A047707 -id:A047707 - cp's OEIS frontend

A084870 Number of 3-multiantichains of an n-set.

1, 2, 6, 28, 190, 1692, 16766, 166028, 1586430, 14580412, 129654526, 1123451628, 9544185470, 79881877532, 661135445886, 5425962250828, 44250287565310, 359161631645052, 2904756409742846, 23429320590259628, 188594431902253950

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.
Index entries for linear recurrences with constant coefficients, signature (28,-315,1820,-5684,9072,-5760).

Cf. A016269, A047707, A051112-A051118, A084869-A084883.

[(8^n - 6*6^n + 6*5^n + 9*4^n - 18*3^n + 14*2^n)/6: n in [0..50]]; // G. C. Greubel, Oct 08 2017

Table[(8^n - 6*6^n + 6*5^n + 9*4^n - 18*3^n + 14*2^n)/6, {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)

for(n=0,50, print1((8^n - 6*6^n + 6*5^n + 9*4^n - 18*3^n + 14*2^n)/6, ", ")) \\ G. C. Greubel, Oct 08 2017

a(n) = (1/3!)*(8^n - 6*6^n + 6*5^n + 9*4^n - 18*3^n + 14*2^n).

G.f.: ( 1-26*x+265*x^2-1330*x^3+3340*x^4-3432*x^5 ) / ( (6*x-1)*(4*x-1)*(3*x-1)*(2*x-1)*(8*x-1)*(5*x-1) ). - R. J. Mathar, Jul 08 2011

A084882 Number of (k,m,n)-multiantichains of multisets with k=3 and m=5.

Original entry on oeis.org

1, 3, 51, 4129, 1439381, 814788851, 395927618035, 155157302244381, 51960586962031617, 15663181302847575559, 4402571746033946222639, 1180812802393866826858193, 306839347397532891662028733

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

nonn

By a (k,m,n)-multiantichain of multisets we mean an m-multiantichain of k-bounded multisets on an n-set. The elements of a multiantichain could have the multiplicities greater than 1. A multiset is called k-bounded if every its element has the multiplicity not greater than k-1.

G. C. Greubel, Table of n, a(n) for n = 0..415
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.

Cf. A016269, A047707, A051112-A051118, A084869-A084883.

Table[(1/5!)*(243^n - 20*162^n + 60*126^n + 20*108^n + 10*98^n - 120*93^n - 120*84^n + 30*81^n + 30*78^n + 120*77^n + 120*70^n - 120*63^n + 20*56^n - 360*54^n + 720*42^n + 120*36^n - 720*31^n + 275*27^n + 180*26^n - 1650*18^n + 1650*14^n + 870*9^n - 1740*6^n + 744*3^n), {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)

a(n) = (1/5!)*(243^n - 20*162^n + 60*126^n + 20*108^n + 10*98^n - 120*93^n - 120*84^n + 30*81^n + 30*78^n + 120*77^n + 120*70^n - 120*63^n + 20*56^n - 360*54^n + 720*42^n + 120*36^n - 720*31^n + 275*27^n + 180*26^n - 1650*18^n + 1650*14^n + 870*9^n - 1740*6^n + 744*3^n).

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

A059090 Triangle T(n,m) giving number of m-element intersecting antichains on a labeled n-set or n-variable Boolean functions with m nonzero values in the Post class F(7,2), m=0,.., A037952(n).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 7, 3, 1, 1, 15, 30, 30, 5, 1, 31, 195, 605, 780, 543, 300, 135, 45, 10, 1, 1, 63, 1050, 9030, 41545, 118629, 233821, 329205, 327915, 224280, 100716, 29337, 5950, 910, 105, 7

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Dec 28 2000

An antichain is called intersecting (or proper) antichain if every two members have a nonempty intersection. Row sums give the number of intersecting antichains on a labeled n-set or n-variable Boolean functions in the Post class F(7,2) or self-dual monotone Boolean functions of n+1 variables. Cf. A001206.

			1;
1, 1;
1, 3;
1, 7, 3, 1;
1, 15, 30, 30, 5;
1, 31, 195, 605, 780, 543, 300, 135, 45, 10, 1;
1, 63, 1050, 9030, 41545, 118629, 233821, 329205, 327915, 224280, 100716, 29337, 5950, 910, 105, 7;

Jovovic V., Kilibarda G., The number of n-variable Boolean functions in the Post class F(7,2), Belgrade, 2001, in preparation.
Pogosyan G., Miyakawa M., Nozaki A., Rosenberg I., The Number of Clique Boolean Functions, IEICE Trans. Fundamentals, Vol. E80-A, No. 8, pp. 1502-1507, 1997/8.

Cf. A001206, A032263, A051303-A051307, A036239, A051180-A051185, A016269, A047707, A051112-A051118, A000372.

T(n, 0)=1, T(n, 1)=2^n-1, T(n, 2)=A032263(n), T(n, 3)=A051303(n), T(n, 4)=A051304(n), T(n, 5)=A051305(n), T(n, 6)=A051306(n), T(n, 7)=A051307(n).

A084871 Number of 4-multiantichains of an n-set.

Original entry on oeis.org

1, 2, 7, 41, 398, 6177, 128232, 2881531, 62769238, 1288737197, 25012685732, 463681018671, 8294783320578, 144410750517217, 2462999084589232, 41359616334934211, 686406989350511918, 11290725888842193237

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 0..825
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.

Cf. A016269, A047707, A051112-A051118, A084869-A084883.

[(16^n - 12*12^n + 24*10^n + 4*9^n - 6*8^n + 6*7^n - 108*6^n + 108*5^n + 83*4^n - 166*3^n + 90*2^n)/24: n in [0..50]]; // G. C. Greubel, Oct 08 2017

Table[(16^n - 12*12^n + 24*10^n + 4*9^n - 6*8^n + 6*7^n - 108*6^n + 108*5^n + 83*4^n - 166*3^n + 90*2^n)/4!, {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)

for(n=0,50, print1((16^n - 12*12^n + 24*10^n + 4*9^n - 6*8^n + 6*7^n - 108*6^n + 108*5^n + 83*4^n - 166*3^n + 90*2^n)/4!, ", ")) \\ G. C. Greubel, Oct 08 2017

a(n) = (16^n - 12*12^n + 24*10^n + 4*9^n - 6*8^n + 6*7^n - 108*6^n + 108*5^n + 83*4^n - 166*3^n + 90*2^n)/4!.
From R. J. Mathar, Jul 08 2011: (Start)
G.f.: (-1 + 80*x - 2813*x^2 + 57293*x^3 - 749139*x^4 + 6577949*x^5 - 39353597*x^6 + 158972472*x^7 - 417774220*x^8 + 651991536*x^9 - 465379200*x^10) / ( (9*x-1) *(6*x-1) *(7*x-1) *(3*x-1) *(5*x-1) *(2*x-1) *(12*x-1) *(10*x-1) *(4*x-1) *(8*x-1) *(16*x-1) ).
a(n) = 82*a(n-1) - 2970*a(n-2) + 62700*a(n-3) - 856713*a(n-4) + 7947786*a(n-5) - 51019100*a(n-6) + 226259000*a(n-7) - 678011136*a(n-8) + 1304341632*a(n-9) - 1445575680*a(n-10) + 696729600*a(n-11). (End)

A084872 Number of 5-multiantichains of an n-set.

Original entry on oeis.org

1, 2, 8, 56, 726, 17938, 722680, 35955180, 1798971434, 83885891894, 3612380896332, 145277787750064, 5534505187364062, 202229611397865690, 7158136006402746464, 247316732670273773108, 8389241054998193347410

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 0..660
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.

Cf. A016269, A047707, A051112-A051118, A084869-A084883.

[(32^n - 20*24^n + 60*20^n + 20*18^n + 10*17^n - 90*16^n - 120*15^n + 150*14^n + 120*13^n - 480*12^n + 20*11^n + 720*10^n + 120*9^n - 445*8^n + 180*7^n - 1650*6^n + 1650*5^n + 870*4^n - 1740*3^n + 744*2^n)/120: n in [0..50]]; // G. C. Greubel, Oct 08 2017

Table[(32^n - 20*24^n + 60*20^n + 20*18^n + 10*17^n - 90*16^n - 120*15^n + 150*14^n + 120*13^n - 480*12^n + 20*11^n + 720*10^n + 120*9^n - 445*8^n + 180*7^n - 1650*6^n + 1650*5^n + 870*4^n - 1740*3^n + 744*2^n)/120, {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)

for(n=0,50, print1((32^n - 20*24^n + 60*20^n + 20*18^n + 10*17^n - 90*16^n - 120*15^n + 150*14^n + 120*13^n - 480*12^n + 20*11^n + 720*10^n + 120*9^n - 445*8^n + 180*7^n - 1650*6^n + 1650*5^n + 870*4^n - 1740*3^n + 744*2^n)/120, ", ")) \\ G. C. Greubel, Oct 08 2017

a(n) = (1/5!)*(32^n - 20*24^n + 60*20^n + 20*18^n + 10*17^n - 90*16^n - 120*15^n + 150*14^n + 120*13^n - 480*12^n + 20*11^n + 720*10^n + 120*9^n - 445*8^n + 180*7^n - 1650*6^n + 1650*5^n + 870*4^n - 1740*3^n + 744*2^n).

A084873 Number of 6-multiantichains of an n-set.

Original entry on oeis.org

1, 2, 9, 73, 1212, 44667, 3251186, 345094227, 39552733796, 4234657495267, 409948262617398, 36190736880911571, 2964860272283578040, 229165985114590010307, 16940021231116707830570

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 0..550
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.

Cf. A016269, A047707, A051112-A051118, A084869-A084883.

Table[(1/6!)*(64^n - 30*48^n + 120*40^n + 60*36^n + 60*34^n - 12*33^n - 315*32^n - 720*30^n + 810*28^n + 120*27^n + 480*26^n + 360*25^n - 1080*24^n - 720*23^n - 240*22^n - 540*21^n + 3180*20^n + 750*19^n + 660*18^n + 90*17^n - 4535*16^n - 5420*15^n + 6750*14^n + 5400*13^n - 13620*12^n + 900*11^n + 16440*10^n + 2740*9^n - 12165*8^n + 4110*7^n - 25650*6^n + 25650*5^n + 10474*4^n - 20948*3^n + 7560*2^n), {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)

a(n) = (1/6!)*(64^n - 30*48^n + 120*40^n + 60*36^n + 60*34^n - 12*33^n - 315*32^n - 720*30^n + 810*28^n + 120*27^n + 480*26^n + 360*25^n - 1080*24^n - 720*23^n - 240*22^n - 540*21^n + 3180*20^n + 750*19^n + 660*18^n + 90*17^n - 4535*16^n - 5420*15^n + 6750*14^n + 5400*13^n - 13620*12^n + 900*11^n + 16440*10^n + 2740*9^n - 12165*8^n + 4110*7^n - 25650*6^n + 25650*5^n + 10474*4^n - 20948*3^n + 7560*2^n).

A084874 Number of (k,m,n)-antichains of multisets with k=3 and m=2.

Original entry on oeis.org

0, 0, 9, 162, 2025, 21870, 219429, 2112642, 19847025, 183642390, 1682955549, 15327821322, 139038251625, 1257873017310, 11360034454869, 102475388237202, 923689006041825, 8321664254958630, 74945757885541389, 674816499677616282

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

nonn

By a (k,m,n)-antichain of multisets we mean an m-antichain of k-bounded multisets on an n-set. A multiset is called k-bounded if every its element has the multiplicity not greater than k-1.

a(n) is also the number of entries that are divisible by 3 in rows 0 through 3^n-1 of Pascal's triangle A007318. - Tim Cieplowski, Nov 25 2014

G. C. Greubel, Table of n, a(n) for n = 0..1000
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.
Index entries for linear recurrences with constant coefficients, signature (18,-99,162).

Cf. A016269, A047707, A051112-A051118, A084869-A084883.

[(9^n - 2*6^n + 3^n)/2: n in [0..50]]; // G. C. Greubel, Oct 08 2017

Table[(9^n - 2*6^n + 3^n)/2, {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)
LinearRecurrence[{18,-99,162},{0,0,9},20] (* Harvey P. Dale, Oct 01 2023 *)

for(n=0,50, print1((9^n - 2*6^n + 3^n)/2, ", ")) \\ G. C. Greubel, Oct 08 2017

a(n) = (1/2!)*(9^n - 2*6^n + 3^n).
G.f.: -9*x^2 / ( (6*x-1)*(3*x-1)*(9*x-1) ). - R. J. Mathar, Jul 08 2011
E.g.f.: (exp(9*x) - 2*exp(6*x) + exp(3*x))/2. - G. C. Greubel, Oct 08 2017

A084875 Number of (k,m,n)-antichains of multisets with k=3 and m=3.

Original entry on oeis.org

0, 0, 1, 350, 24025, 1061570, 38306701, 1238697950, 37547263825, 1093418309690, 31035659056501, 866306577308150, 23915774118612025, 655397866616830610, 17872808187862527901, 485794481046271815950, 13175146525408965630625

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

nonn

By a (k,m,n)-antichain of multisets we mean an m-antichain of k-bounded multisets on an n-set. A multiset is called k-bounded if every its element has the multiplicity not greater than k-1.

G. C. Greubel, Table of n, a(n) for n = 0..695
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.
Index entries for linear recurrences with constant coefficients, signature (77,-2277,32895,-242514,854388,-1102248).

Cf. A016269, A047707, A051112-A051118, A084869-A084883.

[(27^n - 6*18^n + 6*14^n + 3*9^n - 6*6^n + 2*3^n)/6: n in [0..50]]; // G. C. Greubel, Oct 08 2017

Table[(27^n - 6*18^n + 6*14^n + 3*9^n - 6*6^n + 2*3^n)/6, {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)
LinearRecurrence[{77,-2277,32895,-242514,854388,-1102248},{0,0,1,350,24025,1061570},20] (* Harvey P. Dale, May 29 2025 *)

for(n=0,50, print1((27^n - 6*18^n + 6*14^n + 3*9^n - 6*6^n + 2*3^n)/6, ", ")) \\ G. C. Greubel, Oct 08 2017

a(n) = (1/3!)*(27^n - 6*18^n + 6*14^n + 3*9^n - 6*6^n + 2*3^n).

G.f.: -x^2*(-1-273*x+648*x^2+24300*x^3) / ( (18*x-1)*(9*x-1)*(6*x-1)*(3*x-1)*(14*x-1)*(27*x-1) ). - R. J. Mathar, Jul 08 2011

cp's OEIS Frontend

A084870 Number of 3-multiantichains of an n-set.

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A084882 Number of (k,m,n)-multiantichains of multisets with k=3 and m=5.

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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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A059090 Triangle T(n,m) giving number of m-element intersecting antichains on a labeled n-set or n-variable Boolean functions with m nonzero values in the Post class F(7,2), m=0,.., A037952(n).

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A084871 Number of 4-multiantichains of an n-set.

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A084872 Number of 5-multiantichains of an n-set.

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A084873 Number of 6-multiantichains of an n-set.

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