A006905 -id:A006905 - cp's OEIS frontend

A000666 Number of symmetric relations on n nodes.

1, 2, 6, 20, 90, 544, 5096, 79264, 2208612, 113743760, 10926227136, 1956363435360, 652335084592096, 405402273420996800, 470568642161119963904, 1023063423471189431054720, 4178849203082023236058229792, 32168008290073542372004082199424

Offset: 0

N. J. A. Sloane

Each node may or may not be related to itself.
Also the number of rooted graphs on n+1 nodes.
The 1-to-1 correspondence is as follows: Given a rooted graph on n+1 nodes, replace each edge joining the root node to another node with a self-loop at that node and erase the root node. The result is an undirected graph on n nodes which is the graph of the symmetric relation.
Also the number of the graphs with n nodes whereby each node is colored or not colored. A loop can be interpreted as a colored node. - Juergen Will, Oct 31 2011

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, pp. 101, 241.
M. D. McIlroy, Calculation of numbers of structures of relations on finite sets, Massachusetts Institute of Technology, Research Laboratory of Electronics, Quarterly Progress Reports, No. 17, Sept. 15, 1955, pp. 14-22.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

David Applegate, Table of n, a(n) for n = 0..80 [Shortened file because terms grow rapidly: see Applegate link below for additional terms]
David Applegate, Table of n, a(n) for n = 0..100
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
R. L. Davis, The number of structures of finite relations, Proc. Amer. Math. Soc. 4 (1953), 486-495.
R. L. Davis, Structure of dominance relations, Bull. Math. Biophys., 16 (1954), 131-140. [Annotated scanned copy]
F. Harary, The number of linear, directed, rooted, and connected graphs, Trans. Am. Math. Soc. 78 (1955) 445-463, eq. (24).
Frank Harary, Edgar M. Palmer, Robert W. Robinson and Allen J. Schwenk, Enumeration of graphs with signed points and lines, J. Graph Theory 1 (1977), no. 4, 295-308.
Adib Hassan, Po-Chien Chung and Wayne B. Hayes, Graphettes: Constant-time determination of graphlet and orbit identity including (possibly disconnected) graphlets up to size 8, arXiv:1708.04341 [cs.DS], 2017.
Mathematics Stack Exchange, Enumerating Graphs with Self-Loops, Jan 23 2014.
M. D. McIlroy, Calculation of numbers of structures of relations on finite sets, Massachusetts Institute of Technology, Research Laboratory of Electronics, Quarterly Progress Reports, No. 17, Sep. 15, 1955, pp. 14-22. [Annotated scanned copy]
W. Oberschelp, Kombinatorische Anzahlbestimmungen in Relationen, Math. Ann., 174 (1967), 53-78.
W. Oberschelp, The number of non-isomorphic m-graphs, Presented at Mathematical Institute Oberwolfach, July 3 1967 [Scanned copy of manuscript]
W. Oberschelp, Strukturzahlen in endlichen Relationssystemen, in Contributions to Mathematical Logic (Proceedings 1966 Hanover Colloquium), pp. 199-213, North-Holland Publ., Amsterdam, 1968. [Annotated scanned copy]
G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
Marko Riedel, Maple code for sequence.
Marko Riedel et al., Enumerating Graphs with Self-Loops
R. W. Robinson, Notes - "A Present for Neil Sloane"
R. W. Robinson, Notes - computer printout
Sage, Common Graphs (Graph Generators)
Lorenzo Sauras-Altuzarra, Graphs
J. M. Tangen and N. J. A. Sloane, Correspondence, 1976-1976
Eric Weisstein's World of Mathematics, Rooted Graph

Cf. A000595, A001172, A001174, A006905, A000250, A054921 (connected relations).

Maple
```
# see Riedel link above
```

Join[{1,2}, Table[CycleIndex[Join[PairGroup[SymmetricGroup[n]], Permutations[Range[n*(n-1)/2+1, n*(n+1)/2]], 2],s] /. Table[s[i]->2, {i,1,n^2-n}], {n,2,8}]] (* Geoffrey Critzer, Nov 04 2011 *)
Table[Module[{eds,pms,leq},
eds=Select[Tuples[Range[n],2],OrderedQ];
pms=Map[Sort,eds/.Table[i->Part[#,i],{i,n}]]&/@Permutations[Range[n]];
leq=Function[seq,PermutationCycles[Ordering[seq],Length]]/@pms;
Total[Thread[Power[2,leq]]]/n!
],{n,0,8}] (* This is after Geoffrey Critzer's program but does not use the (deprecated) Combinatorica package. - Gus Wiseman, Jul 21 2016 *)
permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[Sum[GCD[v[[i]], v[[j]]], {j, 1, i-1}], {i, 2, Length[v]}] + Sum[Quotient[v[[i]], 2] + 1, {i, 1, Length[v]}];
a[n_] := a[n] = (s = 0; Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!);
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 17}] (* Jean-François Alcover, Nov 13 2017, after Andrew Howroyd *)

permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i],v[j]))) + sum(i=1, #v, v[i]\2 + 1)}
a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*2^edges(p)); s/n!} \\ Andrew Howroyd, Oct 22 2017

from itertools import combinations
from math import prod, factorial, gcd
from fractions import Fraction
from sympy.utilities.iterables import partitions
def A000666(n): return int(sum(Fraction(1<>1)+1)*r+(q*r*(r-1)>>1) for q, r in p.items()),prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n))) # Chai Wah Wu, Jul 02 2024

Euler transform of A054921. - N. J. A. Sloane, Oct 25 2018
Let G_{n+1,k} be the number of rooted graphs on n+1 nodes with k edges and let G_{n+1}(x) = Sum_{k=0..n(n+1)/2} G_{n+1,k} x^k. Thus a(n) = G_{n+1}(1). Let S_n(x_1, ..., x_n) denote the cycle index for Sym_n. (cf. the link in A000142).
Compute x_1*S_n and regard it as the cycle index of a set of permutations on n+1 points and find the corresponding cycle index for the action on the n(n+1)/2 edges joining those points (the corresponding "pair group"). Finally, by replacing each x_i by 1+x^i gives G_{n+1}(x). [Harary]
Example, n=2. S_2 = (1/2)*(x_1^2+x_2), x_1*S_2 = (1/2)*(x_1^3+x_1*x_2). The pair group is (1/2)*(x_1^2+x_1*x_2) and so G_3(x) = (1/2)*((1+x)^3+(1+x)*(1+x^2)) = 1+2*x+2*x^2+x^3; set x=1 to get a(2) = 6.
a(n) ~ 2^(n*(n+1)/2)/n! [McIlroy, 1955]. - Vaclav Kotesovec, Dec 19 2016

Description corrected by Christian G. Bower
More terms from Vladeta Jovovic, Apr 18 2000
Entry revised by N. J. A. Sloane, Mar 06 2007

A088316 a(n) = 13*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 13.

Original entry on oeis.org

2, 13, 171, 2236, 29239, 382343, 4999698, 65378417, 854919119, 11179326964, 146186169651, 1911599532427, 24996980091202, 326872340718053, 4274337409425891, 55893258663254636, 730886700031736159, 9557420359075824703, 124977351368017457298, 1634262988143302769577

Offset: 0

Nikolay V. Kosinov, Dmitry V. Polyakov (kosinov(AT)unitron.com.ua), Nov 06 2003

For more information about this type of recurrence follow the Khovanova link and see A086902 and A054413. - Johannes W. Meijer, Jun 12 2010

G. C. Greubel, Table of n, a(n) for n = 0..890
Tanya Khovanova, Recursive Sequences
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (13,1).

Cf. A006905, A041318, A054413, A086902, A088316, A097845, A140455.

I:=[2,13]; [n le 2 select I[n] else 13*Self(n-1) +Self(n-2): n in [1..31]]; // G. C. Greubel, Dec 13 2022

LinearRecurrence[{13,1}, {2,13}, 31] (* Stefano Spezia, Sep 20 2022 *)

A088316=BinaryRecurrenceSequence(13,1,2,13)
[A088316(n) for n in range(31)] # G. C. Greubel, Dec 13 2022

a(n) = ((13+sqrt(173))/2)^n + ((13-sqrt(173))/2)^n.
Lim_{n -> oo} a(n+1)/a(n) = (13 + sqrt(173))/2.
Lim_{n -> oo} a(n)/a(n+1) = 2/(13+sqrt(173)).
G.f.: (2-13*x)/(1-13*x-x^2). - Philippe Deléham, Nov 02 2008
From Johannes W. Meijer, Jun 12 2010: (Start)
a(2*n+1) = 13*A097845(n).
a(3*n+1) = A041318(5n), a(3n+2) = A041318(5n+3), a(3n+3) = 2*A041318(5n+4).
Limit_{k->oo} a(n+k)/a(k) = (A088316(n) + A140455(n)*sqrt(173))/2.
Limit_{n->oo} A088316(n)/A140455(n) = sqrt(173). (End)

A121337 Number of idempotent relations on n labeled elements.

Original entry on oeis.org

1, 2, 11, 123, 2360, 73023, 3465357

Offset: 0

Florian Kammüller (flokam(AT)cs.tu-berlin.de), Aug 28 2006

A relation r is idempotent if r ; r = r, where ; denotes sequential composition.
From Geoffrey Critzer, Oct 18 2023 : (Start)
a(n) is also the number of maximal subgroups in the semigroup of binary relations on [n].  See Butler and Markowski link.
A binary relation is idempotent iff it is both dense (A355730) and transitive (A006905).
A binary relation is idempotent iff it is both limit dominating (A366194) and limit dominated (A366722).  See Gregory, Kirkland, and Pullman link.
A binary relation R on [n] is idempotent iff the following biconditional statement holds for all x,y in [n]:  There is a cyclic traverse from x to y in G(R) iff (x,y) is in R.  Here, G(R) is the directed graph with self loops allowed (A002416) corresponding to R.  See Rosenblatt link.
Let Q be a quasi-order (A000798) on [n].  Let D(X) be the relation {(x,x):x is in X}.  Let S be a subset of [n] such that: (i) For all x in S, the class in the equivalence relation Q intersect Q^(-1) containing (x,x) is a singleton and (ii) for all x,y in S, the component containing x is not covered by the component containing y in the condensation of G(Q) .  Here, the condensation of G(Q) is the acyclic digraph (A003024) obtained from G(Q) by replacing every strongly connected component (SCC) by a single vertex and all directed edges from one SCC to another with a single directed edge.  Then a relation is idempotent iff it is of the form Q-D(S).  See Schein link.  (End)

			a(2) = 11 because given a set {a,b} of two elements, of the 2^(2*2) = 16 relations on the set, only 5 are not idempotent. - _Michael Somos_, Jul 28 2013
These 5 relations that are not idempotent are: {(a,b)}, {(b,a)}, {(a,b),(b,a)}, {(a,b),(b,a),(b,b)}, {(a,a),(a,b),(b,a)}. - _Geoffrey Critzer_, Aug 07 2016

F. Kammüller, Interactive Theorem Proving in Software Engineering, Habilitationsschrift, Technische Universitaet Berlin (2006).
Ki Hang Kim, Boolean Matrix Theory and Applications, Marcel Decker, 1982.

G. Brinkmann and B. D. McKay, Counting unlabeled topologies and transitive relations, J. Integer Sequences, Volume 8, 2005.
K. K.-H. Butler and G. Markowsky, The Number of Maximal Subgroups of the Semigroup of Binary Relations, Kyungpook Math. J. Vol 12, June 1972.
D. A. Gregory, S. Kirkland, and N. J. Pullman, Power convergent Boolean matrices, Linear Algebra and its Applications, Volume 179, 15 January 1993, Pages 105-117.
F. Kammüller, Counting idempotent relations.
F. Kammüller and J. W. Sanders, Idempotent relations in Isabelle/HOL, International Colloquium on Theoretical Aspects of Computing, ICTAC'04. Volume 3407 of Lecture Notes in Computer Science, Springer-Verlag (2005).
G. Pfeiffer, Counting transitive relations, J. Integer Sequences, Volume 7, 2004, #3.
D. Rosenblatt, On the graphs of finite Boolean relation matrices, Journal of Research of the National Bureau of Standards, 67B No. 4, 1963.
B. M. Schein, A construction for idempotent binary relations, Proc. Japan Acad., Vol. 46, No. 3 (1970), pp. 246-247.

Cf. A000798 (labeled quasi-orders (or topologies)), A001930 (unlabeled quasi-orders), A001035 (labeled partial orders), A000112 (unlabeled partial orders), A002416, A003024, A366722, A366194, A355730, A006905.

Row sums of A360984.

Prepend[Table[Length[Select[Tuples[Tuples[{0, 1}, n], n], (MatrixPower[#, 2] /. x_ /; x > 0 -> 1) == # &]], {n, 1, 4}], 1] (* Geoffrey Critzer, Aug 07 2016 *)

Offset corrected by James Mitchell, Jul 28 2013

a(1) corrected by Philippe Beaudoin, Aug 11 2015

A169720 a(n) = (32^(n-1)-1)(3*2^n-1).

Original entry on oeis.org

1, 10, 55, 253, 1081, 4465, 18145, 73153, 293761, 1177345, 4713985, 18865153, 75479041, 301953025, 1207885825, 4831690753, 19327057921, 77308821505, 309236465665, 1236948221953, 4947797606401, 19791199862785, 79164818325505, 316659311050753, 1266637319700481

Offset: 0

Alice V. Kleeva (alice27353(AT)gmail.com), Jan 19 2010

A subsequence of the triangular numbers A000217.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Alice V. Kleeva, Grid for this sequence
Alice V. Kleeva, Illustration of initial terms
Robert Munafo, Sequence A169720, and two others by Alice V. Kleeva
Robert Munafo, Sequence A169720, and two others by Alice V. Kleeva [Cached copy, in pdf format, included with permission]
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A169721-A169727, A033484, A000217.

I:=[1, 10, 55]; [n le 3 select I[n] else 7*Self(n-1)-14*Self(n-2)+8*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Dec 03 2012

CoefficientList[Series[(1 + 3*x - x^2)/((1-x)*(1-2*x)*(1-4*x)), {x, 0, 30}], x] (* or *) LinearRecurrence[{7, -14, 8}, {1, 10, 55}, 30] (* Vincenzo Librandi, Dec 03 2012 *)

a(n)=polcoeff((1+3*x-x^2)/((1-x)*(1-2*x)*(1-4*x)+x*O(x^n)),n) \\ Paul D. Hanna, Apr 29 2010

G.f.: (1 + 3*x - x^2)/((1-x)*(1-2*x)*(1-4*x)). - Paul D. Hanna, Apr 29 2010
a(n) = A000217(A033484(n)). - Mitch Harris, Dec 02 2012
a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3). - Vincenzo Librandi, Dec 03 2012
a(n) = (3*A169726(n)-1)/2. - L. Edson Jeffery, Dec 03 2012
a(n) = A006095(n+2) +3*A006095(n+1) - A006905(n). - R. J. Mathar, Dec 04 2016

A340264 T(n, k) = Sum_{j=0..k} binomial(n, k - j)*Stirling2(n - k + j, j). Triangle read by rows, 0 <= k <= n.

Original entry on oeis.org

1, 0, 2, 0, 1, 4, 0, 1, 6, 8, 0, 1, 11, 24, 16, 0, 1, 20, 70, 80, 32, 0, 1, 37, 195, 340, 240, 64, 0, 1, 70, 539, 1330, 1400, 672, 128, 0, 1, 135, 1498, 5033, 7280, 5152, 1792, 256, 0, 1, 264, 4204, 18816, 35826, 34272, 17472, 4608, 512

Offset: 0

Peter Luschny, Jan 08 2021

A006905(n) = Sum_{k=0..n} A001035(k) * T(n, k). - Michael Somos, Jul 18 2021

T(n, k) is the number of idempotent relations R on [n] containing exactly k strongly connected components such that the following conditional statement holds for all x, y in [n]: If x, y are in distinct strongly connected components of R then (x, y) is not in R. - Geoffrey Critzer, Jan 10 2024

			[0] 1;
[1] 0, 2;
[2] 0, 1,   4;
[3] 0, 1,   6,    8;
[4] 0, 1,  11,   24,    16;
[5] 0, 1,  20,   70,    80,    32;
[6] 0, 1,  37,  195,   340,   240,    64;
[7] 0, 1,  70,  539,  1330,  1400,   672,   128;
[8] 0, 1, 135, 1498,  5033,  7280,  5152,  1792,  256;
[9] 0, 1, 264, 4204, 18816, 35826, 34272, 17472, 4608, 512;

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
Štefan Schwarz, On idempotent binary relations on a finite set, Czechoslovak Mathematical Journal, Vol. 20 (1970), No. 4, 696-702.
Eric Weisstein's World of Mathematics, Bell Polynomial.

Sum of row(n) is A000110(n+1).
Sum of row(n) - 2^n is A058681(n).
Alternating sum of row(n) is A109747(n).
Cf. A001035, A006905, A137375, A186081, A121337.

egf := exp(t*(exp(-x) - x - 1));
ser := series(egf, x, 22):
p := n -> coeff(ser, x, n);
seq(seq((-1)^n*n!*coeff(p(n), t, k), k=0..n), n = 0..10);
# Alternative:
T := (n, k) -> add(binomial(n, k - j)*Stirling2(n - k + j, j), j=0..k):
seq(seq(T(n, k), k = 0..n), n=0..9); # Peter Luschny, Feb 09 2021

T[ n_, k_] := Sum[ Binomial[n, k-j] StirlingS2[n-k+j, j], {j, 0 ,k}]; (* Michael Somos, Jul 18 2021 *)

T(n, k) = sum(j=0, k, binomial(n, j)*stirling(n-j, k-j, 2)); /* Michael Somos, Jul 18 2021 */

T(n, k) = (-1)^n * n! * [t^k] [x^n] exp(t*(exp(-x) - x - 1)).

n-th row polynomial R(n,x) = exp(-x)*Sum_{k >= 0} (x + k)^n * x^k/k! = Sum_{k = 0..n} binomial(n,k)*Bell(k,x)*x^(n-k), where Bell(n,x) denotes the n-th Bell polynomial. - Peter Bala, Jan 13 2022

New name from Peter Luschny, Feb 09 2021

A085628 Number of antisymmetric transitive binary relations on n labeled points.

Original entry on oeis.org

1, 2, 12, 152, 3504, 135392, 8321472, 784621952, 110521185024, 22789653765632, 6769730814753792, 2859584874712881152, 1699286839524775931904, 1407801166901961190203392, 1613567168628788544015286272, 2541721059997800475952740401152, 5470980000021882982488097199161344

Offset: 0

Goetz Pfeiffer (goetz.pfeiffer(AT)nuigalway.ie), Jan 21 2004

nonn

Jean-François Alcover, Table of n, a(n) for n = 0..18
G. Pfeiffer, Counting Transitive Relations, preprint, 2004.

Cf. A079265 (unlabeled antisymmetric transitive relations), A001035 (labeled partial orders), A000798 (labeled reflexive transitive relations), A006905 (labeled transitive relations).

A001035 = Cases[Import["https://oeis.org/A001035/b001035.txt", "Table"], {, }][[All, 2]];
a[n_] := 2^n A001035[[n + 1]];
a /@ Range[0, 18] (* Jean-François Alcover, Jan 01 2020 *)

a(n) = 2^n * A001035(n) = A000079(n) * A001035(n)

E.g.f.: A(2*x) where A(x) is the e.g.f. for A001035. - Geoffrey Critzer, Jul 28 2014

2 more terms from Charles R Greathouse IV, Aug 31 2006

A091073 Number of transitive relations on n unlabeled points.

Original entry on oeis.org

1, 2, 8, 39, 242, 1895, 19051, 246895, 4145108, 90325655, 2555630036, 93810648902, 4461086120602, 274339212258846, 21775814889230580, 2226876304576948549

Offset: 0

Goetz Pfeiffer (goetz.pfeiffer(AT)nuigalway.ie), Jan 21 2004

a(13)-a(15) are from Brinkmann's and McKay's paper. - Vladeta Jovovic, Jan 07 2006

Gunnar Brinkmann and Brendan D. McKay, Counting unlabelled topologies and transitive relations.
Gunnar Brinkmann and Brendan D. McKay, Counting Unlabelled Topologies and Transitive Relations, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.1.
G. Pfeiffer, Counting Transitive Relations, preprint, 2004.
G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.

Cf. A079265 (antisymmetric transitive relations), A001930 (reflexive transitive relations), A000112 (partial orders), A006905 (labeled transitive relations).

More terms from Vladeta Jovovic, Jan 07 2006

A355730 Number of binary relations R on [n] such that R is contained in R^2.

Original entry on oeis.org

1, 2, 13, 333, 34924, 15339497, 28399641433

Offset: 0

Geoffrey Critzer, Jul 15 2022

Equivalently, a(n) is the number of binary relations R on [n] such that for all x,y in [n], if (x,y) is in R then there is a z in [n] such that (x,z) and (z,y) are both in R.  A relation having this property is sometimes called a dense relation.
Almost all relations are dense.
A relation is idempotent if and only if it is both transitive and dense.  A transitive relation R is dense (hence idempotent) if and only if there does not exist a pair (C_1,C_2) of edgeless components such that C_1 covers C_2 in the condensation of G(R).  Here, G(R) is the directed graph (with self loops allowed) associated to R.  The condensation of G(R) is the acyclic digraph obtained from G(R) by replacing every strongly connected component (SCC) by a single vertex and all directed edges from one SCC to another with a single directed edge.   See Schein reference.
If R is contained in R^2 then the maximal cyclic nets in R are primitive (A070322) so that R is convergent, i.e., the period of R is equal to 1. Moreover, R converges to its transitive closure so that the index of R is at most n.  See Rosenblatt reference. - Geoffrey Critzer, Sep 05 2023

			a(2) = 13 because all 16 binary relations on [2] are dense except for {{0,1},{0,0}}, {{0,0},{1,0}}, {{0,1},{1,0}}.

G. Markowsky, Bounds on the index and period of a binary relation on a finite set, Semigroup Forum, Vol 13 (1977), 253-259.
D. Rosenblatt, On the graphs of finite Boolean relation matrices, Journal of Research of the National Bureau of Standards, 67B No. 4, 1963.
B. M. Schein, A construction for idempotent binary relations, Proc. Japan Acad., Vol. 46, No. 3 (1970), pp. 246-247.
Wikipedia, Dense order.

Cf. A006905, A121337, A070322.

Table[B = Tuples[Tuples[{0, 1}, nn], nn]; subsetQ[matrix1_, matrix2_] :=
  Apply[And, Map[! MemberQ[#, 1] &, matrix1 - matrix2]];Select[B, subsetQ[#, Clip[#.#]] &] // Length, {nn, 0, 4}]

Limit_{n->oo} a(n)/2^(n^2) = 1.

a(5)-a(6) from Pontus von Brömssen, Jul 19 2022

Comments clarified by Geoffrey Critzer, Oct 16 2023

cp's OEIS Frontend

A174137 Partial sums of A006905.

Views

Author

Keywords

Comments

Crossrefs

Formula

A348151 First differences of A006905.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A000666 Number of symmetric relations on n nodes.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

Extensions

A088316 a(n) = 13*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 13.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A121337 Number of idempotent relations on n labeled elements.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Extensions

A169720 a(n) = (3*2^(n-1)-1)*(3*2^n-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A340264 T(n, k) = Sum_{j=0..k} binomial(n, k - j)*Stirling2(n - k + j, j). Triangle read by rows, 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A169720 a(n) = (32^(n-1)-1)(3*2^n-1).