Search: a003087 -id:a003087
|
| |
| |
|
|
|
1, 2, 4, 10, 41, 343, 6327, 249995, 20536020, 3445474030, 1169394086932, 798731069436512, 1094825607339329108, 3006942191342795594938, 16533858553282474307626008, 181924651566316766693192987410
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
Partial sums of number of acyclic digraphs with n unlabeled nodes; equivalently partial sums of the number of equivalence classes of n X n real (0,1)-matrices with all eigenvalues positive, up to conjugation by permutations. The subsequence of primes in this partial sum begins: 2, 41, no more through a(18).
|
|
|
LINKS
|
|
|
|
FORMULA
|
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A003024
|
|
Number of acyclic digraphs (or DAGs) with n labeled nodes.
(Formerly M3113)
|
|
+10
71
|
|
|
|
1, 1, 3, 25, 543, 29281, 3781503, 1138779265, 783702329343, 1213442454842881, 4175098976430598143, 31603459396418917607425, 521939651343829405020504063, 18676600744432035186664816926721, 1439428141044398334941790719839535103, 237725265553410354992180218286376719253505
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
Also the number of n X n real (0,1)-matrices with all eigenvalues positive. - Conjectured by Eric W. Weisstein, Jul 10 2003 and proved by McKay et al. 2003, 2004
Also the number of n X n real (0,1)-matrices with permanent equal to 1, up to permutation of rows/columns, cf. A089482. - Vladeta Jovovic, Oct 28 2009
Also the number of nilpotent elements in the semigroup of binary relations on [n]. - Geoffrey Critzer, May 26 2022
Also the number of sets of n nonempty subsets of {1..n} such that there is a unique way to choose a different element from each. For example, non-isomorphic representatives of the a(3) = 25 set-systems are:
{{1},{2},{3}}
{{1},{2},{1,3}}
{{1},{2},{1,2,3}}
{{1},{1,2},{1,3}}
{{1},{1,2},{2,3}}
{{1},{1,2},{1,2,3}}
(End)
|
|
|
REFERENCES
|
Archer, K., Gessel, I. M., Graves, C., & Liang, X. (2020). Counting acyclic and strong digraphs by descents. Discrete Mathematics, 343(11), 112041.
S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 310.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 19, Eq. (1.6.1).
R. W. Robinson, Counting labeled acyclic digraphs, pp. 239-273 of F. Harary, editor, New Directions in the Theory of Graphs. Academic Press, NY, 1973.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P Stanley, Enumerative Combinatorics I, 2nd. ed., p. 322.
|
|
|
LINKS
|
Laphou Lao, Zecheng Li, Songlin Hou, Bin Xiao, Songtao Guo, and Yuanyuan Yang, A Survey of IoT Applications in Blockchain Systems: Architecture, Consensus and Traffic Modeling, ACM Computing Surveys (CSUR, 2020) Vol. 53, No. 1, Article No. 18.
|
|
|
FORMULA
|
a(0) = 1; for n > 0, a(n) = Sum_{k=1..n} (-1)^(k+1)*C(n, k)*2^(k*(n-k))*a(n-k).
1 = Sum_{n=>0} a(n)*x^n/(1 + 2^n*x)^(n+1). - Paul D. Hanna, Oct 17 2009
1 = Sum_{n>=0} a(n)*C(n+m-1,n)*x^n/(1 + 2^n*x)^(n+m) for m>=1. - Paul D. Hanna, Apr 01 2011
log(1+x) = Sum_{n>=1} a(n)*(x^n/n)/(1 + 2^n*x)^n. - Paul D. Hanna, Apr 01 2011
Let E(x) = Sum_{n >= 0} x^n/(n!*2^C(n,2)). Then a generating function for this sequence is 1/E(-x) = Sum_{n >= 0} a(n)*x^n/(n!*2^C(n,2)) = 1 + x + 3*x^2/(2!*2) + 25*x^3/(3!*2^3) + 543*x^4/(4!*2^6) + ... (Stanley). Cf. A188457. - Peter Bala, Apr 01 2013
a(n) ~ n!*2^(n*(n-1)/2)/(M*p^n), where p = 1.488078545599710294656246... is the root of the equation Sum_{n>=0} (-1)^n*p^n/(n!*2^(n*(n-1)/2)) = 0, and M = Sum_{n>=1} (-1)^(n+1)*p^n/((n-1)!*2^(n*(n-1)/2)) = 0.57436237330931147691667... Both references to the article "Acyclic digraphs and eigenvalues of (0,1)-matrices" give the wrong value M=0.474! - Vaclav Kotesovec, Dec 09 2013 [Response from N. J. A. Sloane, Dec 11 2013: The value 0.474 has a typo, it should have been 0.574. The value was taken from Stanley's 1973 paper.]
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 2*x^2 + 10*x^3 + 146*x^4 + 6010*x^5 + ... appears to have integer coefficients (cf. A188490). - Peter Bala, Jan 14 2016
|
|
|
EXAMPLE
|
For n = 2 the three (0,1)-matrices are {{{1, 0}, {0, 1}}, {{1, 0}, {1, 1}}, {{1, 1}, {0, 1}}}.
|
|
|
MAPLE
|
p:=evalf(solve(sum((-1)^n*x^n/(n!*2^(n*(n-1)/2)), n=0..infinity) = 0, x), 50); M:=evalf(sum((-1)^(n+1)*p^n/((n-1)!*2^(n*(n-1)/2)), n=1..infinity), 40); # program for evaluation of constants p and M in the asymptotic formula, Vaclav Kotesovec, Dec 09 2013
|
|
|
MATHEMATICA
|
a[0] = a[1] = 1; a[n_] := a[n] = Sum[ -(-1)^k * Binomial[n, k] * 2^(k*(n-k)) * a[n-k], {k, 1, n}]; Table[a[n], {n, 0, 13}](* Jean-François Alcover, May 21 2012, after PARI *)
Table[2^(n*(n-1)/2)*n! * SeriesCoefficient[1/Sum[(-1)^k*x^k/k!/2^(k*(k-1)/2), {k, 0, n}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 19 2015 *)
Table[Length[Select[Subsets[Subsets[Range[n]], {n}], Length[Select[Tuples[#], UnsameQ@@#&]]==1&]], {n, 0, 5}] (* Gus Wiseman, Jan 01 2024 *)
|
|
|
PROG
|
(PARI) a(n)=if(n<1, n==0, sum(k=1, n, -(-1)^k*binomial(n, k)*2^(k*(n-k))*a(n-k)))
(PARI) {a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k/(1+2^k*x+x*O(x^n))^(k+1)), n)} \\ Paul D. Hanna, Oct 17 2009
|
|
|
CROSSREFS
|
Cf. A055165, which counts nonsingular {0, 1} matrices and A085656, which counts positive definite {0, 1} matrices.
These are the reverse-alternating sums of rows of A046860.
The weakly connected case is A082402.
|
|
|
KEYWORD
|
nonn,easy,nice
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A000595
|
|
Number of binary relations on n unlabeled points.
(Formerly M1980 N0784)
|
|
+10
45
|
|
|
|
1, 2, 10, 104, 3044, 291968, 96928992, 112282908928, 458297100061728, 6666621572153927936, 349390545493499839161856, 66603421985078180758538636288, 46557456482586989066031126651104256, 120168591267113007604119117625289606148096, 1152050155760474157553893461743236772303142428672
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
Number of orbits under the action of permutation group S(n) on n X n {0,1} matrices. The action is defined by f.M(i,j)=M(f(i),f(j)).
Equivalently, the number of digraphs on n unlabeled nodes with loops allowed but no more than one arc with the same start and end node. - Andrew Howroyd, Oct 22 2017
|
|
|
REFERENCES
|
F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 76 (2.2.30)
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.
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).
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) = sum {1*s_1+2*s_2+...=n} (fixA[s_1, s_2, ...] / (1^s_1*s_1!*2^s_2*s_2!*...)) where fixA[s_1, s_2, ...] = 2^sum {i, j>=1} (gcd(i, j)*s_i*s_j). - Christian G. Bower, Jan 05 2004
|
|
|
EXAMPLE
|
Non-isomorphic representatives of the a(2) = 10 relations:
{}
{1->1}
{1->2}
{1->1, 1->2}
{1->1, 2->1}
{1->1, 2->2}
{1->2, 2->1}
{1->1, 1->2, 2->1}
{1->1, 1->2, 2->2}
{1->1, 1->2, 2->1, 2->2}
(End)
|
|
|
MATHEMATICA
|
Join[{1, 2}, Table[CycleIndex[Join[PairGroup[SymmetricGroup[n], Ordered], Permutations[Range[n^2-n+1, n^2]], 2], s] /. Table[s[i]->2, {i, 1, n^2-n}], {n, 2, 7}]] (* Geoffrey Critzer, Nov 02 2011 *)
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[2*GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[v];
a[n_] := (s=0; Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!);
dinorm[m_]:=If[m=={}, {}, If[Union@@m!=Range[Max@@Flatten[m]], dinorm[m/.Apply[Rule, Table[{(Union@@m)[[i]], i}, {i, Length[Union@@m]}], {1}]], First[Sort[dinorm[m, 1]]]]];
dinorm[m_, aft_]:=If[Length[Union@@m]<=aft, {m}, With[{mx=Table[Count[m, i, {2}], {i, Select[Union@@m, #1>=aft&]}]}, Union@@(dinorm[#1, aft+1]&)/@Union[Table[Map[Sort, m/.{par+aft-1->aft, aft->par+aft-1}, {0}], {par, First/@Position[mx, Max[mx]]}]]]];
Table[Length[Union[dinorm/@Subsets[Tuples[Range[n], 2]]]], {n, 0, 3}] (* Gus Wiseman, Jun 17 2019 *)
|
|
|
PROG
|
(GAP) NSeq := function ( n ) return Sum(List(ConjugacyClasses(SymmetricGroup(n)), c -> (2^Length(Orbits(Group(Representative(c)), CartesianProduct([1..n], [1..n]), OnTuples))) * Size(c)))/Factorial(n); end; # Dan Hoey, May 04 2001
(PARI)
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, 2*gcd(v[i], v[j]))) + sum(i=1, #v, v[i])}
a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*2^edges(p)); s/n!} \\ Andrew Howroyd, Oct 22 2017
(Python)
from itertools import product
from math import prod, factorial, gcd
from fractions import Fraction
from sympy.utilities.iterables import partitions
def A000595(n): return int(sum(Fraction(1<<sum(p[r]*p[s]*gcd(r, s) for r, s in product(p.keys(), repeat=2)), prod(q**p[q]*factorial(p[q]) for q in p)) for p in partitions(n))) # Chai Wah Wu, Jul 02 2024
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,nice
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
Still more terms from Dan Hoey, May 04 2001
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A003025
|
|
Number of n-node labeled acyclic digraphs with 1 out-point.
(Formerly M2083)
|
|
+10
16
|
|
|
|
1, 2, 15, 316, 16885, 2174586, 654313415, 450179768312, 696979588034313, 2398044825254021110, 18151895792052235541515, 299782788128536523836784628, 10727139906233315197412684689421
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
Also the number of n-element sets of finite nonempty subsets of {1..n}, including a unique singleton, such that there is exactly one way to choose a different element from each. For example, the a(0) = 0 through a(3) = 15 set-systems are:
. {{1}} {{1},{1,2}} {{1},{1,2},{1,3}}
{{2},{1,2}} {{1},{1,2},{2,3}}
{{1},{1,3},{2,3}}
{{2},{1,2},{1,3}}
{{2},{1,2},{2,3}}
{{2},{1,3},{2,3}}
{{3},{1,2},{1,3}}
{{3},{1,2},{2,3}}
{{3},{1,3},{2,3}}
{{1},{1,2},{1,2,3}}
{{1},{1,3},{1,2,3}}
{{2},{1,2},{1,2,3}}
{{2},{2,3},{1,2,3}}
{{3},{1,3},{1,2,3}}
{{3},{2,3},{1,2,3}}
These set-systems are all connected.
The case of labeled graphs is A000169.
(End)
|
|
|
REFERENCES
|
R. W. Robinson, Counting labeled acyclic digraphs, pp. 239-273 of F. Harary, editor, New Directions in the Theory of Graphs. Academic Press, NY, 1973.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
|
LINKS
|
|
|
|
FORMULA
|
|
|
|
EXAMPLE
|
a(2) = 2: o-->--o (2 ways)
a(3) = 15: o-->--o-->--o (6 ways) and
o ... o o-->--o
.\ . / . \ . /
. v v ... v v
.. o ..... o
(3 ways) (6 ways)
|
|
|
MATHEMATICA
|
Table[Length[Select[Subsets[Subsets[Range[n]], {n}], Count[#, {_}]==1&&Length[Select[Tuples[#], UnsameQ@@#&]]==1&]], {n, 0, 4}] (* Gus Wiseman, Jan 02 2024 *)
|
|
|
PROG
|
(PARI) \\ see Marcel et al. link (formula for a').
seq(n)={my(a=vector(n)); a[1]=1; for(n=2, #a, a[n]=sum(k=1, n-1, (-1)^(k-1) * binomial(n, k) * (2^(n-k)-1)^k * a[n-k])); a} \\ Andrew Howroyd, Jan 01 2022
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A122078
|
|
Triangle read by rows: T(n,k) is the number of unlabeled acyclic digraphs with n >= 0 nodes and n-k outnodes (0 <= k <= n).
|
|
+10
14
|
|
|
|
1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 11, 16, 0, 1, 4, 25, 108, 164, 0, 1, 5, 47, 422, 2168, 3341, 0, 1, 6, 78, 1251, 15484, 88747, 138101, 0, 1, 7, 120, 3124, 79836, 1215783, 7409117, 11578037, 0, 1, 8, 174, 6925, 333004, 11620961, 199203464, 1252610909, 1961162564, 0
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,8
|
|
|
REFERENCES
|
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
Triangle T(n,k) begins:
1:
1, 0;
1, 1, 0;
1, 2, 3, 0;
1, 3, 11, 16, 0;
1, 4, 25, 108, 164, 0;
1, 5, 47, 422, 2168, 3341, 0;
1, 6, 78, 1251, 15484, 88747, 138101, 0;
...
|
|
|
PROG
|
(PARI) \\ See link for program code.
{ my(T=AcyclicDigraphsByNonSources(8)); for(n=1, #T, print(T[n])) } \\ Andrew Howroyd, Dec 31 2021
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A326225
|
|
Number of Hamiltonian unlabeled n-vertex digraphs (without loops).
|
|
+10
11
|
| |
|
|
|
OFFSET
|
0,4
|
|
|
COMMENTS
|
A digraph is Hamiltonian if it contains a directed cycle passing through every vertex exactly once.
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
Non-isomorphic representatives of the a(3) = 4 digraph edge-sets:
{12,23,31}
{12,13,21,32}
{12,13,21,23,31}
{12,13,21,23,31,32}
|
|
|
CROSSREFS
|
Non-Hamiltonian unlabeled digraphs (without loops) are A326222.
|
|
|
KEYWORD
|
nonn,hard,more
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A326226
|
|
Number of unlabeled n-vertex Hamiltonian digraphs (with loops).
|
|
+10
10
|
| |
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
A digraph is Hamiltonian if it contains a directed cycle passing through every vertex exactly once.
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
Non-isomorphic representatives of the a(2) = 3 digraph edge-sets:
{12,21}
{11,12,21}
{11,12,21,22}
|
|
|
MATHEMATICA
|
dinorm[m_]:=If[m=={}, {}, If[Union@@m!=Range[Max@@Flatten[m]], dinorm[m/. Apply[Rule, Table[{(Union@@m)[[i]], i}, {i, Length[Union@@m]}], {1}]], First[Sort[dinorm[m, 1]]]]];
dinorm[m_, aft_]:=If[Length[Union@@m]<=aft, {m}, With[{mx=Table[Count[m, i, {2}], {i, Select[Union@@m, #1>=aft&]}]}, Union@@(dinorm[#1, aft+1]&)/@Union[Table[Map[Sort, m/. {par+aft-1->aft, aft->par+aft-1}, {0}], {par, First/@Position[mx, Max[mx]]}]]]];
Table[Length[Select[Union[dinorm/@Subsets[Tuples[Range[n], 2]]], FindHamiltonianCycle[Graph[Range[n], DirectedEdge@@@#]]!={}&]], {n, 0, 4}] (* Mathematica 8.0+. Warning: Using HamiltonianGraphQ instead of FindHamiltonianCycle returns a(4) = 867 which is incorrect *)
|
|
|
CROSSREFS
|
The undirected case is A003216 (without loops) or A326215 (with loops).
Unlabeled non-Hamiltonian digraphs are A326223.
Unlabeled digraphs with a Hamiltonian path are A326221.
|
|
|
KEYWORD
|
nonn,hard,more
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A350415
|
|
Number of acyclic digraphs on n unlabeled nodes with a global source (or sink).
|
|
+10
10
|
|
|
|
1, 1, 3, 16, 164, 3341, 138101, 11578037, 1961162564, 668678055847, 457751797355605, 628137837068751147, 1726130748679532455689, 9493834992383031007906911, 104476428350838383854529661007, 2299979227717819421763629684068904
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,3
|
|
|
COMMENTS
|
A local source (also called an out-node) is a node whose in-degree is zero. In the case of an acyclic digraph with only one local source, the source is also a global source.
|
|
|
LINKS
|
|
|
|
PROG
|
(PARI) A350415seq(16) \\ See PARI link in A122078 for program code.
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A326221
|
|
Number of unlabeled n-vertex digraphs (with loops) containing a Hamiltonian path.
|
|
+10
9
|
| |
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
A directed path is Hamiltonian if it passes through every vertex exactly once.
|
|
|
LINKS
|
|
|
|
FORMULA
|
|
|
|
CROSSREFS
|
The undirected case is A057864 (without loops).
Unlabeled digraphs not containing a Hamiltonian path are A326224.
Unlabeled digraphs containing a Hamiltonian cycle are A326226.
|
|
|
KEYWORD
|
nonn,more
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A326224
|
|
Number of unlabeled n-vertex digraphs (with loops) not containing a Hamiltonian path.
|
|
+10
9
|
| |
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
A directed path is Hamiltonian if it passes through every vertex exactly once.
|
|
|
LINKS
|
|
|
|
FORMULA
|
|
|
|
CROSSREFS
|
The undirected case is A283420 (without loops).
Unlabeled digraphs containing a Hamiltonian path are A326221.
Unlabeled digraphs not containing a Hamiltonian cycle are A326223.
|
|
|
KEYWORD
|
nonn,more
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
Search completed in 0.017 seconds
|
