A003182 -id:A003182 - cp's OEIS frontend

A326574 Number of antichains of subsets of {1..n} with equal edge-sums.

2, 3, 5, 10, 22, 61, 247, 2096, 81896, 52260575

Offset: 0

Gus Wiseman, Jul 18 2019

An antichain is a finite set of finite sets, none of which is a subset of any other. The edge-sums are the sums of vertices in each edge, so for example the edge sums of {{1,3},{2,5},{3,4,5}} are {4,7,12}.

			The a(0) = 2 through a(4) = 22 antichains:
  {}    {}     {}       {}           {}
  {{}}  {{}}   {{}}     {{}}         {{}}
        {{1}}  {{1}}    {{1}}        {{1}}
               {{2}}    {{2}}        {{2}}
               {{1,2}}  {{3}}        {{3}}
                        {{1,2}}      {{4}}
                        {{1,3}}      {{1,2}}
                        {{2,3}}      {{1,3}}
                        {{1,2,3}}    {{1,4}}
                        {{3},{1,2}}  {{2,3}}
                                     {{2,4}}
                                     {{3,4}}
                                     {{1,2,3}}
                                     {{1,2,4}}
                                     {{1,3,4}}
                                     {{2,3,4}}
                                     {{1,2,3,4}}
                                     {{3},{1,2}}
                                     {{4},{1,3}}
                                     {{1,4},{2,3}}
                                     {{2,4},{1,2,3}}
                                     {{3,4},{1,2,4}}

Set partitions with equal block-sums are A035470.
Antichains with different edge-sums are A326030.
MM-numbers of multiset partitions with equal part-sums are A326534.
The covering case is A326566.
Cf. A000372, A003182, A006126, A307249, A321455, A321717, A321718, A326518, A326565, A326572.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
cleqset[set_]:=stableSets[Subsets[set],SubsetQ[#1,#2]||Total[#1]!=Total[#2]&];
Table[Length[cleqset[Range[n]]],{n,0,5}]

a(9) from Andrew Howroyd, Aug 13 2019

A302250 The number of antichains in the lattice of set partitions of an n-element set.

Original entry on oeis.org

2, 3, 10, 347, 79814832

Offset: 1

John Machacek, Apr 04 2018

Computing terms in this sequence is analogous to Dedekind's problem which asks for the number of antichains in the Boolean algebra.

This count includes the empty antichain consisting of no set partitions.

			For n = 3 the a(3) = 10 antichains are:
  {}
  {1/2/3}
  {1/23}
  {12/3}
  {13/2}
  {1/23, 12/3}
  {1/23, 13/2}
  {12/3, 13/2}
  {1/23, 12/3, 13/2}
  {123}.
Here we have used the usual shorthand notation for set partitions where 1/23 denotes {{1}, {2,3}}.

Dmitry I. Ignatov, A Note on the Number of (Maximal) Antichains in the Lattice of Set Partitions. In: Ojeda-Aciego, M., Sauerwald, K., Jäschke, R. (eds) Graph-Based Representation and Reasoning. ICCS 2023. Lecture Notes in Computer Science(). Springer, Cham.

Equals A302251 + 1, Cf. A000372, A007153, A003182, A014466.

[Posets.SetPartitions(n).antichains().cardinality() for n in range(4)]

A326571 Number of covering antichains of nonempty, non-singleton subsets of {1..n}, all having different sums.

Original entry on oeis.org

1, 0, 1, 5, 61, 2721, 788221

Offset: 0

Gus Wiseman, Jul 18 2019

An antichain is a finite set of finite sets, none of which is a subset of any other. It is covering if its union is {1..n}. The edge-sums are the sums of vertices in each edge, so for example the edge sums of {{1,3},{2,5},{3,4,5}} are {4,7,12}.

			The a(3) = 5 antichains:
  {{1,2,3}}
  {{1,3},{2,3}}
  {{1,2},{2,3}}
  {{1,2},{1,3}}
  {{1,2},{1,3},{2,3}}
The a(4) = 61 antichains:
  {1234}  {12}{34}    {12}{13}{14}   {12}{13}{14}{24}   {12}{13}{14}{24}{34}
          {13}{24}    {12}{13}{24}   {12}{13}{14}{34}   {12}{13}{23}{24}{34}
          {12}{134}   {12}{13}{34}   {12}{13}{23}{24}
          {12}{234}   {12}{14}{34}   {12}{13}{23}{34}
          {13}{124}   {12}{23}{24}   {12}{13}{24}{34}
          {13}{234}   {12}{23}{34}   {12}{14}{24}{34}
          {14}{123}   {12}{24}{34}   {12}{23}{24}{34}
          {14}{234}   {13}{14}{24}   {13}{14}{24}{34}
          {23}{124}   {13}{23}{24}   {13}{23}{24}{34}
          {23}{134}   {13}{23}{34}   {12}{13}{14}{234}
          {24}{134}   {13}{24}{34}   {12}{23}{24}{134}
          {34}{123}   {14}{24}{34}   {123}{124}{134}{234}
          {123}{124}  {12}{13}{234}
          {123}{134}  {12}{14}{234}
          {123}{234}  {12}{23}{134}
          {124}{134}  {12}{24}{134}
          {124}{234}  {13}{14}{234}
          {134}{234}  {13}{23}{124}
                      {14}{34}{123}
                      {23}{24}{134}
                      {12}{134}{234}
                      {13}{124}{234}
                      {14}{123}{234}
                      {23}{124}{134}
                      {123}{124}{134}
                      {123}{124}{234}
                      {123}{134}{234}
                      {124}{134}{234}

Antichain covers are A006126.
Set partitions with different block-sums are A275780.
MM-numbers of multiset partitions with different part-sums are A326535.
Antichain covers with equal edge-sums and no singletons are A326565.
Antichain covers with different edge-sizes and no singletons are A326569.
The case with singletons allowed is A326572.
Antichains with equal edge-sums are A326574.
Cf. A000372, A003182, A035470, A307249, A321469, A326519, A326566, A326570, A326573.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
cleq[n_]:=Select[stableSets[Subsets[Range[n],{2,n}],SubsetQ[#1,#2]||Total[#1]==Total[#2]&],Union@@#==Range[n]&];
Table[Length[cleq[n]],{n,0,5}]

A336737 Number of factorizations of n whose factors have pairwise intersecting prime signatures.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 2, 3, 1, 5, 1, 2, 2, 2, 2, 7, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 6, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 9, 1, 2, 3, 4, 2, 5, 1, 3, 2, 5, 1, 9, 1, 2, 3, 3, 2, 5, 1, 6, 3, 2, 1, 9, 2, 2, 2

Offset: 1

Gus Wiseman, Aug 06 2020

nonn

First differs from A327400 at a(72) = 9, A327400(72) = 10.

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

			The a(n) factorizations for n = 2, 4, 12, 24, 30, 36, 60:
  (2)  (4)    (12)     (24)       (30)     (36)       (60)
       (2*2)  (2*6)    (2*12)     (5*6)    (4*9)      (2*30)
              (2*2*3)  (2*2*6)    (2*15)   (6*6)      (3*20)
                       (2*2*2*3)  (3*10)   (2*18)     (5*12)
                                  (2*3*5)  (3*12)     (6*10)
                                           (2*3*6)    (2*5*6)
                                           (2*2*3*3)  (2*2*15)
                                                      (2*3*10)
                                                      (2*2*3*5)

A001055 counts factorizations.
A118914 is sorted prime signature.
A124010 is prime signature.
A336736 counts factorizations with disjoint signatures.
Cf. A003182, A051185, A305843, A305844, A305854, A306006, A319752, A319787, A319789, A321469, A336424.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
prisig[n_]:=If[n==1,{},Last/@FactorInteger[n]];
Table[Length[Select[facs[n],stableQ[#,Intersection[prisig[#1],prisig[#2]]=={}&]&]],{n,100}]

A326030 Number of antichains of subsets of {1..n} with different edge-sums.

Original entry on oeis.org

2, 3, 6, 19, 132, 3578, 826949

Offset: 0

Gus Wiseman, Jul 18 2019

An antichain is a finite set of finite sets, none of which is a subset of any other. The edge-sums are the sums of vertices in each edge, so for example the edge sums of {{1,3},{2,5},{3,4,5}} are {4,7,12}.

			The a(0) = 2 through a(3) = 19 antichains:
  {}    {}     {}         {}
  {{}}  {{}}   {{}}       {{}}
        {{1}}  {{1}}      {{1}}
               {{2}}      {{2}}
               {{1,2}}    {{3}}
               {{1},{2}}  {{1,2}}
                          {{1,3}}
                          {{2,3}}
                          {{1},{2}}
                          {{1,2,3}}
                          {{1},{3}}
                          {{2},{3}}
                          {{1},{2,3}}
                          {{2},{1,3}}
                          {{1,2},{1,3}}
                          {{1,2},{2,3}}
                          {{1},{2},{3}}
                          {{1,3},{2,3}}
                          {{1,2},{1,3},{2,3}}

Set partitions with different block-sums are A275780.
MM-numbers of multiset partitions with different part-sums are A326535.
The covering case is A326572.
Antichains with equal edge-sums are A326574.
Cf. A000372, A003182, A006126, A321469, A326519, A326571, A326573.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
cleqset[set_]:=stableSets[Subsets[set],SubsetQ[#1,#2]||Total[#1]==Total[#2]&];
Table[Length[cleqset[Range[n]]],{n,0,5}]

A326569 Number of covering antichains of subsets of {1..n} with no singletons and different edge-sizes.

Original entry on oeis.org

1, 0, 1, 1, 13, 121, 2566, 121199, 13254529

Offset: 0

Gus Wiseman, Jul 18 2019

An antichain is a finite set of finite sets, none of which is a subset of any other. It is covering if its union is {1..n}. The edge-sizes are the numbers of vertices in each edge, so for example the edge sizes of {{1,3},{2,5},{3,4,5}} are {2,2,3}.

			The a(2) = 1 through a(4) = 13 antichains:
  {{1,2}}  {{1,2,3}}  {{1,2,3,4}}
                      {{1,2},{1,3,4}}
                      {{1,2},{2,3,4}}
                      {{1,3},{1,2,4}}
                      {{1,3},{2,3,4}}
                      {{1,4},{1,2,3}}
                      {{1,4},{2,3,4}}
                      {{2,3},{1,2,4}}
                      {{2,3},{1,3,4}}
                      {{2,4},{1,2,3}}
                      {{2,4},{1,3,4}}
                      {{3,4},{1,2,3}}
                      {{3,4},{1,2,4}}

Antichain covers are A006126.
Set partitions with different block sizes are A007837.
The case with singletons is A326570.
Cf. A000372, A003182, A306021, A307249, A326565, A326571, A326572, A326573.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
cleq[n_]:=Select[stableSets[Subsets[Range[n],{2,n}],SubsetQ[#1,#2]||Length[#1]==Length[#2]&],Union@@#==Range[n]&];
Table[Length[cleq[n]],{n,0,6}]

a(n) = A326570(n) - n*a(n-1) for n > 0. - Andrew Howroyd, Aug 13 2019

a(8) from Andrew Howroyd, Aug 13 2019

A326570 Number of covering antichains of subsets of {1..n} with different edge-sizes.

Original entry on oeis.org

2, 1, 1, 4, 17, 186, 3292, 139161, 14224121

Offset: 0

Gus Wiseman, Jul 18 2019

An antichain is a finite set of finite sets, none of which is a subset of any other. It is covering if its union is {1..n}. The edge-sizes are the numbers of vertices in each edge, so for example the edge-sizes of {{1,3},{2,5},{3,4,5}} are {2,2,3}.

			The a(0) = 2 through a(4) = 17 antichains:
  {}    {{1}}  {{1,2}}  {{1,2,3}}    {{1,2,3,4}}
  {{}}                  {{1},{2,3}}  {{1},{2,3,4}}
                        {{2},{1,3}}  {{2},{1,3,4}}
                        {{3},{1,2}}  {{3},{1,2,4}}
                                     {{4},{1,2,3}}
                                     {{1,2},{1,3,4}}
                                     {{1,2},{2,3,4}}
                                     {{1,3},{1,2,4}}
                                     {{1,3},{2,3,4}}
                                     {{1,4},{1,2,3}}
                                     {{1,4},{2,3,4}}
                                     {{2,3},{1,2,4}}
                                     {{2,3},{1,3,4}}
                                     {{2,4},{1,2,3}}
                                     {{2,4},{1,3,4}}
                                     {{3,4},{1,2,3}}
                                     {{3,4},{1,2,4}}

Antichain covers are A006126.
Set partitions with different block sizes are A007837.
The case without singletons is A326569.
(Antichain) covers with equal edge-sizes are A306021.
Cf. A000372, A003182, A307249, A326026, A326514, A326566, A326572.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
cleq[n_]:=Select[stableSets[Subsets[Range[n]],SubsetQ[#1,#2]||Length[#1]==Length[#2]&],Union@@#==Range[n]&];
Table[Length[cleq[n]],{n,0,6}]

a(8) from Andrew Howroyd, Aug 13 2019

A348260 Number of inequivalent maximal antichains of the Boolean lattice on a set of n elements.

Original entry on oeis.org

1, 2, 3, 5, 10, 30, 233, 35925

Offset: 0

Dmitry I. Ignatov, Oct 13 2021

a(n) is the number of orbits for the corresponding families of maximal antichains (see also A326358) of the powerset of {1,2,...,n} under the action of the symmetry group S_n.

			The a(0)=1 maximal antichains is {}.
The a(1)=2 maximal antichains are {}, {1}.
The a(2)=3 maximal antichains {}, {1}{2}, {12}.
Representatives of the a(3)=5 maximal antichains are: {}, {1}{2}{3}, {12}{3}, {12}{13}{23}, {123}.
Representatives of the a(4)=10 maximal antichains are:
   {},                       {1}{2}{3}{4},
   {12}{3}{4},               {12}{13}{23}{4},
   {123}{4},                 {12}{13}{24}{14}{24}{34},
   {123}{14}{24}{34},        {123}{124}{34},
   {123}{124}{134}{234},     {1234}.

Dmitry I. Ignatov, On the Number of Maximal Antichains in Boolean Lattices for n up to 7. Lobachevskii J. Math., 44 (2023), 137-146.
Dmitry I. Ignatov, Supporting iPython code and input files for generating and counting inequivalent maximal antichains for n up to 7 and related sequences, Github repository, sections 0,1,4
Dmitry I. Ignatov, Supporting iPython notebook: A348260.ipynb
Dmitry I. Ignatov, PDF of the supporting iPython notebook

Cf. A003182 (not necessarily maximal), A326358 (labeled case).

Cf. A326359, A326360.

A336736 Number of factorizations of n whose distinct factors have disjoint prime signatures.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 2, 2, 1, 1, 1, 4, 5, 1, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Aug 06 2020

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

			The a(n) factorizations for n = 36, 360, 720, 192, 288:
  (36)     (360)    (720)     (192)      (288)
  (6*6)    (5*72)   (8*90)    (3*64)     (8*36)
  (2*2*9)  (8*45)   (9*80)    (4*48)     (9*32)
  (3*3*4)  (9*40)   (10*72)   (6*32)     (16*18)
           (10*36)  (16*45)   (12*16)    (2*144)
           (5*8*9)  (5*144)   (3*8*8)    (6*6*8)
                    (5*9*16)  (4*6*8)    (2*2*72)
                    (8*9*10)  (3*4*16)   (2*9*16)
                              (3*4*4*4)  (3*3*32)
                                         (2*2*8*9)
                                         (3*3*4*8)
                                         (2*2*2*36)
                                         (2*2*2*2*2*9)

A001055 counts factorizations.
A118914 is sorted prime signature.
A124010 is prime signature.
A336737 counts factorizations with intersecting signatures.
Cf. A000372, A003182, A006126, A109298, A112798, A293606, A294068, A305844, A321469, A336424.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
prisig[n_]:=If[n==1,{},Last/@FactorInteger[n]];
Table[Length[Select[facs[n],stableQ[#,Intersection[prisig[#1],prisig[#2]]!={}&]&]],{n,100}]

A372495 Number of inequivalent unate functions of n or fewer variables.

Original entry on oeis.org

2, 4, 10, 34, 200, 3466, 829744

Offset: 0

Aniruddha Biswas, May 03 2024

A Boolean function is unate in a variable if it is either nondecreasing or nonincreasing with respect to that variable. Therefore in the circuit representation of unate functions, each variable appears either in its original form or in complemented form. Thus x⊕y = (x∧¬y)∨(¬x∧y) is not a unate function.

Moreover, two Boolean functions are said to be equivalent if they are equivalent under the permutation of variables. For example, f(x,y)=x is equivalent to f(x,y)=y under the permutation of input variables.

			The list of all 2-variable inequivalent unate functions f(x,y) is 0,1,x,¬x,x∧y,¬x∧y,¬x∧¬y,x∨y,¬x∨y,¬x∨¬y. So a(2)=10.

Aniruddha Biswas and Palash Sarkar, Counting unate and balanced monotone Boolean functions, arXiv:2304.14069 [math.CO], 2023.
Aniruddha Biswas and Palash Sarkar, Counting Unate and Monotone Boolean Functions Under Restrictions of Balancedness and Non-Degeneracy, J. Int. Seq. (2025) Vol. 28, Art. No. 25.3.4. See pp. 4, 16.
Index entries for sequences related to Boolean functions

Cf. A000372, A003182, A245079.

cp's OEIS Frontend

A326574 Number of antichains of subsets of {1..n} with equal edge-sums.

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A302250 The number of antichains in the lattice of set partitions of an n-element set.

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A326571 Number of covering antichains of nonempty, non-singleton subsets of {1..n}, all having different sums.

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A336737 Number of factorizations of n whose factors have pairwise intersecting prime signatures.

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Mathematica

A326030 Number of antichains of subsets of {1..n} with different edge-sums.

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A326569 Number of covering antichains of subsets of {1..n} with no singletons and different edge-sizes.

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A326570 Number of covering antichains of subsets of {1..n} with different edge-sizes.

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A348260 Number of inequivalent maximal antichains of the Boolean lattice on a set of n elements.

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A336736 Number of factorizations of n whose distinct factors have disjoint prime signatures.

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