2,3,4}}, {{4,5},{3,5},{1,2}}, {{4,5},{3,5},{1,2,5}}, {{4,5},{3,5},{1,2,4}}, {{4,5},{3,5},{1,2,3,4}}, {{4,5},{2,3},{1,3,5}}, {{4,5},{2,3,5},{2,3,4}}, {{4,5},{2,3,5},{1,3,5}}, {{4,5},{2,3,5},{1,3,4}}, {{4,5},{2,3,5},{1,2,3}}, {{4,5},{2,3,5},{1,2,3,4}}, {{4,5},{1,2,3,5},{1,2,3,4}}, {{3,4,5},{2,4,5},{2,3,5}}, {{3,4,5},{2,4,5},{1,4,5}}, {{3,4,5},{2,4,5},{1,3,5}}, {{3,4,5},{2,4,5},{1,2,3}}, {{3,4,5},{2,4,5},{1,2,3,5}}, {{3,4,5},{1,2,5},{1,2,3,4}}, {{3,4,5},{1,2,4,5},{1,2,3,5}}, {{2,3,4,5},{1,3,4,5},{1,2,4,5}}.

V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6)
V. Jovovic, G. Kilibarda, On enumeration of the class of all monotone Boolean functions, in preparation.

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Boolean functions.
Index entries for linear recurrences with constant coefficients, signature (4,-4,-2,2,4,3,-12,3,4,2,-2,-4,4,-1).

Cf. A056005, A047707, A055484, A055485.

seq(n)=Vec((2 + x + 2*x^2 + 4*x^3 - x^5 - 2*x^6)/((1 - x)^8*(1 + x)^2*(1 + x + x^2)^2) + O(x^(n-2)), -(n+1)) \\ Andrew Howroyd, Feb 02 2024

G.f.: x^3*(2 + x + 2*x^2 + 4*x^3 - x^5 - 2*x^6)/((1 - x)^8*(1 + x)^2*(1 + x + x^2)^2). - Andrew Howroyd, Feb 02 2024

a(8) onwards from Andrew Howroyd, Feb 02 2024

A056782 Number of 3-element proper antichains (i.e., antichains such that every two members have nonempty intersection) on an unlabeled n-element set.

Original entry on oeis.org

0, 0, 0, 1, 5, 18, 53, 135, 305, 633, 1220, 2217, 3834, 6359, 10172, 15776, 23807, 35075, 50585, 71576, 99551, 136332, 184084, 245384, 323260, 421256, 543484, 694709, 880393, 1106798, 1381049, 1711231, 2106469, 2577049, 3134488, 3791677, 4562974, 5464339, 6513448

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Aug 18 2000

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Vladeta Jovovic, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (4,-4,-2,2,4,3,-12,3,4,2,-2,-4,4,-1).

Cf. A001206, A047707, A051303 (labeled case), A055484, A055485, A056005.

seq(n)=Vec((1 + x + 2*x^2 + 3*x^3 + 3*x^4 - x^5 - 3*x^7)/((1 - x)^8*(1 + x)^2*(1 + x + x^2)^2) + O(x^(n-2)), -(n+1)) \\ Andrew Howroyd, Feb 02 2024

G.f.: x^3*(1 + x + 2*x^2 + 3*x^3 + 3*x^4 - x^5 - 3*x^7)/((1 - x)^8*(1 + x)^2*(1 + x + x^2)^2). - Andrew Howroyd, Feb 02 2024

a(8) onwards from Andrew Howroyd, Feb 02 2024

A056164 Number of ordered antichain covers of an unlabeled n-set; labeled T_1-hypergraphs (without empty hyperedges) with n hyperedges.

Original entry on oeis.org

1, 2, 6, 109, 191177

Offset: 1

Vladeta Jovovic, Goran Kilibarda, Jul 31 2000

A T_1-hypergraph is a hypergraph (not necessarily without empty hyperedges or multiple hyperedges) which for every ordered pair of distinct nodes has a hyperedge containing one but not the other node.

			There are 6 ordered antichain covers on an unlabeled 3-set: ({1,2,3}), ({1},{2,3}), ({2,3},{1}), ({1,2},{1,3}), ({1},{2},{3}), ({1,2},{1,3},{2,3}).
a(3)=1+3+2=6; a(4)=1+6+17+25+30+30=109; a(5)=1+10+71+429+2176+8310+20580+38640+60480+60480=191177.

V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6)
V. Jovovic and G. Kilibarda, On enumeration of the class of all monotone Boolean functions, in preparation.

K. S. Brown, Dedekind's problem
Eric Weisstein's World of Mathematics, Antichain covers

Cf. A056074, A056090, A056093, A000372, A056005, A056069-A056071, A056073, A056046-A056049, A056052, A056101, A056104, A051112-A051118.

a(n)=Sum_{k=1..C(n, floor(n/2))}b(k, n) where b(k, n) is the number of k-element ordered antichains covers of an unlabeled n-set.

cp's OEIS Frontend

A056069 Number of 4-element ordered antichains on an unlabeled n-element set; T_1-hypergraphs with 4 labeled nodes and n hyperedges.

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A056071 Number of 6-element ordered antichains on an unlabeled n-element set; T_1-hypergraphs with 6 labeled nodes and n hyperedges.

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A056073 Number of 7-element ordered antichains on an unlabeled n-element set; T_1-hypergraphs with 7 labeled nodes and n hyperedges.

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A056078 Number of proper T_1-hypergraphs with 3 labeled nodes and n hyperedges.

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A056070 Number of 5-element ordered antichains on an unlabeled n-element set; T_1-hypergraphs with 5 labeled nodes and n hyperedges.

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A056163 Number of ordered antichains on an unlabeled n-set; labeled T_1-hypergraphs with n hyperedges.

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A056778 Number of 3-element antichains on an unlabeled n-element set; equivalence classes of monotone Boolean functions of n variables with 3 mincuts under action of symmetric group S_n.

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A056782 Number of 3-element proper antichains (i.e., antichains such that every two members have nonempty intersection) on an unlabeled n-element set.

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