A037240 -id:A037240 - cp's OEIS frontend

A333361 Array read by antidiagonals: T(n,k) is the number of directed loopless multigraphs with n arcs and k vertices.

1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 2, 0, 0, 1, 1, 5, 2, 0, 0, 1, 1, 6, 10, 3, 0, 0, 1, 1, 6, 20, 24, 3, 0, 0, 1, 1, 6, 23, 69, 42, 4, 0, 0, 1, 1, 6, 24, 110, 196, 83, 4, 0, 0, 1, 1, 6, 24, 126, 427, 554, 132, 5, 0, 0, 1, 1, 6, 24, 129, 603, 1681, 1368, 222, 5, 0, 0

Offset: 0

Andrew Howroyd, Mar 16 2020

			==================================================
n\k | 0 1 2   3    4     5     6      7      8
----+---------------------------------------------
  0 | 1 1 1   1    1     1     1      1      1 ...
  1 | 0 0 1   1    1     1     1      1      1 ...
  2 | 0 0 2   5    6     6     6      6      6 ...
  3 | 0 0 2  10   20    23    24     24     24 ...
  4 | 0 0 3  24   69   110   126    129    130 ...
  5 | 0 0 3  42  196   427   603    668    684 ...
  6 | 0 0 4  83  554  1681  2983   3811   4116 ...
  7 | 0 0 4 132 1368  5881 13681  20935  24979 ...
  8 | 0 0 5 222 3240 19448 59680 112943 154504 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (antidiagonals 0..50)

Columns k=0..4 are A000007, A000007, A008619, A037240, A050930.

Cf. A052170, A138107, A192517, A290428.

\\ here G(k,x) gives column k as rational function.
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, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^(2*g))) * prod(i=1, #v, t(v[i])^(v[i]-1))}
G(n, x)={my(s=0); forpart(p=n, s+=permcount(p)/edges(p, i->1-x^i)); s/n!}
T(n)={Mat(vector(n+1, k, Col(O(y*y^n) + G(k-1, y + O(y*y^n)))))}
{my(A=T(10)); for(n=1, #A, print(A[n, ]))}

T(n,k) = A052170(n) for k >= 2*n.

A052170 Number of directed loopless multigraphs with n arcs.

Original entry on oeis.org

1, 1, 6, 24, 130, 688, 4211, 26840, 184153, 1328155, 10078617, 79926478, 660616432, 5671793248, 50459837996, 464139053799, 4405521306315, 43077862741114, 433275511964227, 4476516495577776, 47451864583578111, 515494036824348917, 5733423512317010811, 65226494052113260251, 758377712833720838677, 9004484581478188581057

Offset: 0

Vladeta Jovovic, Jan 26 2000

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A050535, A007717, A037240, A050930, A333361.

a(n)={polcoef(G(2*n, x + O(x*x^(n))), n)} \\ Needs G from A333361. - Andrew Howroyd, Mar 16 2020

\\ faster, also needs G from A333361.
EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
InvEulerMT(u)={my(n=#u, p=log(1+x*Ser(u)), vars=variables(p)); Vec(sum(i=1, n, moebius(i)*substvec(p + O(x*x^(n\i)), vars, apply(v->v^i, vars))/i))}
seq(n)={concat([1], EulerT(Vec(-1 + vecsum(InvEulerMT(vector(n+1, k, G(k, y + O(y*y^n))))))))} \\ Andrew Howroyd, Mar 16 2020

a(n) = A333361(n, 2*n). - Andrew Howroyd, Mar 16 2020

a(16)-a(25) from Max Alekseyev, Jun 21 2011

A050930 Number of directed loopless multigraphs on 4 nodes with n arcs.

Original entry on oeis.org

1, 1, 6, 20, 69, 196, 554, 1368, 3240, 7110, 14896, 29624, 56755, 104468, 186484, 322800, 544590, 896259, 1443834, 2278640, 3531647, 5380064, 8069672, 11926928, 17392983, 25042836, 35637168, 50152068, 69853604, 96344440, 131669538

Offset: 0

Vladeta Jovovic, Dec 30 1999

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Column k=4 of A333361.

Cf. A037240.

G.f.: (2*x^18 + 9*x^17 + 20*x^16 + 46*x^15 + 98*x^14 + 133*x^13 + 196*x^12 + 226*x^11 + 254*x^10 + 240*x^9 + 207*x^8 + 128*x^7 + 92*x^6 + 43*x^5 + 24*x^4 + 8*x^3 + 2*x^2-x + 1)/((x^4-1)^3*(x^3-1)^4*(x^2-1)^3*(x-1)^2).

A294085 a(n) is the number of self-symmetric anonymous and neutral equivalence classes of preference profiles with 3 alternatives and n agents (IANC model).

Original entry on oeis.org

1, 1, 3, 4, 8, 10, 17, 20, 30, 35, 49, 56, 75, 84, 108, 120, 150, 165, 202, 220, 264, 286, 338, 364, 425, 455, 525, 560, 640, 680, 771, 816, 918, 969, 1083, 1140, 1267, 1330, 1470, 1540, 1694, 1771, 1940, 2024, 2208, 2300, 2500, 2600, 2817, 2925, 3159, 3276, 3528, 3654, 3925

Offset: 0

Alexander Karpov, Apr 12 2018

Colin Barker, Table of n, a(n) for n = 0..1000
A. Karpov, An Informational Basis for Voting Rules, NRU Higher School of Economics. Series WP BRP "Economics/EC". 2018. No. 188
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1,1,-1,-2,2,1,-1).

Cf. A037240, A005513.

For odd n, it is A000292.

Vec((1 + x^3 + x^4) / ((1 - x)^4*(1 + x)^3*(1 - x + x^2)*(1 + x + x^2)) + O(x^60)) \\ Colin Barker, May 11 2018

a(n) = 2*A005513(n-6) - A037240(n).
If n is odd, a(n) = (n+5)*(n+3)*(n+1)/48;
If n is even, a(n) = ceiling((n+4)^2*(n+2)/48).
From Colin Barker, May 11 2018: (Start)
G.f.: (1 + x^3 + x^4) / ((1 - x)^4*(1 + x)^3*(1 - x + x^2)*(1 + x + x^2)).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) + a(n-6) - a(n-7) - 2*a(n-8) + 2*a(n-9) + a(n-10) - a(n-11) for n>10.
(End)

A296260 Number of preference profiles with 4 alternatives and n agents (IANC model).

Original entry on oeis.org

1, 1, 17, 111, 762, 4095, 19941, 84825, 329214, 1168740, 3858348, 11920740, 34773590, 96282900, 254473884, 644637204, 1571330916, 3697182450, 8421423582, 18615637950, 40023753924, 83859017814, 171530071362, 343059613650, 671825586021, 1289904147324, 2430974136780

Offset: 1

Alexander Karpov, Dec 15 2017

nonn

Ö. Egecioglu, Uniform generation of anonymous and neutral preference profiles for social choice rules, Monte Carlo Methods and Applications, 15(3), Jan 2009, 241-255.

Cf. A037240 for 3 alternatives.

Array[Binomial[# + 23, 23]/24 + Which[Divisible[#1, 12], 3 Binomial[#1/2 + 11, 11]/8 + Binomial[#1/3 + 7, 7]/3 + Binomial[#1/4 + 5, 5]/4, MemberQ[{1, 5, 7, 11}, #2], 0, MemberQ[{2, 10}, #2], 3 Binomial[#1/2 + 11, 11]/8, MemberQ[{3, 9}, #2], Binomial[#1/3 + 7, 7]/3, MemberQ[{4, 8}, #2], 3 Binomial[#1/2 + 11, 11]/8 + Binomial[#1/4 + 5, 5]/4, True, 3 Binomial[#1/2 + 11, 11]/8 + Binomial[#1/3 + 7, 7]/3 ] & @@ {#, Mod[#, 12]} &, 26] (* Michael De Vlieger, Dec 18 2017 *)

if n ==  0  mod 12, a(n) = C(n+23,23)/24 + C(n/2+11,11)*3/8 + C(n/3+7,7)/3+C(n/4+5,5)/4;
if n ==  1,5,7,11  mod 12, a(n) = C(n+23,23)/24;
if n == 2,10 mod 12, a(n) = C(n+23,23)/24 + C(n/2+11,11)*3/8;
if n == 3,9 mod 12, a(n) = C(n+23,23)/24  + C(n/3+7,7)/3;
if n == 4,8 mod 12, a(n) = C(n+23,23)/24 + C(n/2+11,11)*3/8 +C(n/4+5,5)/4;
if n == 6 mod 12, a(n) = C(n+23,23)/24 + C(n/2+11,11)*3/8 + C(n/3+7,7)/3.

More terms from Michael De Vlieger, Dec 18 2017

cp's OEIS Frontend

A333361 Array read by antidiagonals: T(n,k) is the number of directed loopless multigraphs with n arcs and k vertices.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A052170 Number of directed loopless multigraphs with n arcs.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

Extensions

A050930 Number of directed loopless multigraphs on 4 nodes with n arcs.

Views

Author

Keywords

Links

Crossrefs

Formula

A294085 a(n) is the number of self-symmetric anonymous and neutral equivalence classes of preference profiles with 3 alternatives and n agents (IANC model).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A296260 Number of preference profiles with 4 alternatives and n agents (IANC model).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions