A006602 -id:A006602 - cp's OEIS frontend

A327359 Triangle read by rows where T(n,k) is the number of unlabeled antichains of nonempty sets covering n vertices with vertex-connectivity exactly k.

1, 1, 0, 1, 1, 0, 2, 1, 2, 0, 6, 4, 4, 6, 0, 23, 29, 37, 37, 54, 0

Offset: 0

Gus Wiseman, Sep 10 2019

An antichain is a set of sets, none of which is a subset of any other. It is covering if there are no isolated vertices.
The vertex-connectivity of a set-system is the minimum number of vertices that must be removed (along with any empty or duplicate edges) to obtain a non-connected set-system or singleton. Note that this means a single node has vertex-connectivity 0.
If empty edges are allowed, we have T(0,0) = 2.

			Triangle begins:
   1
   1  0
   1  1  0
   2  1  2  0
   6  4  4  6  0
  23 29 37 37 54  0
Row n = 4 counts the following antichains:
{1}{234}      {14}{234}        {134}{234}           {1234}
{12}{34}      {13}{24}{34}     {13}{14}{234}        {12}{134}{234}
{1}{2}{34}    {14}{24}{34}     {12}{13}{24}{34}     {124}{134}{234}
{1}{24}{34}   {14}{23}{24}{34} {13}{14}{23}{24}{34} {12}{13}{14}{234}
{1}{2}{3}{4}                                        {123}{124}{134}{234}
{1}{23}{24}{34}                                     {12}{13}{14}{23}{24}{34}

Row sums are A261005, or A006602 if empty edges are allowed.
Column k = 0 is A327426.
Column k = 1 is A327436.
Column k = n - 1 is A327425.
The labeled version is A327351.
Cf. A003465, A006126, A014466, A048143, A293993, A323818, A326704, A327125, A327334, A327336, A327350, A327358.

A306550 Array read by antidiagonals where A(n,k) is the number of labeled k-antichains covering n vertices.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 6, 0, 0, 0, 0, 1, 25, 2, 0, 0, 0, 0, 1, 90, 56, 0, 0, 0, 0, 0, 1, 301, 790, 25, 0, 0, 0, 0, 0, 1, 966, 8380, 1895, 6, 0, 0, 0, 0, 0, 1, 3025, 76482, 70370, 2116, 1, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Feb 23 2019

			Array begins:
    n=0: n=1: n=2: n=3: n=4: n=5:
---------------------------------
k=0:  1    0    0    0    0    0
k=1:  1    1    1    1    1    1
k=2:  0    0    1    6   25   90
k=3:  0    0    0    2   56  790
k=4:  0    0    0    0   25 1895
k=5:  0    0    0    0    6 2116
Column n = 3 counts the following antichains:
  {{123}}  {{1}{23}}   {{1}{2}{3}}
           {{2}{13}}   {{12}{13}{23}}
           {{3}{12}}
           {{12}{13}}
           {{12}{23}}
           {{13}{23}}

Column sums are A006126. Row k = 2 is A000392. Rows k = 3-9 are A056046-A056049, A056052, A056101, A056104.

Cf. A000372, A006126, A006602, A014466, A261005, A305000, A305844, A317674.

nn=8;
stableSets[u_,Q_,k_]:=If[k==0,{{}},If[Length[u]==0,{},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q,k],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q,k-1]]]]];
ae[n_,k_]:=Length[Select[stableSets[Subsets[Range[n]],SubsetQ,k],Union@@#==Range[n]&]];
Table[ae[k,n-k],{n,0,nn},{k,0,n}]

A079263 Number of constrained mixed models with n factors.

Original entry on oeis.org

2, 6, 22, 101, 576, 4162, 38280, 451411, 6847662, 133841440

Offset: 1

N. J. A. Sloane, Feb 16 2003

A. Hess and H. Iyer, Enumeration of mixed linear models and SAS macro for computation of confidence intervals for variance components, presented at Applied Statistics in Agriculture Conference at Kansas State University 2001.

R. Bayon, N. Lygeros and J.-S. Sereni, New progress in enumeration of mixed models, Applied Mathematics E-Notes, 5 (2005), 60-65.
R. Bayon, N. Lygeros and J.-S. Sereni, Nouveaux progrès dans l'énumération des modèles mixtes, in Knowledge discovery and discrete mathematics : JIM'2003, INRIA, Université de Metz, France, 2003, pp. 243-246.
R. Fraïssé and N. Lygeros, Petits posets: dénombrement, représentabilité par cercles et "compenseurs", C. R. Acad. Sci. Paris, Vol. 313, series I, pp. 417-420, 1991.
Ann Marie Hess, Mixed Models Site

Cf. A000112, A079265, A006126, A006602, A006896-A006898.

a(10) from Bayon, Lygeros, and Sereni (2005) added by Sean A. Irvine, Aug 05 2025

A327436 Number of connected, unlabeled antichains of nonempty subsets of {1..n} covering n vertices with at least one cut-vertex (vertex-connectivity 1).

Original entry on oeis.org

0, 0, 1, 1, 4, 29

Offset: 0

Gus Wiseman, Sep 11 2019

nonn

			Non-isomorphic representatives of the a(2) = 1 through a(5) = 29 antichains:
  {12}  {12}{13}  {12}{134}         {12}{1345}
                  {12}{13}{14}      {123}{145}
                  {12}{13}{24}      {12}{13}{145}
                  {12}{13}{14}{23}  {12}{13}{245}
                                    {13}{24}{125}
                                    {13}{124}{125}
                                    {14}{123}{235}
                                    {12}{13}{14}{15}
                                    {12}{13}{14}{25}
                                    {12}{13}{24}{35}
                                    {12}{13}{14}{235}
                                    {12}{13}{23}{145}
                                    {12}{13}{45}{234}
                                    {12}{14}{23}{135}
                                    {12}{15}{134}{234}
                                    {15}{23}{124}{134}
                                    {15}{123}{124}{134}
                                    {15}{123}{124}{234}
                                    {12}{13}{14}{15}{23}
                                    {12}{13}{14}{23}{25}
                                    {12}{13}{14}{23}{45}
                                    {12}{13}{15}{24}{34}
                                    {12}{13}{14}{15}{234}
                                    {12}{13}{14}{25}{234}
                                    {12}{13}{14}{15}{23}{24}
                                    {12}{13}{14}{15}{23}{45}
                                    {12}{13}{14}{23}{24}{35}
                                    {15}{123}{124}{134}{234}
                                    {12}{13}{14}{15}{23}{24}{34}

Column k = 1 of A327359.
The labeled version is A327356.
Cf. A006602, A014466, A048143, A261005, A326704, A326786, A327112, A327114, A327426, A327334, A327336, A327350, A327351, A327358.

a(n > 2) = A261006(n) - A305028(n).

A373697 Number of nondegenerate balanced unate functions of n or fewer variables.

Original entry on oeis.org

0, 2, 0, 8, 256, 16832, 31287424, 10393552784640

Offset: 0

Aniruddha Biswas, Jun 13 2024

Aniruddha Biswas and P. 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, 13.
Index entries for sequences related to Boolean functions

Cf. A006126, A006602, A341633, A245079.

A003183 Number of NPN-equivalence classes of unate Boolean functions of n or fewer variables.

Original entry on oeis.org

1, 2, 3, 6, 17, 112, 8282

Offset: 0

N. J. A. Sloane

Number of inequivalent (under the group of permutations and "inversion of variables") monotone Boolean functions of n of fewer variables.

Given f, a function of n variables, we define the "inversion of variables", i, by (i.f)(x1,...,xn)=1+f(1+x1,...,1+xn) (we can write (i.f)(x)=1+f(1+x) where the second "1" denotes (1,...,1)). It turns out that if f is monotone, then i.f is also monotone. On the other hand, a permutation of n elements, p, acts on f by (p.f)(x)=f(p(x)). It turns out that if f is monotone, then p.f is also monotone. We define p.i by (p.i)(f)=p.(i.f) and i.p by (i.p)(f)=i.(p.f). If we define a.b by (a.b).f=a.(b.f) for a,b elements of G, it turns out that G={p.i,p: p is a permutation of n elements} is a group. In this context, f and g are equivalent if there exists b (an element of G) such that b.f=g.

			a(2)=3 because m(x,y)=x, n(x,y)=y, k(x,y)=0, h(x,y)=1, f(x,y)=x*y, g(x,y)=x+y+xy are the six monotone Boolean functions of 2 or fewer variables; m and n are equivalent, k and h are equivalent, f and g are equivalent. Then we have 3 inequivalent monotone Boolean functions of 2 or fewer variables.

S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38, Table 2.3.2. - Row 18.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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. 2, 8, 17.
Saburo Muroga, Threshold Logic and Its Applications, Wiley, NY, 1971. [Annotated scans of a few pages]
Index entries for sequences related to Boolean functions

Cf. A120608, A120587, A006602.

Additional comments from Alan Veliz-Cuba (alanavc(AT)vt.edu), Jun 18 2006

A037843 Number of matrices with n columns whose rows do not cover each other; ordered antichains of subsets of an n-set.

Original entry on oeis.org

2, 3, 7, 39, 2551, 22928343, 6641112790058484007

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Jul 23 2000

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
Index entries for sequences related to Boolean functions

Cf. A003182, A051112-A051117, A007411, A006602.

a(n)=Sum_{k=0..C(n, floor(n/2))}k!*M(n, k) where M(n, k) is the number of distinct monotone Boolean functions of n variables with k mincuts.

A120618 Number of inequivalent (under "inversion of variables") monotone Boolean functions of n or fewer variables.

Original entry on oeis.org

1, 2, 4, 12, 90, 3831

Offset: 0

Alan Veliz-Cuba (alanavc(AT)vt.edu), Jun 18 2006

We define the "inversion of variables", i, by (i.f)(x1,...,xn)=1+f(1+x1,...,1+xn). Note that {i,identity function} is a group. It turns out that if f is a monotone function, then i.f is also a monotone function. f is equivalent to g if f=g or f=i.g.

			a(1)=2 because m(x)=0,n(x)=1,k(x)=x are the three monotone Boolean functions (of 1 or fewer variables) and m,n are equivalent.

Cf. A120608, A120587, A006602.

cp's OEIS Frontend

A327359 Triangle read by rows where T(n,k) is the number of unlabeled antichains of nonempty sets covering n vertices with vertex-connectivity exactly k.

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A306550 Array read by antidiagonals where A(n,k) is the number of labeled k-antichains covering n vertices.

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Mathematica

A079263 Number of constrained mixed models with n factors.

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A327436 Number of connected, unlabeled antichains of nonempty subsets of {1..n} covering n vertices with at least one cut-vertex (vertex-connectivity 1).

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Formula

A373697 Number of nondegenerate balanced unate functions of n or fewer variables.

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A003183 Number of NPN-equivalence classes of unate Boolean functions of n or fewer variables.

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Comments

Examples

References

Links

Crossrefs

Extensions

A037843 Number of matrices with n columns whose rows do not cover each other; ordered antichains of subsets of an n-set.

Views

Author

Keywords

References

Links

Crossrefs

Formula

A120618 Number of inequivalent (under "inversion of variables") monotone Boolean functions of n or fewer variables.

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Author

Keywords

Comments

Examples

Crossrefs