A085656 -id:A085656 - cp's OEIS frontend

A003024 Number of acyclic digraphs (or DAGs) with n labeled nodes.

1, 1, 3, 25, 543, 29281, 3781503, 1138779265, 783702329343, 1213442454842881, 4175098976430598143, 31603459396418917607425, 521939651343829405020504063, 18676600744432035186664816926721, 1439428141044398334941790719839535103, 237725265553410354992180218286376719253505

Offset: 0

N. J. A. Sloane

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
From Gus Wiseman, Jan 01 2024: (Start)
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}}
These set-systems have ranks A367908, subset of A367906, for multisets A368101.
The version for no ways is A368600, any length A367903, ranks A367907.
The version for at least one way is A368601, any length A367902.
(End)

			For n = 2 the three (0,1)-matrices are {{{1, 0}, {0, 1}}, {{1, 0}, {1, 1}}, {{1, 1}, {0, 1}}}.

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.

Alois P. Heinz, Table of n, a(n) for n = 0..77 (first 41 terms from T. D. Noe)
T. E. Allen, J. Goldsmith, and N. Mattei, Counting, Ranking, and Randomly Generating CP-nets, 2014.
Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, thesis, 2002.
Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, thesis, 2002 [Local copy, with permission]
Eunice Y.-J. Chen, A. Choi, and A. Darwiche, On Pruning with the MDL Score, JMLR: Workshop and Conference Proceedings vol 52, 98-109, 2016.
S. Engstrom, Random acyclic orientations of graphs, Master's thesis written at the department of Mathematics at the Royal Institute of Technology (KTH) in Stockholm, Jan. 2013.
Zehao Jin, Mario Pasquato, Benjamin L. Davis, Tristan Deleu, Yu Luo, Changhyun Cho, Pablo Lemos, Laurence Perreault-Levasseur, Yoshua Bengio, Xi Kang, Andrea Valerio Maccio, and Yashar Hezaveh, A Data-driven Discovery of the Causal Connection between Galaxy and Black Hole Evolution, arXiv:2410.00965 [astro-ph.GA], 2024. See p. 33.
Zehao Jin, Mario Pasquato, Benjamin L. Davis, Andrea V. Macciò, and Yashar Hezaveh, Beyond Causal Discovery for Astronomy: Learning Meaningful Representations with Independent Component Analysis, arXiv:2410.14775 [astro-ph.GA], 2024. See pp. 2, 4, 10.
Zehao Jin, Mario Pasquato, Benjamin L. Davis, Tristan Deleu, Yu Luo, Changhyun Cho, Pablo Lemos, Laurence Perreault-Levasseur, Yoshua Bengio, and Xi Kang, Causal Discovery in Astrophysics: Unraveling Supermassive Black Hole and Galaxy Coevolution, Astrophys. J. (2025) Vol. 979, 212. See 4.1.
P. L. Krapivsky, Hiring Strategies, arXiv:2412.10490 [cs.GT], 2024. See p. 12.
Qipeng Kuang, Ondřej Kuželka, Yuanhong Wang, and Yuyi Wang, Bridging Weighted First Order Model Counting and Graph Polynomials, arXiv:2407.11877 [cs.LO], 2024. See p. 33.
Jack Kuipers and Giusi Moffa, Uniform generation of random acyclic digraphs, arXiv preprint arXiv:1202.6590 [stat.CO], 2012. - _N. J. A. Sloane_, Sep 14 2012
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.
Tobias Ellegaard Larsen, Claus Thorn Ekstrøm, and Anne Helby Petersen, Score-Based Causal Discovery with Temporal Background Information, arXiv:2502.06232 [stat.ME], 2025. See p. 9.
Daniel Mallia, Towards an Unsupervised Bayesian Network Pipeline for Explainable Prediction, Decision Making and Discovery, Master's Thesis, CUNY Hunter College (2023).
B. D. McKay, F. E. Oggier, G. F. Royle, N. J. A. Sloane, I. M. Wanless and H. S. Wilf, Acyclic digraphs and eigenvalues of (0,1)-matrices, J. Integer Sequences, 7 (2004), #04.3.3.
B. D. McKay, F. E. Oggier, G. F. Royle, N. J. A. Sloane, I. M. Wanless and H. S. Wilf, Acyclic digraphs and eigenvalues of (0,1)-matrices, arXiv:math/0310423 [math.CO], Oct 28 2003.
A. Motzek and R. Möller, Exploiting Innocuousness in Bayesian Networks, Preprint 2015.
Yisu Peng, Y. Jiang, and P. Radivojac, Enumerating consistent subgraphs of directed acyclic graphs: an insight into biomedical ontologies, arXiv preprint arXiv:1712.09679 [cs.DS], 2017.
J. Peters, J. Mooij, D. Janzing, and B. Schölkopf, Causal Discovery with Continuous Additive Noise Models, arXiv preprint arXiv:1309.6779 [stat.ML], 2013.
R. W. Robinson, Enumeration of acyclic digraphs, Manuscript. (Annotated scanned copy)
V. I. Rodionov, On the number of labeled acyclic digraphs, Disc. Math. 105 (1-3) (1992) 319-321
I. Shpitser, T. S. Richardson, J. M. Robins and R. Evans, Parameter and Structure Learning in Nested Markov Models, arXiv preprint arXiv:1207.5058 [stat.ML], 2012.
I. Shpitser, R. J. Evans, T. S. Richardson, and J. M. Robins, Introduction to nested Markov models, Behaviormetrika, Behaviormetrika Vol. 41, No. 1, 2014, 3-39.
R. P. Stanley, Acyclic orientation of graphs, Discrete Math. 5 (1973), 171-178. North Holland Publishing Company.
Christian Toth, Christian Knoll, Franz Pernkopf, and Robert Peharz, Rao-Blackwellising Bayesian Causal Inference, arXiv:2402.14781 [cs.LG], 2024.
Sumanth Varambally, Yi-An Ma, and Rose Yu, Discovering Mixtures of Structural Causal Models from Time Series Data, arXiv:2310.06312 [cs.LG], 2023.
S. Wagner, Asymptotic enumeration of extensional acyclic digraphs, in Proceedings of the SIAM Meeting on Analytic Algorithmics and Combinatorics (ANALCO12).
Daniel Waxman, Kurt Butler, and Petar M. Djuric, Dagma-DCE: Interpretable, Non-Parametric Differentiable Causal Discovery, arXiv:2401.02930 [cs.LG], 2024.
Eric Weisstein's World of Mathematics, (0,1)-Matrix
Eric Weisstein's World of Mathematics, Acyclic Digraph
Eric Weisstein's World of Mathematics, Positive Eigenvalued Matrix
Eric Weisstein's World of Mathematics, Weisstein's Conjecture
Jun Wu and Mathias Drton, Partial Homoscedasticity in Causal Discovery with Linear Models, arXiv:2308.08959 [math.ST], 2023.
Chris Ying, Enumerating Unique Computational Graphs via an Iterative Graph Invariant, arXiv:1902.06192 [cs.DM], 2019.
Index entries for sequences related to binary matrices

Cf. A086510, A081064 (refined by # arcs), A307049 (by # descents).
Cf. A055165, which counts nonsingular {0, 1} matrices and A085656, which counts positive definite {0, 1} matrices.
Cf. A188457, A135079, A137435 (acyclic 3-multidigraphs), A188490.
For a unique sink we have A003025.
The unlabeled version is A003087.
These are the reverse-alternating sums of rows of A046860.
The weakly connected case is A082402.
A reciprocal version is A334282.
Row sums of A361718.
Cf. A003465, A058877, A058891, A059201, A088957, A133686, A323818.

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

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 *)

a(n)=if(n<1,n==0,sum(k=1,n,-(-1)^k*binomial(n,k)*2^(k*(n-k))*a(n-k)))

{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

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)*exp(-2^n*x)*x^n/n!. - Vladeta Jovovic, Jun 05 2005
a(n) = Sum_{k=1..n} (-1)^(n-k)*A046860(n,k) = Sum_{k=1..n} (-1)^(n-k)*k!*A058843(n,k). - Vladeta Jovovic, Jun 20 2008
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

A080858 Triangle read by rows: T(n,k), n >=1, 0 <= k <= C(n,k), = number of n X n symmetric positive definite matrices with 2's on the main diagonal and 1's and 0's elsewhere and with k 1's above the diagonal.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 1, 6, 15, 20, 12, 6, 1, 1, 10, 45, 120, 190, 192, 140, 80, 30, 10, 1, 1, 15, 105, 455, 1290, 2382, 3385, 3195, 2880, 1860, 1098, 435, 240, 60, 15, 1, 1, 21, 210, 1330, 5775, 17157, 36092, 60210, 75075, 87185, 68775, 64470, 38395, 26355, 13125, 7987, 2394, 1365, 560, 105, 21, 1, 1, 28, 378, 3276, 19985, 86772, 265286, 595136, 1104642, 1499680, 2036412, 2057496, 2115855, 1729672, 1508580, 912128, 755300, 378336, 260848, 120120, 80388, 26384, 16856, 4200, 3640, 840, 168, 28, 1

Offset: 1

N. J. A. Sloane, Jul 13 2003

			1;
1,1;
1,3,3,1;
1,6,15,20,12,6,1;
...

Index entries for sequences related to binary matrices

Rows sums give A085657.

A085656(n) = Sum_{k=0..C(n,2)} 2^k * T(n,k).

Rows n=6..8 added by Max Alekseyev, Jun 04 2024

A085657 Number of n X n symmetric positive definite matrices with 2's on the main diagonal and 1's and 0's elsewhere.

Original entry on oeis.org

1, 2, 8, 61, 819, 17417, 506609, 15582436

Offset: 1

N. J. A. Sloane, Jul 12 2003

nonn

Of course the total number of symmetric matrices of this type (not necessarily positive definite) is 2^C(n,2).

This gives the number of different values of A + A' where A runs through the matrices counted in A085656. - Max Alekseyev, Dec 13 2005

			The singular matrix
2 0 1 1
0 2 1 1
1 1 2 0
1 1 0 2
is one of the three 4 X 4 matrices which are not positive definite.

Index entries for sequences related to binary matrices

Cf. A085659, A080858, A083029.

Cf. A085656, A114601.

{ a(n) = M=matrix(n,n,i,j,2*(i==j)); r=0; b(1); r } { b(k) = if(k>n, r++; return); forvec(x=vector(k-1,i,[0,1]), for(i=1,k-1,M[k,i]=M[i,k]=x[i]); if( matdet(vecextract(M,2^k-1,2^k-1),1)>0, b(k+1) ) ) } (Max Alekseyev)

More terms from Max Alekseyev, Dec 13 2005

A038379 Number of real {0,1} n X n matrices A such that M = A + A' has 2's on the main diagonal, 0's and 1's elsewhere and is positive semi-definite.

Original entry on oeis.org

1, 3, 27, 729, 52649, 9058475, 3383769523, 2520512534065

Offset: 1

N. J. A. Sloane, Jul 13 2003

Necessarily A has all 1's on the main diagonal.
A real matrix M is positive semi-definite if its eigenvalues are >= 0.
For n <= 4, a(n) equals the upper bound 3^C(n,2).
For the number of different values of symmetric parts A + A', see A085658. - Max Alekseyev, Nov 11 2006

Index entries for sequences related to binary matrices

Cf. A055165, which counts nonsingular {0, 1} matrices, A003024, which counts {0, 1} matrices with positive eigenvalues, A085656 (positive definite matrices).

Cf. A085657, A085658, A080858, A083029.

a(n) = Sum_{k=0..C(n,2)} 2^k * A083029(n,k).

Definition corrected Nov 10 2006
a(6)-a(8) from Max Alekseyev, Nov 11 2006
Edited by Max Alekseyev, Jun 05 2024

A086510 Number of n X n real (0,1)-matrices with all eigenvalues >= 0.

Original entry on oeis.org

1, 2, 13, 261, 15418, 2566333

Offset: 0

Frederique Oggier (frederique.oggier(AT)epfl.ch) and N. J. A. Sloane, Sep 10 2003

			a(2)=13 because only 3 of the 16 possible matrices have eigenvalues < 0:
.
  0  1
  1  0
  with eigenvalues {1,-1}
and
  1 1
  1 0
.
  0 1
  1 1
  both with eigenvalues {1.61803..(Golden ratio),-0.61803...}

B. D. McKay, F. E. Oggier, G. F. Royle, N. J. A. Sloane, I. M. Wanless and H. S. Wilf, Acyclic digraphs and eigenvalues of (0,1)-matrices, arXiv:math/0310423 [math.CO], 2003.
B. D. McKay, F. E. Oggier, G. F. Royle, N. J. A. Sloane, I. M. Wanless and H. S. Wilf, Acyclic digraphs and eigenvalues of (0,1)-matrices, J. Integer Sequences, 7 (2004), #04.3.3.
Index entries for sequences related to binary matrices

Cf. A003024, A055165, A085656, A098148.

a[0] = 1; a[n_] := Module[{M, iter, cnt = 0}, M = Table[a[i, j], {i, 1, n}, {j, 1, n}]; iter = Thread[{Flatten[M], 0, 1}]; Do[If[AllTrue[Eigenvalues[ M], NonNegative], cnt++], Evaluate[Sequence @@ iter]]; cnt];
Do[Print[n, " ", a[n]], {n, 0, 5}] (* Jean-François Alcover, Dec 09 2018 *)

a(5) from Hugo Pfoertner, Sep 26 2017

A086215 Number of (-1,0,1) n X n matrices M that are positive definite.

Original entry on oeis.org

1, 7, 311, 79505, 105311665, 642005451319, 15477341239385927

Offset: 1

Eric W. Weisstein, Jul 12 2003

M need not be symmetric. For the number of different values of M + M' see A114601. - Max Alekseyev, Dec 13 2005

Eric Weisstein's World of Mathematics, Positive Definite Matrix

Cf. A114601, A085656.

Table[Count[Tuples[{-1, 0, 1}, {n, n}], ?PositiveDefiniteMatrixQ], {n, 3}] (* _Eric W. Weisstein, Jan 03 2021 *)

{ a(n) = M=matrix(n,n,i,j,2*(i==j)); r=0; b(1); r } { b(k) = local(z,t); if(k>n, z=t=0; for(i=1,n, for(j=1,i-1, if(M[ i,j ]==0,z++); if(abs(M[ i, j ])==1,t++); )); r+=3^z*2^t; return; ); forvec(x=vector(k-1,i,[ -1,1 ]), for(i=1,k-1,M[ k,i ]=M[ i,k ]=x[ i ]); if( matdet(vecextract(M,2^k-1, 2^k-1),1)>0, b(k+1) ) ) } /* Max Alekseyev */

a(4) from Wouter Meeussen, Sep 05 2003
a(5)-a(6) from Max Alekseyev, Dec 13 2005
a(7) from Max Alekseyev, Nov 09 2006

A098148 Number of real (0,1) n X n matrices such that some eigenvalues are strictly complex.

Original entry on oeis.org

0, 0, 52, 22196, 21005094

Offset: 1

Hugo Pfoertner, Sep 07 2004

			The 3 X 3 matrix ((0,1,0),(0,0,1),(1,1,1)) has real eigenvalue 1.83929 and the complex pair -0.41964+-0.60629*i. There are 12 (0,1) 3 X 3 matrices with these eigenvalues. There are 6 groups of 6 matrices having eigenvalues (1.3472,-0.66236+-0.56228*i), (1.46557,-0.23279+-0.79255*i),..., (2.32472,0.33764+-0.56228*i). Two matrices (e.g. ((0,0,1),(1,0,0),(0,1,0)) ) have eigenvalues (1,-0.5+-0.5*sqrt(3)*i). Two matrices (e.g. ((1,1,0),(0,1,1),(1,0,1)) ) have eigenvalues (2,0.5+-0.5*sqrt(3)*i). Total: 12+6*6+2+2=52=a(3).

Index entries for sequences related to binary matrices

Cf. other counts for (0, 1) matrices: A003024 (positive eigenvalues), A055165 (nonsingular), A085656 (positive definite), A086510 (nonnegative eigenvalues).

a[n_] := Module[{M, iter, cnt=0}, M = Table[a[i, j], {i, 1, n}, {j, 1, n}]; iter = Thread[{Flatten[M], 0, 1}]; Do[If[AnyTrue[Eigenvalues[M], Im[#] != 0&], cnt++], Evaluate[Sequence @@ iter]]; cnt];
Do[Print[n, " ", a[n]], {n, 1, 4}] (* Jean-François Alcover, Dec 09 2018 *)

a(5) corrected by Hugo Pfoertner, Sep 26 2017

A127502 Number of n X n positive definite matrices with 1's on the main diagonal and -1's and 0's elsewhere.

Original entry on oeis.org

1, 3, 19, 201, 3001, 55291, 1115003, 21837649, 373215601, 8282131891

Offset: 1

Max Alekseyev, Jan 16 2007

A real matrix M is positive-definite if x M x' > 0 for all nonzero real vectors x. Equivalently, all eigenvalues of M + M' are positive.

M need not be symmetric. For the number of different values of M + M' see A084552.

			For n = 2 the three matrices are {{{1, 0}, {0, 1}}, {{1, 0}, {-1, 1}}, {{1, -1}, {0, 1}}}.

Cf. A085656, A085657, A085658, A086215, A038379, A080858, A083029, A127503.

{ a(n) = M=matrix(n,n,i,j,2*(i==j)); r=0; b(1); r } { b(k) = local(t); if(k> n, t=0; for(i=1,n, for(j=1,i-1, if(M[i,j]==1,t++); )); r+=2^t; return; ); forvec(x=vector(k-1,i,[ -1,0]), for(i=1,k-1,M[k,i]=M[i,k]=x[i]); if( matdet(vecextract(M,2^k-1,2^k-1),1)>0, b(k+1) ) ) } (Alekseyev)

A087488 Number of n X n (-1,1)-matrices with all eigenvalues >= 0.

Original entry on oeis.org

1, 1, 6, 64, 4744, 536736

Offset: 0

Frederique Oggier (frederique.oggier(AT)epfl.ch) and N. J. A. Sloane, Oct 24 2003

			For n = 2 the six matrices are (+ means +1, - means -1):
++ +- -- -+ +- ++
-- +- ++ -+ -+ ++
with eigenvalues
00 00 00 00 20 20 respectively.

B. D. McKay, F. E. Oggier, G. F. Royle, N. J. A. Sloane, I. M. Wanless and H. S. Wilf, Acyclic digraphs and eigenvalues of (0,1)-matrices, arXiv:math/0310423 [math.CO], 2003.
B. D. McKay, F. E. Oggier, G. F. Royle, N. J. A. Sloane, I. M. Wanless and H. S. Wilf, Acyclic digraphs and eigenvalues of (0,1)-matrices, J. Integer Sequences, 7 (2004), #04.3.3.
Index entries for sequences related to binary matrices

Cf. A086510, A003024, A055165, A085656.

a[n_] := Select[Partition[#, n] & /@ Tuples[{-1, 1}, {n^2}], AllTrue[ Eigenvalues[#], NonNegative]&] // Length; a[0] = 1;
Do[Print[n, " ", a[n]], {n, 0, 5}] (* Jean-François Alcover, Feb 13 2019 *)

a(5) from Jean-François Alcover, Feb 13 2019

cp's OEIS Frontend

A003024 Number of acyclic digraphs (or DAGs) with n labeled nodes.

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A080858 Triangle read by rows: T(n,k), n >=1, 0 <= k <= C(n,k), = number of n X n symmetric positive definite matrices with 2's on the main diagonal and 1's and 0's elsewhere and with k 1's above the diagonal.

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A085657 Number of n X n symmetric positive definite matrices with 2's on the main diagonal and 1's and 0's elsewhere.

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PARI

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A038379 Number of real {0,1} n X n matrices A such that M = A + A' has 2's on the main diagonal, 0's and 1's elsewhere and is positive semi-definite.

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A086510 Number of n X n real (0,1)-matrices with all eigenvalues >= 0.

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A086215 Number of (-1,0,1) n X n matrices M that are positive definite.

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A098148 Number of real (0,1) n X n matrices such that some eigenvalues are strictly complex.

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A127502 Number of n X n positive definite matrices with 1's on the main diagonal and -1's and 0's elsewhere.

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